BACKGROUND NOTES ON p-DIVISIBLE GROUPS OVER LOCAL FIELDS

BRIAN CONRAD

The aim of these notes is to present an overview of the background needed for the theory of formal group schemes and p-divisible groups over the integer rings of local fields, from the point of view developed by Fontaine in his book [3]. This work of Fontaine is vitally important at the foundation of some current work in number theory (for example, it lies at the heart of Wiles’ method for understanding supersingular reduction cases) and so it deserves to be understood by a wider audience!

Some of the proofs we give are modifications of standard ones, and other things included here seem not to be written up anywhere in an approachable and comprehensive form (at least not anywhere that I could find).

Finally, it should be added that these are rough notes and so this is not intended to be a polished manuscript. Hopefully the exposition is clear, but there may be occasional clumsiness in the writing style.

§1. Preliminary Algebra

Before we can carry out a study of p-divisible groups and commutative formal group schemes over rings such as Z_{p}, we need to develop an appropriate amount of ‘pro-algebra’ in order to serve our needs. This material is for the most part quite boring, but it is best to discuss a certain amount of it at the start in order to avoid awkwardness later on.

The primary example to always keep in mind is a commutative formal group law in n parameters over Z_{p}, which is a certain type of structure imposed on Z_{p}[[T_{1}, …, T_{n}]]. The contravariant language of formal spectra will enable us to view such an algebraic notion as a group object in an appropriate geometric category. In order to make this precise, we will carry out a very general study of rings such as Z_{p}[[T_{1}, …, T_{n}]]. However, we will need a theory wide enough to encompass certain rings that are extremely non-noetherian. One reason is that a fundamental representability theorem of Grothendieck (Theorem 1.4.7), which constitutes an essential ingredient in Fontaine’s approach to the theory of Dieudonne modules, requires vast generality in order for the proof to work. Also, the study of formal étale group schemes naturally leads to a consideration of rings such as ∏ Z_{p}, with the product indexed by an uncountable set. Thus, the reader should beware that we definitely make no noetherian hypotheses in what follows.

§1.1 Basic Notions

Definition 1.1.1 A ring A is linearly topologized if it is a topological ring with a base of neighborhoods of 0 consisting of open ideals. If A is a linearly topologized ring, we let ℐ_{A} denote the set of open ideals of A and we define

Â = \underleftarrow{lim}_{𝔞∈ℐ_{A}}A/𝔞

(where ℐ_{A} is ordered by inclusion). This is a linearly topologized ring by using the inverse limit topology (so the kernels of the maps Â → A/𝔞 form a base of neighborhoods of 0).

Linearly topologized rings form a category in an obvious manner and there is a natural map of linearly topologized rings

A → Â.

Definition 1.1.2 A linearly topologized ring A is called separated if A → Â is injective (i.e., A is Hausdorff) and is called separated and complete if the map A → Â is an isomorphism of rings.

Note that we do not define the notion of ‘complete’ by itself. Also, if A is separated and complete and 𝔞 ∈ ℐ_{A}, then the image of 𝔞 under A ≃ Â is precisely the open ideal ker(Â → A/𝔞), so A ≃ Â is a topological embedding. Thus, it is a topological isomorphism. It is obvious also that Â is itself separated and complete, for any linearly topologized ring A. It should be emphasized, however, that ℐ_{A} could be enormous (even in applications), and it is partly for this reason that one has to be a little careful when working with general linearly topologized rings.

Now fix a linearly topologized ring A.

Definition 1.1.3 A linearly topologized A-module is a topological A-module M with a base of neighborhoods of 0 which consist of open A-submodules. Let Λ_{M} denote the set of open A-submodules of M and define

M̂ = \underleftarrow{lim}_{M′∈Λ_{M}} M/M′.

If M → M̂ is injective, we say that M is separated. If A ≃ Â and the above map of modules is an isomorphism, we say that M is separated and complete. Finally, a linearly topologized A-algebra is a linearly topologized ring B together with a morphism A → B of linearly topologized rings.

By the continuity of A×M → M → M/M′ for M′ ∈ Λ_{M}, we see that for all m ∈ M, there exists 𝔞 ∈ ℐ_{A} such that 𝔞m ⊆ M′. Thus, each M/M′ has the structure of an Â-module, easily topological, so M̂ is a topological Â-module, obviously separated and complete as such.

Definition 1.1.4 A linearly topologized ring A is pre-compact if A/𝔞 is an artinian ring for all 𝔞 ∈ ℐ_{A}. If A is also separated and complete, then we say that A is pseudocompact. For a pre-compact ring A and a linearly topologized A-module M, we say that M is pre-finite if M/M′ is a finitely generated A-module for all M′ ∈ Λ_{M}. If instead all M/M′ are artinian A-modules, then we say that M is pre-artinian. We similarly define a pre-finite (respectively pre-artinian) A-algebra to be a linearly topologized A-algebra which is pre-finite (respectively pre-artinian) as an A-module.

The main example of a pre-artinian (even artinian) module that is not pre-finite is Q_{p}/Z_{p} as a Z_{p}-module. This does arise in practice, so one really should remember the distinction between pre-finite and pre-artinian, even though we will usually be dealing only with prefinite objects.

Definition 1.1.5 A pre-finite module over a pseudocompact ring A is call profinite if it is separated and complete. A pro-artinian module over A is defined similarly.

Be careful to note that although complete local noetherian rings with their maximal-adic topology are pseudocompact, in general a pseudocompact local ring (A, 𝔪) need not have the property that the higher powers 𝔪^{n} of 𝔪 are open (or even closed) for n > 1 (we’ll later see that 𝔪 is necessarily open). For example, one could consider A to be a formal power series ring over a (discretely topologized) field in countably many indeterminates, with an obvious ‘product’ topology structure (i.e., a base of neighborhoods of 0 are the ideals generated by almost all of the indeterminates, together with some powers of the finitely many other indeterminates). The following assertion is obvious.

Theorem 1.1.6 Let A be a pre-compact ring and M a pre-finite (resp. pre-artinian) Amodule. Then Â is pseudocompact and M̂ is profinite (resp. pro-artinian) as an Â-module.

Our interest will be exclusively with pseudocompact rings and pro-artinian (and especially profinite) modules over them. Some important examples of pseudocompact rings are complete local noetherian rings (with their maximal-adic topology), as well as arbitrary products of such rings. We will see later (Corollary 1.2.7) that a noetherian pseudocompact ring is necessarily a product of finitely many complete local noetherian rings, with the semi-local topology.

Definition 1.1.7 Let A be a linearly topologized ring and M, N two linearly topologized A-modules. We define

M ˆ⊗_{A}N = \underleftarrow{lim}_{(𝔞,M′,N′)}(M/M′) ⊗_{A/𝔞} (N/N′),

where the inverse limit is taken over all triples (𝔞, M′, N′) with 𝔞 ∈ ℐ_{A}, M′ ∈ Λ_{M}, and N′ ∈ Λ_{N} such that 𝔞M ⊆ M′ and 𝔞N ⊆ N′ (and the inverse system is formed in the obvious manner). This is naturally a linearly topologized Â-module.

It is obvious that the natural map of linearly topologized Â-modules M ˆ⊗_{A}N → M̂ˆ⊗ÂN̂ is an isomorphism.

For a general M′ ∈ Λ_{M}, there’s no reason to expect there to exist any 𝔞 ∈ ℐ_{A} for which 𝔞M ⊆ M′, so the conditions defining the above inverse system are somewhat restrictive in general. However, if A is pre-compact and M is pre-finite, then M/M′ is finitely generated over A by \{\overline{m}_{i}\}, with each \overline{m}_{i} annihilated by some 𝔞_{i} ∈ ℐ_{A}, so 𝔞 = ∩𝔞_{i} ∈ ℐ_{A} annihilates M/M′. Also, if B is a linearly topologized A-algebra, M is a linearly topologized A-module, and C is a linearly topologized B-algebra, then there is a natural map of linearly topologized Ĉ-modules

(M ˆ⊗_{A}B)ˆ⊗B̂Ĉ → M ˆ⊗_{A}C,

as well as a map in the other direction which is an inverse, so this map is an isomorphism. We’ll often use this ‘associativity of extension of scalars’ without comment.

Theorem 1.1.8

1. Let A be pre-compact and M, N be pre-finite A-modules. Then M ˆ⊗_{A}N is a profinite Â-module.

2. If B is a pre-finite A-algebra and C is a topological A-algebra which is pre-finite as a ring, then B ˆ⊗_{A}C is pseudocompact and is a topological algebra over Â, B̂, and Ĉ. It is profinite as a Ĉ-algebra.

3. If M is a pre-finite A-module and C is a topological A-algebra which is pre-finite as a ring, then M ˆ⊗_{A}C is a profinite Ĉ-module.

Proof. 1. By definition, M ˆ⊗_{A}N is the completion of M ⊗_{A} N with respect to the linearized A-module topology which has as a base of neighborhoods of 0 the submodules image(M′ ⊗_{A} N) + image(M ⊗_{A} N′) over triples (𝔞, M′, N′) of the usual sort. Since the quotient by this submodule is (M/M′) ⊗_{A/𝔞} (N/N′), if this is a finite A/𝔞-module then by Theorem 1.1.6 we’ll be done. Thus, it suffices to check that for an artinian ring R and M_{1}, M_{2} finite-length R-modules, M_{1} ⊗_{R} M_{2} is also a finite-length R-module. This is easily proven via induction and the right exactness of tensoring; in fact, one checks the more precise statement

ℓ_{R}(M_{1} ⊗_{R} M_{2}) ≤ ℓ_{R}(M_{1})ℓ_{R}(M_{2}).

Since it is not clear whether or not the artinian property is preserved under tensoring, we have required pre-finiteness and not just pre-artinianness for our modules (I suspect there are counterexamples, but I have not found any).

2. It is obvious that B ˆ⊗_{A}C is a separated and complete ring. Choose 𝔞 ∈ ℐ_{A}, 𝔟 ∈ ℐ_{B}, and 𝔠 ∈ ℐ_{C} so that 𝔞B ⊆ 𝔟 and 𝔞C ⊆ 𝔠. Since

B ˆ⊗_{A}C → (B/𝔟) ⊗_{A/𝔞} (C/𝔠)

is surjective (using B ⊗_{A} C → B ˆ⊗_{A}C), it remains to check that the ring on the right is artinian. But B/𝔟 is a finite A/𝔞-module and C/𝔠 is an artinian ring, so we’re done.

3. We can use the same method as in (2).

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The operation ˆ⊗ has the standard functoriality properties (commutativity, associativity, unit object). However, we observe in passing that in the setting of pseudocompact rings and profinite modules, Theorem 1.1.8(3) shows that there is a good notion of ‘extension of scalars’. Another interesting (and useful!) fact is that ˆ⊗ is ‘right exact’ on the category of profinite modules over a (fixed) pseudocompact ring. In order to make this precise, we first need to check that the category of such modules is abelian in a reasonable manner. This requires a more careful study of pseudocompact rings, which we take up in the next section.

§1.2 Pseudocompact Rings

From now on, we fix a pseudocompact ring A. We define P F_{A} to be the category of profinite A-modules and P A_{A} to be the category of pro-artinian A-modules.

Theorem 1.2.1 1. For any ideal 𝔞, the closure \overline{𝔞} is the intersection of all open ideals containing 𝔞.

2. All closed maximal ideals of A are open. If A is local, then the maximal ideal is open.

3. For any closed ideal 𝔞_{0}, A/𝔞_{0} with the quotient topology is a pseudocompact ring.

Proof. 1. One inclusion is clear. Conversely, if x ∈ A is not in \overline{𝔞}, then for some open 𝔟_{0} ∈ ℐ_{A}, we have (x + 𝔟_{0}) ∩ 𝔞 = ∅. But then 𝔟 = 𝔞 + 𝔟_{0} is an open ideal in A which contains 𝔞 and does not contain x.

2. The first part is obvious, by (1). Since A is separated and complete, thus Hausdorff, if A is local (so in particular A ≠ 0) then there is a proper ideal 𝔞 ∈ ℐ_{A}, yet this is necessarily inside of 𝔪, so 𝔪 is open.

3. Certainly the quotient topology makes A/𝔞_{0} into a linearly topologized ring. Also, A/𝔞_{0} is clearly pre-finite, since A → A/𝔞_{0} is open. By (1) and the fact that 𝔞_{0} is closed, the only obstacle to whether or not A/𝔞_{0} is separated and complete (and thus pseudocompact) is whether the natural map

A/𝔞_{0} → \underleftarrow{lim}_{𝔞⊇𝔞_{0}}A/𝔞

is surjective (where as usual we only consider 𝔞 ∈ ℐ_{A} in the inverse limit). Since A = \underleftarrow{lim} A/𝔞, we are asking if the natural map

\underleftarrow{lim} A/𝔞 → \underleftarrow{lim}_{𝔞⊇𝔞_{0}}A/𝔞

is surjective. This is a map of profinite A-modules with a dense image, so it is enough to know that the image is closed. This is guaranteed by (3) in the next theorem.

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We now prove the fundamental result concerning profinite modules over a pseudocompact ring. In the end, it reduces to applications of Zorn’s Lemma.

Theorem 1.2.2

1. Let M be a pro-artinian A-module, {M_{α}} a collection of closed submodules, forming a directed set with respect to reverse inclusion. The map of A-modules

M → \underleftarrow{lim} M/M_{α}

is surjective and for any closed submodule M′ in M,

M′ + (⋂ M_{α}) = ⋂(M′ + M_{α}).

2. Let N be a closed submodule of a pro-artinian (resp. profinite) A-module M. Give N the induced topology and M/N the quotient topology. These are then pro-artinian (resp. profinite) A-modules.

3. Let u : M → N be a map between pro-artinian A-modules. Then u is a closed map.

4. The categories P A_{A} and P F_{A} are abelian by using the natural topological definitions as in (2). Monomorphisms are the injective maps and epimorphisms are the surjective maps. If

0 → M′ → M → M′′ → 0

is a short exact sequence in P A_{A}, then M is in P F_{A} if and only if M′ and M′′ are in P F_{A}.

Proof. 1. The essential input is Zorn’s Lemma, which allows one to prove the following general fact: if {M_{α}} and {N_{α}} are two inverse systems of modules over a ring R and u_{α} : M_{α} → N_{α} are compatible surjections with ker(u_{α}) an artinian R-module, then the induced inverse limit map \underleftarrow{lim} M_{α} → \underleftarrow{lim} N_{α} is surjective. Let’s grant this for a moment.

Let {ℳ_{i}} be the open submodules in M. Consider the commutative diagram of A-modules

M →^{∼} \underleftarrow{lim} M/ ℳ_{i}
f_{1} ↓  ↓ f_{2}
\underleftarrow{lim} M/M_{α} →^{π} \underleftarrow{lim} M/(ℳ_{i} +M_{α})

Since the lower right entry can be identified with \underleftarrow{lim}_{i} \underleftarrow{lim}_{α}M/(ℳ_{i} +M_{α}) and M/ ℳ_{i} is artinian, clearly

\underleftarrow{lim}_{α}M/(ℳ_{i} +M_{α}) ≃ M/(ℳ_{i} +M_{α})

for large α (depending on i), so

M/ ℳ_{i} → \underleftarrow{lim}_{α}M/(ℳ_{i} +M_{α})

is surjective for each i, with kernel

(ℳ_{i} +M_{α})/ ℳ_{i} ⊆ M/ ℳ_{i}

which is artinian over A. Thus, by the above-mentioned general fact (which we’ll prove shortly), f_{2} is surjective and so π is surjective.

However,

ker π = \underleftarrow{lim}_{α} \underleftarrow{lim}_{i}(ℳ_{i} +M_{α})/M_{α} = \underleftarrow{lim}_{α} (⋂_{i}(ℳ_{i} +M_{α})) /M_{α}.

By an argument analogous to the proof of Theorem 1.2.1(1), we see that ∩_{i}(ℳ_{i} +M_{α}) is the closure of M_{α}, which is just M_{α}. This shows that ker π = 0, so π is an isomorphism of A-modules, so f_{1} is surjective, as desired.

For a closed submodule M′ in M, consider the diagram of A-modules with exact rows

0 → M′  →  M  →  M/M′  → 0
ϕ_{1} ↓  ↓ f_{1}  ↓
0 → \underleftarrow{lim} M′/(M ∩ M_{α}) → \underleftarrow{lim} M/M_{α} → \underleftarrow{lim} M/(M′ + M_{α})

We know that f_{1} is surjective. It is trivial to check that because M′ is closed, if we give it the induced topology from M then M′ is pro-artinian (not just pre-artinian). Thus, ϕ_{1} is surjective. Thus, by the snake lemma it follows that the kernel (∩(M′ + M_{α}))/M′ of the right column coincides with the image in M/M′ of ker f_{1}, which is ((∩M_{α})+M′)/M′.

It remains to justify the general surjective inverse limit assertion stated at the start. Using the notation introduced there, pick n = (n_{α}) ∈ \underleftarrow{lim} N_{α} and let E_{α} = u_{α}^{−1}(n_{α}) ⊆ M_{α}. Define Σ_{α} to be the set consisting of ∅ and all translated submodules of M_{α} which lies inside of E_{α} (e.g., E_{α} ∈ Σ_{α}). Note that if X_{1}, …, X_{n} ∈ Σ_{α}, x ∈ X_{n}, and

X_{1} ⊇ X_{2} ⊇ ⋯ ⊇ X_{n},

then {−x+X_{i}} is a descreasing chain of submodules inside of the artinian ker(u_{α}), so it terminates. Since X, X′ ∈ Σ_{α} implies that X ∩X′ ∈ Σ_{α} (recall that ∅ ∈ Σ_{α}), clearly Σ_{α} is stable under arbitrary intersections and satisfies the finite intersection property (as in the definition of a compact topological space). Also, if x_{α} ∈ E_{α} and f_{βα} : E_{β} → E_{α} is the obvious transition map, then f_{βα}^{−1}(x_{α}) ⊆ E_{β} is an element of Σ_{β}; if X_{β} ∈ Σ_{β}, then f_{βα}(X_{β}) ∈ Σ_{α}. Since all E_{α} ≠ ∅ (as all u_{α} are surjective), by [2, Thm 1, Ch III, §7.4] it follows that \underleftarrow{lim} E_{α} ≠ ∅. This gives rise to an element in \underleftarrow{lim} M_{α} lying over n ∈ \underleftarrow{lim} N_{α}.

2. The case of closed submodules is easy, as was remarked in the proof of (1). As for

quotients, since

⋂(N + ℳ_{i}) = N,

it remains to check that M/N → \underleftarrow{lim} M/(N + ℳ_{i}) is surjective, so we want

M ≃ \underleftarrow{lim} M/ ℳ_{i} → \underleftarrow{lim} M/(N + ℳ_{i})

to be surjective. Since M/ ℳ_{i} → M/(N + ℳ_{i}) is surjective with kernel N/(N ∩ ℳ_{i}) ⊆ M/ ℳ_{i} artinian over A, by the general fact stated at the beginning of the proof of (1) we are done.

3. Since ker(u) is closed, by (2) we know that M/ ker(u) is pro-artinian with respect to the quotient topology and so it suffices to check the assertion when u is injective. Let N′ = u(M). We must show that N′ ⊆ N is closed and that its induced topology from N coincides with the topology induced on it by M. Note that both topologies on N′ make N′ a linearly topologized A-module. Define 𝒩 = u(M′), with a fixed M′ ∈ Λ_{M}, and let {N′_{α}} be the open submodules of N′ via the induced topology from N. We’ll show that 𝒩 ⊇ N′_{α} for large α, so u is a topological embedding.

As A-modules (𝒩 +N′_{α})/ 𝒩 ⊆ N′/ 𝒩 ≃ M/M′, the latter isomorphism arising from the injective u. Since M/M′ is artinian, we see that for large α, the submodule 𝒩 +N′_{α} is ‘constant’ in α. Thus, for large α, we have

𝒩 +N′_{α} = ⋂(𝒩 +N′_{α})
≃ ⋂(M′ + u^{−1}(N′_{α}))
= M′ + ⋂ u^{−1}(N′_{α}) by (1)
≃ 𝒩 + ⋂ N′_{α}
= 𝒩,

so indeed N′_{α} ⊆ 𝒩 for large α.

Now that we know that u : M ↪ N is a topological embedding, it remains to check that u(M) is closed in N. Choose n ∈ N lying in the closure of u(M). For each N_{α} ∈ Λ_{N}, we have that M_{α} =^{def} u^{−1}(N_{α}) ∈ Λ_{M} and there exists m_{α} ∈ M such that

n − u(m_{α}) ∈ N_{α}.

If M_{α} ⊆ M_{β}, then u(m_{α} − m_{β}) = (u(m_{α}) − n) + (n − u(m_{β})) ∈ N_{β}, so m_{α} − m_{β} ∈ M_{β}. Hence, since M is separated and complete, there exists m ∈ M such that m−m_{α} ∈ M_{α}, so u(m) − u(m_{α}) ∈ N_{α} for all α. Thus, u(m) − n ∈ N_{α} for all α, so n = u(m) ∈ u(M).

4. This is obvious.

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Corollary 1.2.3

1. If {M_{i}} is an inverse system of pro-artinian (resp. profinite) modules over A, then \underleftarrow{lim} M_{i} is pro-artinian (resp. profinite) with respect to the inverse limit topology. This serves as an inverse limit in the category P A_{A} (resp. P F_{A}).

2. If {M_{i}} is an inverse system of pro-artinian A-modules and there are compatible continuous surjective maps M_{i} → M, for M a pro-artinian A-module, then the map \underleftarrow{lim} M_{i} → M is surjective.

3. If \{A_{i}\} is an inverse system of profinite A-algebras, then \underleftarrow{lim} A_{i} is a profinite A-algebra.

Proof. 1. This is clear.

2. The image is dense, so use (1) and Theorem 1.2.2(3).

3. This is easy.

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Lemma 1.2.4 1. If A is artinian, then A is necessarily discrete.

2. If M is a pro-artinian A-module with ℓ_{A}(M) finite, then M is discrete.

Proof. 1. Since {0} = ∩_{𝔞∈ℐ_{A}}𝔞, yet A is artinian as an A-module, we easily see that the

intersection can be taken over finitely many 𝔞 ∈ ℐ_{A}, so {0} is open.

2. Let M′ be a minimal element in Λ_{M}. Since M is separated and complete, M′ = {0}.

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For any open maximal ideal 𝔪 ∈ ℐ_{A}, we define a ‘completed local ring’

A_{𝔪} = \underleftarrow{lim}_{𝔞⊆𝔪}(A/𝔞)_{𝔪/𝔞} ,

where the inverse limit is taken over open ideals 𝔞. There is no risk of confusion with the usual localization notation, since when A is artinian (i.e., A is discrete), this agrees with the usual notion and otherwise we shall never use the algebraic notion of localization of A at a prime ideal. The ring A_{𝔪} is a local ring and with the inverse limit topology it is profinite as an A-module (by Corollary 1.2.3). Thus, A_{𝔪} is clearly pseudocompact as a ring, and is even a profinite A-algebra (and in fact any profinite A-algebra is clearly pseudocompact as a ring).

The canonical map of discrete rings

A/𝔞 ↠ (A/𝔞)_{𝔪/𝔞}

is open and surjective for 𝔞 ∈ ℐ_{A} with 𝔞 ⊆ 𝔪, but

A = \underleftarrow{lim} A/𝔞 → \underleftarrow{lim}_{a⊆𝔪}A/𝔞 → A_{𝔪}

is a continuous map of profinite A-modules with a dense image, so by Theorem 1.2.2(3) it is surjective. Now it is clear that A → A_{𝔪} is an open map. We can be even more precise, as in the following theorem.

Theorem 1.2.5 The natural map of pseudocompact rings

A → ∏_{𝔪∈ℐ_{A}} A_{𝔪}

is an isomorphism. In particular, A_{𝔪} is the algebraic localization of A at 𝔪.

Proof. Both sides are profinite as A-modules, so by Theorem 1.2.2 it suffices to check that this is a bijection of sets. But using the canonical decomposition of A/𝔞 into a product of artin local rings (for 𝔞 ∈ ℐ_{A}) and passing to the inverse limit, we’re done.

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Corollary 1.2.6 1. The ideal 𝔯_{A} of topologically nilpotent elements of A is 𝔯_{A} = ∩_{𝔪∈ℐ_{A}}𝔪, so 𝔯_{A} is closed.

2. Define k_{𝔪} = A/𝔪 for an open maximal ideal 𝔪 (so k_{𝔪} has the discrete topology). As pseudocompact rings (and profinite A-modules),

A/𝔯_{A} ≃ ∏ k_{𝔪}.

We can now show that noetherian pseudocompact rings are what we think they ought to be.

Corollary 1.2.7 Let A be a noetherian pseudocompact ring. Then A is a product of finitely many complete local noetherian rings, with the usual semi-local topology. Moreover, if 𝔞 is an ideal in A (necessarily closed!) and M is a profinite A-module, then 𝔞M is closed in M and the natural A-module map M ⊗_{A} (A/𝔞) → M ˆ⊗_{A}(A/𝔞) is an isomorphism.

Proof. By Theorem 1.2.5 and the fact that A is noetherian, clearly there are only finitely many open maximal ideals and so without loss of generality (for the first part) A is local. Since A = \underleftarrow{lim} A/𝔞 with A/𝔞 an artin local ring, easily A → \underleftarrow{lim} A/𝔪^{n} is a ring isomorphism, so A is a complete local noetherian ring and 𝔪 is open (by Theorem 1.2.1(2)).

If x_{1}, …, x_{n} generate 𝔪, then for each i ≥ 1 we have an A-linear map

∏_{1≤j≤n} 𝔪^{i} → A

given by

(y_{j}) ↦ ∑ x_{j}y_{j},

which has image 𝔪^{i+1}. This shows by induction on i (and Theorem 1.2.2(3)) that 𝔪^{i} is closed for all i. Hence, by Theorem 1.2.1(3), A/𝔪^{i} with the quotient topology is a pseudocompact artinian ring, so it is discrete by Lemma 1.2.4(1). Thus, 𝔪^{i} is open in A for all i. Therefore, A → \underleftarrow{lim} A/𝔪^{i} is a bijective map between profinite A-modules, with each A/𝔪^{i} discrete, so this is a topological isomorphism.

For the second part of the assertion, we again return to the case in which A might be non-local. Since all ideals 𝔞 in A are finitely generated as A-modules, 𝔞M is the image of a map from ∏ M to M, with the product indexed by a (finite) set of A-module generators of 𝔞. Thus, 𝔞M is closed in M by Theorem 1.2.2(3). In particular, 𝔞 is closed in A (which we knew already, essentially by the Artin-Rees Lemma). The fact that M ⊗_{A} (A/𝔞) ≃ M ˆ⊗_{A}(A/𝔞) follows from Theorem 1.3.2(1) below.

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We conclude this section with a somewhat curious result on profinite modules that can be sometimes used to show that a linearly topologized module is pro-artinian or profinite.

Theorem 1.2.8 Let A be a pseudocompact ring and M′ and M′′ pro-artinian A-modules. Let M be a linearly topologized A-module and assume there is a short exact sequence of linearly topologized A-modules

0 → M′ →^{f} M →^{g} M′′ → 0.

Then M is pro-artinian. The same holds if we replace ‘pro-artinian’ by ‘pro-finite’.

Proof. For U ⊆ M an open submodule, the image g(U) is an open submodule of M′′ (since an additive group quotient map such as g is open) and so the collection of such submodules {g(U)} forms a base of open neighborhoods of 0 in M′′. Thus,

M′′ ≃ \underleftarrow{lim} M′′/g(U)

as linearly topologized A-modules. Similarly,

M′ ≃ \underleftarrow{lim} M′/f^{−1}(U)

as linearly topologized A-modules. But the short exact sequence

0 → M′/f^{−1}(U) → M/U → M′′/g(U) → 0

shows that M/U is an artinian (resp. finite) A-module. From the commutative diagram of A-modules

0 →  M′  →  M  →  M′′  → 0
≃↓  ↓  ↓≃
0 → \underleftarrow{lim} M′/f^{−1}(U) → \underleftarrow{lim} M/U → \underleftarrow{lim} M′′/g(U) → 0

(Zorn yields surjectivity in the second row) we see that the continuous map M → \underleftarrow{lim} M/U is injective. Thus, M is separated and the middle column above is a topological embedding. But it is an isomorphism of A-modules, so it is a topological isomorphism and so M is pro-artinian (resp. profinite).

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§1.3 Properties of ˆ⊗

Theorem 1.3.1 1. The functor M ˆ⊗_{A}(·) : P F_{A} → P F_{A} is right exact.

2. There are natural isomorphisms

M ˆ⊗_{A} (∏_{i∈I} N_{i}) ≃ ∏_{i∈I}(M ˆ⊗_{A}N_{i})

and

M ˆ⊗_{A}(\underleftarrow{lim} N_{i}) ≃ \underleftarrow{lim}(M ˆ⊗_{A}N_{i}).

3. For an open maximal ideal 𝔪, define M_{𝔪} = M ˆ⊗_{A}A_{𝔪} (which lies in P F_{A_{𝔪}} by Theorem 1.1.8(3)). Then there is a natural isomorphism

M ≃ ∏ M_{𝔪}

in P F_{A} and a natural isomorphism of A-modules

Hom_{P F_{A}}(M, N) ≃ ∏ Hom_{P F_{A𝔪}}(M_{𝔪}, N_{𝔪}).

In particular, M_{𝔪} is the algebraic localization of M at 𝔪, so M ⊗_{A} A_{𝔪} ≃ M ˆ⊗_{A}A_{𝔪} as A-modules.

Proof. 1. This is proven by interpreting M ˆ⊗_{A}N as ‘pro-representing’ a certain ‘bilinear maps’ functor on the category of profinite A-modules with finite A-length and then using the fact that the usual tensor product is right exact. For complete details (which are not very difficult), see [7, §0.3.1 − 0.3.6].

2. This is also proven in [7, §0.3.1 − 0.3.6]. Note that the assertion for products follows from the assertion for inverse limits.

3. Use (2), together with M ≃ M ˆ⊗_{A}A and A ≃ ∏ A_{𝔪}.

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Theorem 1.3.2

1. For a closed ideal 𝔞 and a profinite A-module M, the natural map M ˆ⊗_{A}(A/𝔞) → M/\overline{𝔞M} is an isomorphism of profinite A-modules.

2. (formal Nakayama’s Lemma) Let M be a profinite A-module and 𝔞 ⊆ 𝔯_{A} any ideal. If \overline{𝔞M} = M, then M = 0. In particular, if A is local and M ˆ⊗_{A}k_{𝔪} = 0 then M = 0.

Proof. 1. Since 𝔞 is closed, we have an exact sequence of profinite A-modules

0 → 𝔞 → A → A/𝔞 → 0

and so by Theorem 1.3.1(1) we get an exact sequence

𝔞ˆ⊗_{A}M → Aˆ⊗_{A}M = M → (A/𝔞)ˆ⊗_{A}M.

The image of 𝔞ˆ⊗_{A}M in M is closed and has 𝔞M as a dense subset, so the image is \overline{𝔞M}.

2. Certainly \overline{𝔯_{A}M} = M, so we can assume 𝔞 = 𝔯_{A}. We have

0 ≃ M ˆ⊗_{A}(A/𝔯_{A}) ≃ M ˆ⊗_{A} (∏ k_{𝔪}) by Corollary 1.2.6(2)
≃ ∏(M ˆ⊗_{A}k_{𝔪}) by Theorem 1.3.1(2)
≃ ∏(M_{𝔪} ˆ⊗_{A_{𝔪}}k_{𝔪})
≃ ∏ M_{𝔪}/\overline{𝔪M_{𝔪}} by (1).

Since M ≃ ∏ M_{𝔪} by Theorem 1.3.1(3), we’re reduced to the local case with 𝔞 = 𝔪. Choose M′ ∈ Λ_{M}. We’ll show that M/M′ = 0, so M = 0 (since M is separated and complete). At the very least, M/M′ is discrete with the quotient topology, so

(M/M′) ⊗_{A} k_{𝔪} ≃ (M/M′)ˆ⊗_{A}k_{𝔪}

as k_{𝔪}-modules. The exactness of

M′ ˆ⊗_{A}k_{𝔪} → M ˆ⊗_{A}k_{𝔪} → (M/M′)ˆ⊗_{A}k_{𝔪} → 0,

together with M ˆ⊗_{A}k_{𝔪} = 0, shows that (M/M′) ⊗_{A} (A/𝔪) = 0. By the usual form of Nakayama’s Lemma, M/M′ = 0.

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Definition 1.3.3 We say that a profinite A-module M is topologically free if M ≃ ∏ Ae_{i}, with the product topology on the right (of course, ∏ Ae_{i} is profinite as an A-module by Corollary 1.2.3(1)). If m_{i} ∈ M corresponds to e_{i} under such an isomorphism, we call {m_{i}} a pseudobasis (or topological basis) of M over A. We say that M is topologically flat if M ˆ⊗_{A}(·) is an exact functor on P F_{A}. We say that M is topologically faithfully flat if sequences in P F_{A} are exact precisely when they are exact after applying M ˆ⊗_{A}(·).

Before giving a characterization of topologically flat modules as precisely the ones which are ‘locally’ topologically free, we need to discuss a convenient duality result which will also be useful in our analysis of commutative formal group schemes over fields later on.

Let k be a local artin ring (pseudocompact with respect to the discrete topology). The functor M ⇝ M^{∗} = Hom_{k}(M, k) is a length-preserving anti-equivalence from the category of finite-length k-modules to itself, with M ≃ M^{∗∗} and

M_{1}^{∗} ⊗ M_{2}^{∗} ≃ (M_{1} ⊗ M_{2})^{∗}.

For M in P F_{k}, define M^{∗} = Hom_{P F_{k}}(M, k), which is a k-module. For any k-module N, define N′ = Hom_{k}(N, k), which we make into a linearly topologized k-module by declaring annihilators of finite-length submodules to be open. It is easy to check that this makes N′ a profinite k-module.

Theorem 1.3.4 1. For a profinite k-module M, M is topologically free if and only if M^{∗} is free.

2. For M in P F_{k}, there is a natural isomorphism

M ≃ (M^{∗})′.

For k-modules N, there is a natural isomorphism

N ≃ (N′)^{∗}.

Moreover, the two dualizing functors M ⇝ M^{∗} and N ⇝ N′ are exact, interchange direct products and direct sums, and there are natural morphisms

M_{1}^{∗} ⊗ M_{2}^{∗} → (M_{1} ˆ⊗M_{2})^{∗}

and

N_{1}′ ˆ⊗N_{2}′ → (N_{1} ⊗ N_{2})′

which are isomorphisms if M_{1} is topologically free and N_{1} is free, respectively.

Proof. 1. This is immediate from the definition of the topology on ∏ ke_{i} and also uses (2)

for the ‘if’ direction.

2. This is a simple exercise with inverse and direct limits, together with the fact that a continous map between profinite k-modules which is an isomorphism of modules is also a topological isomorphism (Theorem 1.2.2(4)). Also, Theorem 1.3.1(2) is needed for the tensor product claims.

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Corollary 1.3.5 If k is a field, then all objects in P F_{k} are topologically free and

M_{1}^{∗} ⊗ M_{2}^{∗} ≃ (M_{1} ˆ⊗M_{2})^{∗}.

The above corollary will be used later to relate affine and formal commutative group schemes over fields. We now come to a result whose importance cannot be overestimated.

Theorem 1.3.6 For M in P F_{A}, the following conditions are equivalent:

1. M is a projective object.

2. M is topologically flat.

3. For all open maximal ideals 𝔪 in A, the object M_{𝔪} in P F_{A_{𝔪}} is topologically free.

In addition, all of these conditions holds when A = k is a field.

Remark. Clearly the equivalence of (2) and (3) is analogous to the usual equivalence of freeness and flatness for finitely generated modules over a local ring, while the equivalence of (1) and (3) is analogous to the equivalence of freeness and projectivity for (not necessarily finitely generated) modules over a local ring with a nilpotent maximal ideal (Nakayama’s Lemma is unnecessary when 𝔪 is nilpotent).

Proof. By Theorem 1.3.1(2), we see that (3) ⇒ (2). Now suppose that A = k is a field. We know that (3) holds by Corollary 1.3.5, so (3) and (2) always hold over a field. Let’s check that (1) always holds over k. Using the exact functor M ⇝ M^{∗}, we are reduced to the assertion that if V_{1}, V_{2}, and V are vector spaces over a field k and V_{2} is a subspace of V_{1}, then any k-linear map V_{2} → V can be extended to a k-linear map V_{1} → V. This is standard.

Now we return to the case in which A is any pseudocompact ring. By Theorem 1.2.5 and Theorem 1.3.1, we immediately are reduced to the case in which A is local, say with maximal ideal 𝔪 (necessarily open, by Theorem 1.2.1(2)). We will now be able to use the formal Nakayama’s Lemma.

First, assume that we know (3) ⇒ (1). Let’s deduce (1) ⇒ (3) and (2) ⇒ (3). Pick a projective object M in P F_{A}. By Theorem 1.1.8(3), M ˆ⊗_{A}k_{𝔪} is an object in P F_{k_{𝔪}}, so by the results over a field we have an isomorphism

\overline{φ} : M ˆ⊗_{A}k_{𝔪} ≃ ∏ k_{𝔪}e_{i}.

But M is projective, so we can construct a map φ : M → ∏ Ae_{i} such that the diagram

M  → ∏ Ae_{i}
↓  ↓
M ˆ⊗_{A}k_{𝔪} →^{∼} ∏ k_{𝔪}e_{i}

with surjective columns is commutative.

Let X = coker(φ). By Theorem 1.3.1(1), we have X ˆ⊗_{A}k_{𝔪} = 0, so X = 0 by Theorem 1.3.2(2). Hence, φ is surjective. By our assumption that (3) ⇒ (1), φ has a section. Thus, applying ˆ⊗_{A}k_{𝔪} to the exact sequence

0 → ker(φ) → M → ∏ Ae_{i} → 0

preserves exactness, so ker(φ)ˆ⊗_{A}k_{𝔪} = 0. By the formal Nakayama’s Lemma, ker(φ) = 0. Thus, M is topologically free (by Theorem 1.2.2(4)). This argument shows more generally that a map between profinite A-modules M → N with N projective and which is an isomorphism on the formal closed fiber must necessarily be an isomorphism. This fact will be used a lot in everything we do.

Now let’s prove (2) ⇒ (3). Again, choose an isomorphism ∏ k_{𝔪}e_{i} ≃ M ˆ⊗_{A}k_{𝔪}. Since ∏ Ae_{i} is projective by (3) ⇒ (1) (which we are assuming for now), we get a commutative diagram

∏ Ae_{i} →^{φ}  M
↓  ↓
∏ k_{𝔪}e_{i} ≃ M ˆ⊗_{A}k_{𝔪}

with surjective columns. By the formal Nakayama’s Lemma, φ is surjective. Let K = ker(φ). We will show that k_{𝔪} ˆ⊗_{A}K = 0, so K = 0, as desired.

Since M is topologically flat, we have a commutative diagram with exact rows and exact columns:

0 ↓

𝔪ˆ⊗_{A}K → 𝔪ˆ⊗_{A}(∏ Ae_{i}) → 𝔪ˆ⊗_{A}M → 0
↓ ↓ ↓
0 → Aˆ⊗_{A}K → Aˆ⊗_{A}(∏ Ae_{i}) → Aˆ⊗_{A}M → 0
↓  ↓  ↓
k_{𝔪} ˆ⊗_{A}K →  ∏ k_{𝔪}e_{i}  →^{∼} k_{𝔪} ˆ⊗_{A}M → 0
↓  ↓  ↓
0  0  0

(the topological flatness of M is needed for the injectivity at the top of the right column). By the snake lemma, the map k_{𝔪} ˆ⊗_{A}K → ∏ k_{𝔪}e_{i} is injective, and so we get k_{𝔪} ˆ⊗_{A}K = 0, as desired.

It remains to prove that ∏ Ae_{i} is a projective object in P F_{A} for A a pseudocompact ring. The continuity conditions make this fact not at all ‘obvious’. This is stated without proof in [7], but a rigorous proof seems to require a slightly careful use of the Zorn’s Lemma method used earlier to prove surjectivity of inverse limit maps under suitable conditions. Since the projectivity of topologically free modules was essential in our proofs of the other implications, it seems worthwhile to give a careful proof. We give the argument now.

Let P = ∏ Ae_{i} and consider a short exact sequence

0 → K → M →^{π} N → 0

in P F_{A}. For M′ ∈ Λ_{M′}, let K′ = K ∩ M′ and N′ = π(M′). Thus, K ≃ \underleftarrow{lim} K/K′ and N ≃ \underleftarrow{lim} N/N′ in P F_{A}. Since K/K′, M/M′, and N/N′ are all finite over discrete artinian quotients of A, it is easy to check that the sequence of groups (even A-modules)

0 → Hom_{P F_{A}}(P, K/K′) → Hom_{P F_{A}}(P, M/M′) →^{φ_{M′}} Hom_{P F_{A}}(P, N/N′) → 0

is exact. Passing to the inverse limit, we get the exact sequence

0 → Hom_{P F_{A}}(P, K) → Hom_{P F_{A}}(P, M) → Hom_{P F_{A}}(P, N),

so we want to show that surjectivity is preserved upon passage to the inverse limit.

Let K_{M′}, M_{M′}, and N_{M′} denote the corresponding Hom objects at the M′th level in the short exact sequences above, so we have short exact sequences

0 → K_{M′} → M_{M′} →^{φ_{M′}} N_{M′} → 0

of A-modules. Since K_{M′} ≃ ⊕_{i∈I}K/K′ (and similarly for M_{M′} and N_{M′}), we see that the kernel K_{M′} is ‘almost’ artinian. For this reason, an argument analogous to the one in the proof of Theorem 1.2.2(1) will yield the surjecivity of the inverse limits.

More precisely, we choose n = ((n_{(i,M′)})_{i∈I})_{M′} ∈ \underleftarrow{lim} N_{M′} (so for each M′, n_{(i,M′)} ∈ N/N′ is 0 for almost all i) and define

E_{M′} = ⨁_{i∈I} φ_{M′}^{−1}(n_{(i,M′)}) ⊆ ⨁_{i∈I} M/M′ = M_{M′}.

Note that φ_{M′}^{−1}(n_{(i,M′)}) is non-empty. In general, if X_{i} ⊆ M/M′ is a non-empty set for each i ∈ I, we define ⊕X_{i} ⊆ ⊕(M/M′) to be the subset consisting of elements whose non-zero coordinates lie in the appropriate X_{i}. Define Σ_{M′} = {∅} ∪ {⊕_{i∈I}X_{(i,M′)}}, where the second set is taken over all choice of non-empty subsets X_{(i,M′)} ⊆ φ_{M′}^{−1}(n_{(i,M′)}) which are translates of submodules of M/M′ (e.g., X_{(i,M′)} = φ_{M′}^{−1}(n_{(i,M′)})). For non-empty subsets X_{i} and Y_{i} in M/M′ we have that the intersection

(⨁_{i∈I} X_{i}) ⋂ (⨁_{i∈I} Y_{i}) = ⨁_{i∈I}(X_{i} ∩ Y_{i})

when all X_{i} ∩ Y_{i} are non-empty, and this is empty otherwise. The same holds for arbitrary intersections. Combining this with the artinian property of M/M′, we see that Σ_{M′} is stable under arbitrary intersections and satisfies the finite intersection property. It is now straightfoward to see that we can apply [2, Theorem 1(a), Ch III, §7.4] in order to conclude that \underleftarrow{lim} E_{M′} is non-empty, as desired.

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Corllary 1.3.7 An object M in P F_{A} is topologically faithfully flat over A if and only if M is topologically flat over A and M_{𝔪} ≠ 0 for all open maximals 𝔪 in A. In particular, if B is a non-zero profinite A-algebra and A → B is topologically flat, then A → B is topologically faithfully flat if and only if A → B is injective.

By ‘base changing’ to a field and using Theorem 1.3.4(2), it is clear that for a topologically free profinite A-module M, the cardinality of a pseudobasis is independent of the choice of pseudobasis (this just reduces to the well-known fact that vector spaces over fields have a well-defined notion of dimension). By a similar argument it is easy to see that any automorphism of M takes a pseudobasis to another pseudobasis and that more precisely the set of pseudobases is a principal homogenous space for the action of the automorphism group of M (this statement can be checked by first reducing to the local case and then passing to the limit on the case of an artin local base ring, for which we have a good duality theory for ‘free’ objects as indicated above).

Corollary 1.3.8 Let f : A → B be a map of pseudocompact rings. The functor M ⇝ B ˆ⊗_{A}M from profinite A-modules to profinite B-modules is right exact. Also, if M is a topologically flat profinite A-module, then B ˆ⊗_{A}M is a topologically flat profinite B-module. The same holds for topological faithful flatness; in fact, if M ≃ ∏ Ae_{i}, then the natural map of profinite B-modules

B ˆ⊗_{A}M → ∏ Be_{i}

is an isomorphism. In particular, if k → k′ is an extension of fields, then M ⇝ k′ ˆ⊗_{k}M is an exact functor from profinite k-modules to profinite k′-modules; moreover, k′ ˆ⊗_{k}M = 0 if and only if M = 0.

Proof. For an open maximal ideal 𝔫 in B, 𝔪 = f^{−1}(𝔫) is an open prime of A and so is necessarily maximal. Thus, f induces a map A_{𝔪} → B_{𝔫}. Since B_{𝔫} ˆ⊗_{B}(B ˆ⊗_{A}M) ≃ B_{𝔫} ˆ⊗_{A_{𝔪}}M_{𝔪} as profinite B_{𝔫}-modules, everything is clearly reduced to the case in which f is a local map between local pseudocompact rings. For M = ∏ Ae_{i}, in order to check that

B ˆ⊗_{A}M → ∏ Be_{i}

is an isomorphism, by the projectivity of the right side (Theorem 1.3.6) and the formal Nakayama’s Lemma, it is enough to check that this is an isomorphism after applying (B/𝔫)ˆ⊗_{B}(·), so we’re reduced to the case in which A = k is a field and B = k′ is an extension field. By the definition of k′ ˆ⊗_{k}(∏ ke_{i}), the map of interest is clearly injective. However, it is easy to see that the image is dense, so the map is an isomorphism.

For the right exactness claim, it is enough to consider a short exact sequence of profinite A-modules

0 → M′ →^{g} M →^{h} M′′ → 0

and to show that the sequence of profinite B-modules

B ˆ⊗_{A}M′ → B ˆ⊗_{A}M → B ˆ⊗_{A}M′′ → 0

is exact. Let {U} run through a base of open submodules of M, so {g^{−1}(U)} runs through a base of open submodules of M′ and {h(U)} runs through a base of open submodules of M′′. Moreover, for a small 𝔞 ∈ ℐ_{A}, we have an exact sequence of finite A/𝔞-modules

0 → M′/g^{−1}(U) → M/U → M′′/h(U) → 0.

Picking an open ideal 𝔟 in B and shrinking 𝔞 so that 𝔞 ⊆ f^{−1}(𝔟), we have a right exact sequence

(B/𝔟) ⊗_{A/𝔞} (M′/g^{−1}(U)) → (B/𝔟) ⊗_{A/𝔞} (M/U) → (B/𝔟) ⊗_{A/𝔞} (M′′/h(U)) → 0.

Letting K_{𝔞,𝔟,U} be the image of the first map, passing to the inverse limit gives an exact sequence of profinite B-modules

0 → K →^{α} B ˆ⊗_{A}M →^{β} B ˆ⊗_{A}M′′

as well as a map γ : B ˆ⊗_{A}M′ → K, where β and γ have a dense image and so are surjective (Theorem 1.2.2(3)), while α∘γ is the canonical map. This yields the desired right exactness.

Since profinite modules over fields are always topologically free (Theorem 1.3.6), the assertion for field extensions is clear via duality and other facts about vector spaces.

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The importance of the final part of Corollary 1.3.8 is that k′/k can be an extension of infinite degree, so in particular k′ need not be profinite as a k-module. It should also be mentioned that Corollary 1.3.8 has a useful application to duality. If k → k′ is a local map of local artin rings and M is a topologically free profinite k-module and N is a free k-module, there are natural maps

k′ ⊗_{k} M^{∗} → (k′ ˆ⊗_{k}M)^{∗}, k′ ˆ⊗_{k}N′ → (k′ ⊗_{k} N)′

which are isomorphisms by Corollary 1.3.8. Thus, duality for topologically free objects behaves well under local artin ring base change.

Theorem 1.3.9 Let A be a pseudocompact local ring, M a topologically free A-module, and {m_{i}} a subset of M. This set is a pseudobasis if and only if m_{i} → 0 (in the sense that any open submodule of M contains all but finitely many m_{i}’s) and the images of the m_{i}’s in (A/𝔪)ˆ⊗_{A}M = M/\overline{𝔪M} give a pseudobasis over A/𝔪.

Also, if M is any profinite A-module and N is the quotient of M by a closed submodule, any subset {n_{i}} in N with n_{i} → 0 lifts to a subset {m_{i}} ⊆ M with m_{i} → 0.

Proof. The ‘only if’ direction in the first part is obvious. Conversely, since m_{i} → 0, there exists a map ∏ Ae_{i} → M sending e_{i} to m_{i} and it is uniquely determined by these properties. By the formal Nakayama’s Lemma this map is surjective. Since M is topologically free and thus projective, this map splits and so by the formal Nakayama’s Lemma we see that the kernel vanishes.

Now let M be any profinite A-module, with N a profinite quotient of M. For {n_{i}} ⊆ N with n_{i} → 0, there exists a unique map of profinite A-modules ∏ Ae_{i} → N with e_{i} ↦ n_{i}. By projectivity, this lifts to a map ∏ Ae_{i} → M. Let m_{i} be the image of e_{i}.

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Finally, over a noetherian pseudocompact base we connect up the notions of flat and topologically flat for profinite modules.

Theorem 1.3.10 Let A be a complete local noetherian ring and M a profinite A-module. Then M is flat over A if and only if M is topologically flat over A. The same holds for faithful flatness and topological faithful flatness.

Proof. By Theorem 1.3.12 below, M is topologically flat over A if and only if M ˆ⊗_{A}(A/𝔪^{n}) is topologically flat over A/𝔪^{n} for all n ≥ 1. Since A is noetherian, we know by Corollary 1.2.7 that M ˆ⊗_{A}(A/𝔪^{n}) ≃ M/𝔪^{n}M as A-modules and that M/𝔪^{n} with its quotient topology is profinite over A (or, equivalently, A/𝔪^{n}). By [8, Thm 22.3], M is flat over A if and only if M/𝔪^{n}M is flat over A/𝔪^{n} for all n ≥ 1. Thus, it is enough (for flatness and topological flatness) to consider the case in which A is a local artin ring. Both flatness and topological flatness of M over A can be determined by tensoring (or completed tenorsing) against finite-length A-modules (which are profinite with respect to the discrete topology). We used Theorem 1.3.1(2) here. But a finite-length A-module N can be written as a finite direct sum of A-modules of the form A/𝔞, so it easily follows from Corollary 1.2.7 that M ⊗_{A} N ≃ M ˆ⊗_{A}N as A-modules. Thus, the functor M ⊗_{A} (·) on finite-length A-modules takes injections to injections if and only if the functor M ˆ⊗_{A}(·) does also.

Now return to the case where A may be non-artinian. Since a topologically flat M is topologically faithfully flat if and only if M ≠ 0, while a flat M is faithfully flat if and only if M ≠ 𝔪M = \overline{𝔪M}, by Corollary 1.2.7 and the formal Nakayama’s Lemma, we get the assertion for (topological) faithful flatness.

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Theorem 1.3.11 Let A be a pseudocompact ring and M a profinite A-module which is finitely generated as an A-module. Then M is topologically flat over A if and only if M is flat over A, and likewise for topological faithful flatness and faithful flatness.

Proof. Recall by Theorem 1.3.1(3) that as A_{𝔪}-modules, M ⊗_{A} A_{𝔪} ≃ M ˆ⊗_{A}A_{𝔪}. Thus, if M is flat over A, then M_{𝔪} is finite and flat over A_{𝔪} and so is finite and free. Hence, M is topologically flat over A. Conversely, suppose that M is topologically flat over A and choose a surjection in P F_{A} of the form A^{n} ↠ M. Since M is a projective object in P F_{A}, this splits, so in particular M is a direct summand of A^{n} as an A-module. Therefore, M is flat over A.

If M is faithfully flat, then M is topologically flat and M ˆ⊗_{A}A_{𝔪} ≃ M ⊗_{A} A_{𝔪} is not zero, so M is topologically faithfully flat. If M is topologically faithfully flat over A, then since there is a surjection from some A^{n} onto M in P F_{A}, we see that M ≃ ∏ A_{𝔪}^{r_{𝔪}} with 1 ≤ r_{𝔪} ≤ n. Thus, A is a direct summand of M in P F_{A}. Since M is topologically flat and therefore flat, the fact that A is a direct summand of M (as an A-module, say) shows that M is faithfully flat over A.

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Theorem 1.3.12 Let A be a pseudocompact ring and {𝔞_{i}} a collection of closed ideals in A such that ∩𝔞_{i} = 0. Let M be a profinite A-module. Then M is topologically flat over A if and only if M ˆ⊗_{A}(A/𝔞_{i}) is topologically flat over A/𝔞_{i} for all i. The same holds for topological faithful flatness.

Proof. We need to check the ‘if’ parts. We clearly can assume that A is local. Since ∩𝔞_{i} = 0, we can suppose that all 𝔞_{i} are proper ideals in A. Choose an isomorphism ∏(A/𝔪)e_{j} ≃ M ˆ⊗_{A}(A/𝔪) and by the formal Nakayama’s Lemma and the projectivity of topologically free modules, this lifts to a surjection π : ∏ Ae_{j} ↠ M which we wish to show is an isomorphism. Since A ≃ \underleftarrow{lim} A/𝔞_{i} and completed tensor product commutes with inverse limits, its enough to check that the above map π is an isomorphism after applying ˆ⊗_{A}(A/𝔞_{i}). But in that case we are reduced to the statement that a surjection between topologically free A-modules which is an isomorphism on the closed fiber is itself an isomorphism. This is clear, by the usual formal Nakayama’s Lemma/projectivity argument.

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§1.4 Functor Theorems

Let A be a pseudocompact ring, ℱ_{A} the category of profinite artinian A-algebras (these objects are all discrete), and 𝒫ℱ_{A} the category of profinite A-algebras.

Definition 1.4.1 For a covariant functor X : ℱ_{A} → Set, we define X′ : 𝒫ℱ_{A} → Set via

X′(B) = \underleftarrow{lim}_{𝔟∈ℐ_{B}} X(B/𝔟).

If X_{i} : ℱ_{A} → Set is a directed system of covariant functors, we define \underrightarrow{lim} X_{i} : ℱ_{A} → Set via (\underrightarrow{lim} X_{i})(R) = \underrightarrow{lim} X_{i}(R).

We have an obvious notion of product of two covariant functors from any category into Set, and the passage from X to X′ above is compatible with formation of products in a natural manner. Since we can define the direct limit of a direct system of covariant functors as above on any category at all, it makes sense to ask if passing from X to X′ as above behaves well with respect to formation of the direct limit functor. This is clearly not the case, since inverse and direct limits don’t generally ‘commute’ with each other; we will primarily only consider the direct limit of functors on ℱ_{A} and not on 𝒫ℱ_{A}.

Since the category 𝒫ℱ_{A} admits inverse limits, it makes sense to ask if a covariant functor Y : 𝒫ℱ_{A} → Set commutes with inverse limits, in the sense that for any inverse system in 𝒫ℱ_{A}, we ask if the natural map of sets

Y (\underleftarrow{lim} B_{i}) → \underleftarrow{lim} Y (B_{i})

is a bijection (in the same way, one can define what it means for a covariant functor X : ℱ_{A} → Set to commute with products). From this point of view, the ‘essential image’ of the construction X ⇝ X′ can be intrinsically described.

Theorem 1.4.2 The association X ⇝ X′ establishes an ‘equivalence’ between the covariant functors ℱ_{A} → Set and the covariant functors 𝒫ℱ_{A} → Set which commute with inverse limits. ‘Restriction’ gives the ‘inverse’ to this association.

Proof. For any inverse system {B_{i}} in 𝒫ℱ_{A} and any covariant functor X : ℱ_{A} → Set, we want to show that

X′(\underleftarrow{lim} B_{i}) → \underleftarrow{lim} X′(B_{i}) = \underleftarrow{lim}_{i} \underleftarrow{lim}_{𝔟_{i}∈ℐ_{B_{i}}}X(B_{i}/𝔟_{i})

is a bijection of sets. Clearly if B is in 𝒫ℱ_{A} and {𝔟} is some base of open ideals around 0 in B, then X′(B) → \underleftarrow{lim} X(B/𝔟) is a bijection of sets. Thus, we have a bijection

X′(\underleftarrow{lim} B_{i}) ≃ \underleftarrow{lim}_{(i,𝔟_{i})}X(B_{i}/𝔟_{i}).

Now all that remains is an easy exercise with iterated inverse limits of sets, and everything else is then clear.

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Note that if X is the restriction of Hom_{𝒫ℱ _{A}}(B, ·) to ℱ_{A}, then X′ ≃ Hom_{𝒫ℱ _{A}}(B, ·). Thus, by Yoneda’s Lemma it makes sense to give the following definition:

Definition 1.4.3 A covariant functor X : ℱ_{A} → Set is call pro-representable if it is isomorphic to the restriction Sp_{A}(B) of Hom_{𝒫ℱ _{A}}(B, ·) to ℱ_{A} for some profinite A-algebra B.

Lemma 1.4.4 1. Sp_{A}(B_{1}) × Sp_{A}(B_{2}) ≃ Sp_{A}(B_{1} ˆ⊗_{A}B_{2}).

2. If A → A′ is a map of pseudocompact rings, with A′ profinite over A, the functor

ℱ_{A′} ↪ ℱ_{A} → Set

which sends R to Sp_{A}(B)(R) is naturally isomorphic to Sp_{A′}(A′ ˆ⊗_{A}B).

3. For \{B_{i}\} an inverse system in 𝒫ℱ_{A} with surjective transition maps, \underrightarrow{lim} Sp_{A}(B_{i}) ≃ Sp_{A}(\underleftarrow{lim} B_{i}).

Note that it is not clear if Lemma 1.4.4(3) is true if we remove the surjectivity hypothesis.

Definition 1.4.5 A formal A-scheme is a covariant functor ℱ_{A} → Set which is prorepresentable.

It is useful to know which covariant functors X : 𝒫ℱ_{A} → Set are representable. By Theorem 1.4.2 and the remark preceding Definition 1.4.3, we see that this is the same as asking which covariant functors X : ℱ_{A} → Set are formal A-schemes. The importance of working with categories such as 𝒫ℱ_{A} is that representability questions are quite easy to answer. In order to give the necessary and sufficient criteria, we recall that a left exact sequence in a subcategory of Set is a diagram

U → V \overrightarrow{→} W

for which U → V is injective and identifies U with the subset of elements of V on which the two maps from V to W coincide.

If X : ℱ_{A} → Set is a covariant functor and R_{1} → R_{2} \overrightarrow{→} R_{3} is a left exact sequence in ℱ_{A}, then we can consider the diagram of sets

X(R_{1}) → X(R_{2}) \overrightarrow{→} X(R_{3})

and we say that X is left exact if the above sequence is always left exact.

It is obvious that if X is a formal A-scheme, then X commutes with products and is left exact. The remarkable fact is that the converse is true.

Theorem 1.4.7 (Grothendieck) A covariant functor X : ℱ_{A} → Set is a formal A-scheme if and only if it is left exact and commutes with products.

Remark. The proof below shows how critical it is to be working in a ‘formal’ category where we can take inverse limits over extremely large index sets. In various contexts (e.g., Fontaine’s approach to the theory of Dieudonne modules) one can sometimes show that the huge inverse limit ring constructed below is actually a finite A-algebra. Nevertheless, it is of essential technical importance to have an a priori large inverse limit ring before we can show that it might actually be finite over the base. Of course, in order to handle this sort of problem, one needs a sufficiently well-developed theory of such huge inverse limit rings; this is precisely the point of the algebraic results we have discussed above.

Proof. Necessity is trivial, we we have already observed. In order to prove sufficiency, we will ‘explicitly’ construct the pro-representing ring as an inverse limit over a very large index set.

To begin, pick an object R in ℱ_{A} and some r ∈ X(R). We first claim that there is a unique minimal A-subalgebra R_{r} ⊆ R such that r ∈ X(R_{r}). Of course, we note that R is annihilated by an open ideal of A and so is finite over an artinian quotient ring of A. Thus, any A-subalgebra R′ ⊆ R is again in ℱ_{A} and the obvious left exact sequence

R′ → R × R′ \overrightarrow{→} R

gives rise to the obvious left exact sequence

X(R′) → X(R) × X(R′) \overrightarrow{→} X(R)

(here we have used both hypotheses on X). In particular, X(R′) → X(R) is injective, so X(R′) can be viewed as a subset of X(R).

By finite-length considerations, the existence of R_{r} reduces to the assertion that for any A-subalgebras R_{1}, R_{2} of R such that r ∈ X(R_{1}), r ∈ X(R_{2}), then r ∈ X(R_{1} ∩R_{2}). Applying X to the left exact sequence

R_{1} ∩ R_{2} → R_{1} × R_{2} \overrightarrow{→} R

yields the corresponding left exact sequence

X(R_{1} ∩ R_{2}) → X(R_{1}) × X(R_{2}) \overrightarrow{→} X(R),

so X(R_{1})∩X(R_{2}) = X(R_{1} ∩R_{2}) (inside of X(R)). This completes the proof of the existence of R_{r}. Observe that this step would completely fail if we had tried to work in the larger category 𝒫ℱ_{A}.

We call a pair (R, r) consisting of R in ℱ_{A} and r ∈ X(R) a minimal pair if R = R_{r}. Note that (possibily non-minimal) pairs (R, r) with r ∈ X(R) form a category in an obvious way. The minimal pairs satisfy a certain critical ‘rigidity’ property which will enable us to circumvent some potentially troublesome set-theoretic issues in our inverse limit construction below. This same type of rigidity holds in the category of pointed connected étale covers of a fixed pointed connected base scheme and plays a role in the construction of the étale fundamental group quite similar to the way in which we will use this rigidity property below.

Lemma 1.4.8 Let (R, r) and (R′, r′) be two minimal pairs. Assume there exists a morphism f : R → R′ such that X(f)(r) = r′. Then such an f is unique and f is surjective.

Proof. The image f(R) ⊆ R′ of f is in ℱ_{A} and so X(R) → X(R′) factors through the injection X(f(R)) ↪ X(R′). Thus, r′ ∈ X(f(R)). Since (R′, r′) is a minimal pair, it follows that f(R) = R′, so f is automatically surjective.

Now assume that we are given two maps f, g : R → R′ with X(f)(r) = X(g)(r) = r′. We wish to prove that f = g. Consider the product map f × g : R → R′ × R′, so

X(f × g)(r) = (X(f)(r), X(g)(r)) = (r′, r′)

inside of X(R′ × R′) ≃ X(R′) × X(R′). We can view R′ as an A-subalgebra of R′ × R′ via the diagonal embedding. Let \overline{R} ⊆ R be the preimage of R′ under f × g. From the left exact sequence

\overline{R} → R \overrightarrow{→} R′,

(where the two right maps are f and g), we see that we have a left exact sequence

X(\overline{R}) → X(R) \overrightarrow{→} X(R′),

where the two right maps are X(f) and X(g). By the left exactness, it follows that r ∈ X(\overline{R}). But (R, r) is a minimal pair, so \overline{R} = R. Hence, f = g.

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Now that Lemma 1.4.8 is proven, we choose a set of minimal pairs {(R_{i}, r_{i})}_{i∈I} such that every minimal pair (R, r) is isomorphic to a unique pair (R_{i}, r_{i}). It is clear from the finiteness conditions built into the definition of ℱ_{A} that such a set really exists! For i, j ∈ I, we say that i ≤ j if and only if there exists a map of minimal pairs (R_{j}, r_{j}) → (R_{i}, r_{i}). By Lemma 1.4.8, if such a map exists, then it is unique and is surjective on the level of Aalgebras. Note that if (R_{1}, r_{1}) and (R_{2}, r_{2}) are minimal pairs, then ((R_{1} × R_{2})_{(r_{1},r_{2})}, (r_{1}, r_{2})) is a minimal pair and is greater than or equal to both (R_{1}, r_{1}) and (R_{2}, r_{2}). Thus, we see that {(R_{i}, r_{i})}_{i∈I} is an inverse system in an unambiguous manner (i.e., our choices of isomorphism class representatives makes no difference whatsoever). Of course, it should be emphasized that I has an incredibly large cardinality (depending on X also), and it is for this reason that we had to allow ourselves to work in extreme generality in the previous sections.

Let B = \underleftarrow{lim} R_{i}, an object in 𝒫ℱ_{A}. This will be the pro-representing ring. Since each R_{j} → R_{i} is surjective for j ≥ i, we know that B → R_{i} is surjective as well (Corollary 1.2.3(2)). This fact will be essential at one point below.

For R in ℱ_{A}, we define a map of sets X(R) → Sp_{A}(B)(R) = Hom_{𝒫ℱ _{A}}(B, R) as follows: for x ∈ X(R), there is a unique (!) i_{x} ∈ I such that (R_{x}, x) ≃ (R_{i_{x}}, r_{i_{x}}) (via a unique isomorphism), so we unambiguously get a map f_{x} : B ↠ R_{i_{x}} ≃ R_{x} ↪ R in Sp_{A}(B)(R). Note that f_{x} has image R_{x}.

Lemma 1.4.9 The morphism of functors X → Sp_{A}(B) defined above is a natural transformation.

Proof. Choose a morphism g : R_{1} → R_{2} in the category ℱ_{A} and pick x_{1} ∈ X(R_{1}). Let x_{2} = X(g)(x_{1}) ∈ X(R_{2}). We want to show that f_{x_{2}} = g ∘ f_{x_{1}}. Let \overline{R}_{1} = g^{−1}((R_{2})_{x_{2}}) ⊆ R_{1}. It is clear that we have a left exact sequence

\overline{R}_{1} → R_{1} × (R_{2})_{x_{2}} \overrightarrow{→} R_{2},

so we obtain a left exact sequence

X(\overline{R}_{1}) → X(R_{1}) × X((R_{2})_{x_{2}}) \overrightarrow{→} X(R_{2})

with the maps being the obvious ones.

The element (x_{1}, x_{2}) ∈ X(R_{1}) × X((R_{2})_{x_{2}}) maps to x_{2} under both right maps in the above diagram, so x_{1} ∈ X(\overline{R}_{1}). Hence, (R_{1})_{x_{1}} ⊆ \overline{R}_{1}. However, g((R_{1})_{x_{1}}) ⊆ R_{2} satisfies x_{2} ∈ X(g((R_{1})_{x_{1}})) since X(g)(x_{1}) = x_{2}, so by minimality we may conclude that (R_{2})_{x_{2}} ⊆ g((R_{1})_{x_{1}}). Since

g((R_{1})_{x_{1}}) ⊆ g(\overline{R}_{1}) ⊆ (R_{2})_{x_{2}},

we obtain the essential relation g((R_{1})_{x_{1}}) = (R_{2})_{x_{2}}. Thus, g induces a map h : (R_{1})_{x_{1}} → (R_{2})_{x_{2}} such that X(h)(x_{1}) = x_{2} and there is a commutative diagram in ℱ_{A}

(R_{1})_{x_{1}} ↪^{j_{1}} R_{1}
h ↓  ↓ g
(R_{2})_{x_{2}} ↪^{j_{2}} R_{2}

By Lemma 1.4.8, if we choose i_{1}, i_{2} ∈ I so that ((R_{n})_{x_{n}}, x_{n}) ≃ (R_{i_{n}}, r_{i_{n}}), then i_{1} ≥ i_{2} and the diagram

R_{i_{1}} ≃ (R_{1})_{x_{1}}
↓  ↓ h
R_{i_{2}} ≃ (R_{2})_{x_{2}}

commutes. If we combine the above two commutative squares with the fact that B → R_{i_{1}} → R_{i_{2}} is the canonical map B → R_{i_{2}}, it follows easily that g ∘ f_{x_{1}} = f_{x_{2}}.

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It remains to check for R in ℱ_{A} that the map of sets

X(R) → Hom_{𝒫F _{A}}(B, R)

is a bijection. First we check surjectivity of this set map. Choose some φ : B → R, so ker(φ) is open. By the definition of the topology on B, there exist i_{1}, …, i_{m} ∈ I (depending on φ) so that if (b_{i})_{i∈I} ∈ B and b_{i_{j}} = 0 for 1 ≤ j ≤ m, then φ((b_{i})) = 0. Choose i_{0} ∈ I with i_{0} ≥ i_{1}, …, i_{m} and let π_{i_{0}} : B → R_{i_{0}} be the canonical (continuous) map.

If π_{i_{0}}((b_{i})) = 0, so b_{i_{0}} = 0, then b_{i_{j}} = 0 for all 1 ≤ j ≤ m, so φ((b_{i})) = 0. Hence, ker(π_{i_{0}}) ⊆ ker(φ), so φ = g ∘ π_{i_{0}} for some g : R_{i_{0}} → R. By the same argument as in the proof of Lemma 1.4.9, if x = X(g)(r_{i_{0}}) ∈ X(R), then g(R_{i_{0}}) = R_{x}. Hence, if i ∈ I is the element for which (R_{x}, x) ≃ (R_{i}, r_{i}), then i_{0} ≥ i and g = h ∘ π_{i_{0},i}, where π_{i_{0},i} : R_{i_{0}} → R_{i} is the canonical transition map and h : R_{i} ≃ R_{x} ↪ R. Thus,

φ = h ∘ π_{i_{0}} ∘ π_{i_{0},i} = h ∘ π_{i} = f_{x}.

This shows that X(R) → Sp_{A}(B)(R) is surjective.

Now we check that f_{x} = f_{y} implies that x = y, for x, y ∈ X(R). Pick i, i′ ∈ I so that (R_{x}, x) ≃ (R_{i}, r_{i}) and (R_{y}, y) ≃ (R_{i′}, r_{i′}). Pick i′′ ∈ I larger than both i and i′, so the equality f_{x} = f_{y} forces the composite A-algebra maps

B ↠ R_{i′′} ↠ R_{i} ≃ R_{x} ↪ R

and

B ↠ R_{i′′} ↠ R_{i′} ≃ R_{y} ↪ R

to coincide. But B ↠ R_{i′′} is surjective, so it follows that R_{i′′} → R_{i} and R_{i′′} → R_{i′} have the same kernel. This gives rise to a commutative diagram of A-algebras

\begin{matrix}R_{i′′} & ↠ & R_{i} \\ ∥ & & ↓≃ \\ R_{i′′} & ↠ & R_{i′} \end{matrix}

and the surjectivity of maps starting from B enables us to conclude that the diagram

R_{i} ≃ R_{x} ↠ R
≃↓  ∥
R_{i′} ≃ R_{y} ↠ R

commutes. This implies that x = y in X(R), as desired.

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§1.5 Étaleness and Smoothness

Let A be a henselian local ring, with residue field k. This means that for all monic P ∈ A[X], any simple k-rational root of \overline{P} ∈ k[X] lifts to an A-rational root of P (which can then be shown to be necessarily unique). We say that A is strictly henselian if k is separably closed. For an excellent brief overview of the basic results on henselian local rings (and sketches of some of the proofs), see [1, §2.3]. The essential point we should note is that the equivalence of the various standard formulations of the henselian condition (see [1, Prop 4]) requires a form of Grothendieck’s generalization of Zariski’s Main Theorem [4, 8.12.8]. From these equivalent formulations (in particular, [1, Prop 4]), we see that a finite local algebra over a henselian local ring is again a henselian local ring. The avoidance of noetherian conditions gives us the technical freedom to state the results below in their most natural setting without the interference of extraneous assumptions.

Definition 1.5.1 Let B be an extension ring of A. We say that an A-subalgebra C in B is a étale subalgebra if C is finite, étale over A; we say that C is a maximal étale subalgebra if, in addition, all A-subalgebras of B which are finite étale over A lie inside of C.

If A is a field and B is a finite extension field, the maximal étale subalgebra exists and is nothing other than the separable closure of A in B. In fact, such a subalgebra always exists when A is a henselian local ring and B is flat over A. Below we’ll often say ‘subalgebra’ when we really mean ‘A-subalgebra’. This should not cause any confusion.

Theorem 1.5.2 Let A be a henselian local ring and let B be a finite flat extension ring. There exists a maximal étale subalgebra C. When B is local, C is the unique étale subalgebra whose residue field maps onto the separable closure of k in B/𝔪_{B}. In general, C is the product of the maximal étale subalgebras of the factor rings of B and C → B is faithfully flat.

Proof. By the primitive element theorem, we can find a monic P in A[X] such that for the finite flat A-algebra C = A[X]/P (X), there is a local k-algebra isomorphism from C/𝔪_{A} to the separable closure of k in B/𝔪_{B} (so C is local and étale over A). By [1, Lemma 7, §2.3], we can lift this uniquely to an A-algebra map C → B which is injective since maps between finite free A-modules that are injective mod 𝔪_{A} are injective to begin with (it is here that flatness of B over the local A is used). Also, it is easy to see by working over A/𝔪_{A} that C → B is finite free, hence faithfully flat. Since any étale subalgebra of B is necessarily local, as B is local, by [1, Lemma 7, §2.3] we see that C is the maximal étale subalgebra of B over A.

In general, write B = ∏ B_{i} as a finite product of local finite A-algebras. Let C_{i} be the maximal étale subalgebra in B_{i} and define C = ∏ C_{i}. Choose any étale subalgebra C′ in B. It suffices to show that the map C′ → B → B_{i} has image inside of C_{i}. Since B_{i} is local and C′ → B_{i} is finite, certainly this map factors through one of the local factors of C′, so we may assume that C′ is local. Invoking [1, Lemma 7, §2.3] again, it follows that C′ → B_{i} factors through C_{i} ↪ B_{i}, as desired.

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We now check an important property of the maximal étale subalgebra.

Theorem 1.5.3 Let B_{1}, B_{2} be finite flat A-algebras, with C_{i} the maximal étale subalgebra of B_{i}. Then C_{1} ⊗_{A} C_{2} → B_{1} ⊗_{A} B_{2} is injective and is the maximal étale subalgebra of B_{1} ⊗_{A} B_{2}.

Proof. We immediately reduce to the case in which B_{1} and B_{2} are local. Thus, C_{1} and C_{2} are local finite étale over A, so on the closed fiber they are fields. Hence, C_{i}/𝔪_{A} → B_{i}/𝔪_{A} is injective (and is even the maximal étale subalgebra), so the injectivity assertion can be checked on the closed fiber over A (as B_{1} ⊗_{A} B_{2} and C_{1} ⊗_{A} C_{2} are finite and A-flat, hence A-free). Since maps between finite étale A-algebras are the same as maps between their closed fibers over A [1, Lemma 7, §2.3], we even see that the other part of the assertion can also be checked on the closed fiber.

Hence, we are reduced to the case in which A = k is a field, so C_{i} = K_{i} is a finite separable field extension of k and B is a finite K_{i}-algebra. Let L_{i} denote the residue field of B_{i}, so L_{i} is a finite extension of k and K_{i} ↪ L_{i} identifies K_{i} with the separable closure of k in L_{i}. The injectivity assertion is now obvious. It remains to check that K_{1} ⊗_{k} K_{2} is the maximal étale subalgebra of B_{1} ⊗_{k} B_{2}. If we decompose B_{1} ⊗_{k} B_{2} into a product of local rings, it is easy to check that that K_{1} ⊗_{k} K_{2} and L_{1} ⊗_{k} L_{2} compatibly decompose, with K_{1} ⊗_{k} K_{2} identified with a product of subfields of each of the residue fields of the local rings. From this, it follows that we may replace each B_{i} by its residue field L_{i} and so we are reduced to the case in which B_{i} = L_{i} is a field, purely inseparable over K_{i} of finite degree.

If k has characteristic 0, then L_{i} = K_{i} and we are done. Now assume that k has characteristic p. In this case, we use a trick. For any finite k-algebra R, we can consider the subalgebra k[R^{p^{n}}] generated over k by the p^{n}th powers of the elements of R. This is a decreasing chain of k-subalgebras, so it terminates. We claim that it terminates at the maximal étale subalgebra. This obviously suffices, and it is clear that it is enough to check this when R is local. Letting L be the residue field of R and K be the separable closure of k in L, we have K = k[K^{p}]. There is a canonical copy of K (as a k-algebra) inside of R and for any r ∈ R, an examination of the residue field shows that a sufficiently high p^{n}th power of r differs from some element of K by a nilpotent element of R. Raising to an even higher p^{m}th power kills the nilpotent part. This finishes the proof.

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Theorem 1.5.4 Let B be a finite flat A-algebra with maximal étale subalgebra C. If A → A′ is a local map of henselian local rings, then C ⊗_{A} A′ → B ⊗_{A} A′ is injective and is the maximal étale subalgebra of B ⊗_{A} A′ over A′.

Proof. Let C′ = C ⊗_{A} A′, B′ = B ⊗_{A} A′, both finite flat A′-algebras (in particular, A′ → B′ is injective). In order to show that C′ → B′ is injective, it is enough to consider the case in which B (and hence C) is local, and we can also then check the injectivity on the closed fiber over A′, so we can assume that A, A′ and C are fields, in which case the injectivity is clear.

Since C′ is certainly finite étale over A′ and C′ → B′ is injective on the closed fiber over A′, it suffices to check the maximality on the closed fiber over A′, so again we can assume that A = k is a field and B is a finite local k-algebra. Thus, A′ = k′ is an extension field of k and C = K is a finite separable extension of k which is separably closed in the residue field of B = L[x_{1}, …, x_{n}]/I, with I containing some power of (x_{1}, …, x_{n}) and K ⊆ L inside of B, with [L : K] < ∞. We want K ⊗_{k} k′ to be the maximal étale subalgebra of B ⊗_{k} k′. Using the same method as in the previous proof (considering k′-subalgebras generated by p^{n}th powers in case k has characteristic p), we are done.

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Corollary 1.5.5 Let C be the maximal étale subalgebra of a finite flat A-algebra B. Then for any finite étale A-algebra R, any map R → B of A-algebras factors through C ↪ B.

Proof. We reduce immediately to the case in which B is local, so we can also assume that R is local. The result then follows from Theorem 1.5.2 and [1, Lemma 7, §2.3].

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We now are interested in the analogue of the above considerations in the setting of topologically flat profinite algebras over a pseudocompact ring. Fix a pseudocompact ring A. The first thing to observe is that if A is local, then it is henselian! Indeed, if P ∈ A[X] is monic and x ∈ k = A/𝔪 is a simple root of \overline{P} ∈ k[X], then for all 𝔞 ∈ ℐ_{A}, x lifts uniquely to a root of P in the local artinian (hence henselian) ring A/𝔞, so passing to the limit shows there exists a lift of x to a root of P in A. Before we can define what it means for a profinite A-algebra R to be (formally) étale over A, we need to review a few basic facts about modules of continuous differentials. Our treatment of formally étale covers is somewhat more general than that given in [3], since we make no hypotheses concerning perfectness of the residue field of A.

For a profinite A-algebra R, let I_{R} = ker(Rˆ⊗_{A}R → R), a closed ideal in the profinite A-algebra Rˆ⊗_{A}R.

Definition 1.5.6 We define the module of continuous differentials Ω̂_{R/A}^{1} to be I_{R}/\overline{I_{R}^{2}}, a profinite A-module. We define the universal continuous derivation d = d_{R/A} : R → Ω̂_{R/A}^{1} by d(r) = r ˆ⊗1 − 1ˆ⊗r.

It is clear from the construction that {dr | r ∈ R} generates a dense R-submodule of Ω̂_{R/A}^{1}. Also, the same proof as in the ‘classical’ case shows that Ω̂_{R/A}^{1} represents the functor from P F_{R} to Set that associates to each profinite R-module M the set of continuous Alinear derivations from R to M. Of course, if A is artinian and R is finite over A (hence discrete), this is just the usual module of differentials with the discrete topology. Also, Theorem 1.3.1(2), the left exactness of inverse limits, and the construction of Ω̂_{R/A}^{1} show that Ω̂_{R/A}^{1} = \underleftarrow{lim} Ω_{(R/𝔯)/A}^{1}, with the inverse limit taken over all open ideals 𝔯 of A (this can also be seen via universal mapping property considerations). Most of the standard facts about the module of differentials still hold in our profinite setting:

Theorem 1.5.7

1. For a map g : R → S of profinite A-algebras, there is an canonical exact sequence of

profinite S-modules

Ω̂_{R/A}^{1} ˆ⊗_{R}S →^{α} Ω̂_{S/A}^{1} →^{β} Ω̂_{S/R}^{1} → 0,

with α and β determined by α(d_{R/A}r ˆ⊗s) = sd_{S/A}g(r), β(d_{S/A}b) = d_{S/R}b.

2. If g : R → S is surjective and J = ker(g), then there is an exact sequence

J/\overline{J^{2}} →^{δ} Ω̂_{R/A}^{1} ˆ⊗_{R}S →^{α} Ω̂_{S/A}^{1} → 0

with δ(x mod \overline{J^{2}}) = d_{R/A}xˆ⊗1.

3. For R a profinite A-algebra and A → A′ a map of pseudocompact rings, so R′ = Rˆ⊗_{A}A′ is a profinite A′-algebra, there are natural isomorphisms of profinite A′-modules

Ω̂_{R′/A′}^{1} ≃ Ω̂_{R/A}^{1} ˆ⊗_{A}A′ ≃ Ω̂_{R/A}^{1} ˆ⊗_{R}R′.

4. For R and S two profinite A-algebras there is an isomorphism of profinite Rˆ⊗_{A}S-

modules

Ω̂_{Rˆ⊗_{A}S /A}^{1} ≃ Ω̂_{Rˆ⊗_{A}S /R}^{1} ⊕ Ω̂_{Rˆ⊗_{A}S /S}^{1}.

Proof. The proofs are the same as the usual ones [8, §25], except one just needs to mention continuity, denseness, and closures in the appropriate places. The only point requiring some care is (3), since A′ is not required to be profinite over A. However, it is not hard to check that all three objects ‘pro-represent’ the same functor on artinian profinite R′-modules and that the resulting isomorphisms are precisely the natural maps one expects.

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We can now define the reasonable analogue of étale in the profinite setting.

Definition 1.5.8 For a pseudocompact ring A, a profinite A-algebra R is said to be formally étale if R is topologically flat over A and Ω̂_{R/A}^{1} = 0.

Note that if A is an artin ring and R is discrete, then this coincides with the usual notion of étale in that setting. By Corollary 1.3.8 and Theorem 1.5.7, it is obvious that for a pseudocompact ring A and two formally étale A-algebras R_{1} and R_{2}, R_{1} ˆ⊗_{A}R_{2} is formally étale over A, and if A → A′ is a map of pseudocompacts (with A′ possibly not profinite over A) then ˆ⊗_{A}A′ takes formally étale A-algebras to formally étale A′-algebras. In fact, we even have an explicit description of formally étale algebras which is analogous to the usual description in the ‘classical’ case:

Theorem 1.5.9 Let R be a profinite A-algebra. Then R is formally étale over A if and only if each factor ring R_{𝔪} (for 𝔪 an open maximal ideal in A) is formally étale over A_{𝔪}. Assume that A is local. Then R is formally étale if and only if R is topologically flat over A and Rˆ⊗_{A}(A/𝔪_{A}) is a product of finite separable field extensions of k_{𝔪} = A/𝔪_{A}. This is equivalent (for local A) to the factor rings of the pseudocompact R being finite étale A-algebras (with their canonical finite free A-module topology.)

Proof. Let’s first address everything aside from the last part of the theorem. We clearly can reduce to the case in which A and R are local. Writing A = \underleftarrow{lim} A/𝔞 as an inverse limit of local artin rings and R = \underleftarrow{lim}(Rˆ⊗_{A}(A/𝔞)), we see that Ω̂_{R/A}^{1} = 0 if and only if Ω̂_{Rˆ⊗_{A}(A/𝔞) /(A/𝔞)}^{1} = 0 for all 𝔞 ∈ ℐ_{A}, so it suffices to treat the case in which A is an artin local ring and then we can pass to the inverse limit.

Since Ω̂_{Rˆ⊗_{A}(A/𝔪_{A}) /(A/𝔪_{A})}^{1} ≃ Ω̂_{R/A}^{1} ˆ⊗_{A}(A/𝔪_{A}), by the formal Nakayama’s Lemma we see that R is formally étale over A if and only if R is topologically flat over A and Rˆ⊗_{A}(A/𝔪_{A}) is formally étale over A/𝔪_{A}. If Rˆ⊗_{A}(A/𝔪_{A}) is a finite separable extension field of A/𝔪_{A}, then by the formal Nakayama’s Lemma we see that R is finite over A (and hence finite free by topological flatness), so it is discrete. Thus, Ω_{R/A}^{1} ≃ Ω̂_{R/A}^{1} = 0, whence R is a finite étale A-algebra.

It remains to check that if R is local and formally étale over a field k, then R is a finite separable extension field of k. If 𝔯 is any proper open ideal of R, then R/𝔯 is local and finite over k, with 0 = Ω̂_{R/k}^{1} → Ω̂_{(R/𝔯)/k}^{1} = Ω_{(R/𝔯)/k}^{1} surjective, so R/𝔯 is a finite separable extension field of k. But for 𝔯_{1} ⊆ 𝔯_{2}, we have a surjection of fields R/𝔯_{2} → R/𝔯_{1}, so this is an isomorphism and therefore 𝔯_{1} = 𝔯_{2}. Since R is separated and complete, we conclude that {0} is the only open proper ideal in the local ring R, so R is itself a finite separable extension field of k.

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Theorem 1.5.10 Let A be a local pseudocompact ring, B a topologically flat profinite Aalgebra. There exists a closed A-subalgebra C which is formally étale over A and which contains all other closed A-subalgebras of B which are formally étale over A. This maximal formally étale subalgebra construction behaves well with respect to base extension A → A′ of local pseudocompact rings, completed tensor products over A, and formation of arbitrary products of rings such as B.

Proof. First assume that B is local. Let C be a closed A-subalgebra of B. We first claim that C is necessarily local. Indeed, C is a profinite A-algebra and thus pseduocompact, so B is a profinite algebra over C also. Since C → B is injective, it is clear that under the decomposition of B indexed by the open maximals in C, each open maximal of C gives rise to a non-zero closed factor ring of B. However, B is a local ring, so it follows that C has a unique open maximal ideal, and thus C is local. Since A is henselian (!), we can now use the same construction as in the proof of Theorem 1.5.2.

More precisely, since B/𝔪_{B} is finite over k, [1, Lemma 7, §2.3] and the primitive element theorem show that we get a maximal formally étale A-subalgebra in B when B is local (here we check the analogous injectivity by using the straightfoward fact that a map between topologically flat A-modules which is an injection after applying ˆ⊗_{A}(A/𝔪_{A}) is itself injective; this latter assertion can be reduced to the case in which A is a local artin ring, and then we can use the duality theory from §1.3).

In general, we write B as a product of local profinite topologically flat A-algebras and can use an argument just as in the proof of Theorem 1.5.2, except we need to invoke continuity at the appropriate places.

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Since local pseudocompact rings are henselian, the method of proof of Corollary 1.5.5 also shows that any map from a formally étale profinite A-algebra R to a topologically flat profinite A-algebra B has its image inside of the maximal formally étale subalgebra C of B. Also, we see by an analysis of the formal closed fiber over A that C → B is topologically faithfully flat (see Theorem 2.3.2 for a more general result).

Theorem 1.5.11 Let A be a local pseudocompact ring, B_{1} and B_{2} topologically flat Aalgebras, C_{i} the maximal formally étale subalgebra of B_{i} over A. Then C_{1} ˆ⊗_{A}C_{2} → B_{1} ˆ⊗_{A}B_{2} is injective and is the maximal formally étale A-subalgebra of the topologically flat A-algebra B_{1} ˆ⊗_{A}B_{2}.

Proof. By being a little careful about continuity, completed tensor products, and formal Nakayama’s lemma, by the same method as in the proof of Theorem 1.5.3 we are reduced to the case in which A = k is a field, B_{i} is local, and C_{i} = K_{i} is a finite separable extension field of k, in which case the injectivity assertion is clear.

In order to prove the maximality, note that since K_{1} ⊗_{k} K_{2} = K_{1} ˆ⊗_{k}K_{2} is finite over k, there exist open ideals 𝔟_{i} in B_{i} such that K_{1} ⊗_{k} K_{2} → (B_{1}/𝔟_{1}) ⊗_{k} (B_{2}/𝔟_{2}) is injective. It is now easy to see that our claim is reduced to the case in which B_{1} and B_{2} are local artin rings finite over k, which we handled in Theorem 1.5.3.

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Theorem 1.5.12 Let B be topologically flat over A with maximal formally étale subalgebra C. If A → A′ is a local map of local pseudocompact rings, then C ˆ⊗_{A}A′ → B ˆ⊗_{A}A′ is an injection between topologically flat A′-algebras and is the maximal formally étale subalgebra.

Proof. It is clear that C′ = C ˆ⊗_{A}A′ is formally étale over A′. Injectivity is checked analogously to the argument in the proof of Theorem 1.5.4. To check maximality, we proceed using the same method as at the end of the proof of Theorem 1.5.11 in order to reduce to the case treated by Theorem 1.5.4.

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Another issue to address is the behavior of the maximal formally étale subalgebra with respect to inverse limits. Let {B_{i}} be an inverse system of topologically flat profinite Aalgebras, so B = \underleftarrow{lim} B_{i} is a topologically flat profinite A-algebra (by Theorem 1.3.1(2)). Let C_{i} be the maximal formally étale subalgebra of B_{i} over A. For j ≥ i, the map C_{j} → B_{j} → B_{i} must factor through C_{i} → B_{i}, by the observation preceding Theorem 1.5.11. In other words, \{C_{i}\} forms an inverse system in the obvious manner, so C = \underleftarrow{lim} C_{i} makes sense as a closed A-subalgebra of B and it is topologically flat over A (since each C_{i} is topologically flat over A). The main result concerning this situation is:

Theorem 1.5.13 For A, B, C as above, C is the maximal formally étale subalgebra of B over A.

Proof. Let C′ ⊆ B be a closed formally étale A-subalgebra. It suffices to show that C′ → B → B_{i} factors through C_{i} ↪ B_{i}. But C′ ˆ⊗_{A}A_{i} is a formally étale profinite A_{i}-algebra and so we can use a modification of the proof above that {C_{i}} forms an inverse system in order to conclude that C′ → B → B_{i} factors through C_{i} ↪ B_{i}, so C′ ⊆ C. It remains to check that C is formally étale over A. By Theorem 1.3.1(2), Theorem 1.5.9, and Theorem 1.5.12, we reduce to the case in which

A = k is a field. If the transition maps among the C_{i}’s were surjective, then via the interpretation of Ω̂^{1} as representating continuous derivations, it would be clear that the vanishing of Ω̂_{C_{i}/A}^{1} would force the vanishing of Ω̂_{C/A}^{1} (just show that Ω̂_{C/A}^{1} admits only the zero map to any profinite discrete A-module, from which we get Hom_{P F_{A}}(Ω̂_{C/A}^{1}, ·) = 0, so Ω̂_{C/A}^{1} = 0).

Thus, it suffices to show that if R is a formally étale profinite k-algebra, with k a field, then any closed k-subalgebra R′ ⊆ R is also formally étale over k (and then we can replace C_{i} in the above inverse limit by the image of the map C → C_{i} of profinite A-algebras). Applying Corollary 1.3.8 with M = Ω̂_{R′/k}^{1}, we see that Ω̂_{R′/k}^{1} = 0 if and only if Ω̂_{R′ ˆ⊗_{k}\overline{k} /\overline{k}}^{1} = 0, so we can base extend to an algebraic closure \overline{k} of k without loss of generality (this preserves injectiveness of R′ → R, by Corollary 1.3.8). In this case, R is a product of copies of k and we want to show that all closed k-subalgebras R′ have this form too. Since R is profinite as an R′-module and a direct factor ring of a formally étale profinite k-algebra is again a formally étale profinite k-algebra, we can decompose R into its factors as a profinite R′-algebra and so can assume that R′ is actually local. Any element in the maximal ideal of R′ is topologically nilpotent in R′ and so is topologically nilpotent in R. But R has no non-zero topologically nilpotent elements, so we see that R′ has maximal ideal {0}. Thus, {0} is open, so R′ is discrete and so is a finite k-algebra which is a field. Since k is algebraically closed, R′ = k as a k-algebra. This completes the proof.

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Now we are ready to consider the notion of smoothness in the setting of pseudocompact rings and profinite algebras.

Definition 1.5.14 Let A be a pseudocompact ring, B a profinite A-algebra. We say that B is formally smooth over A if for all profinite artinian A-algebras R (i.e., finite algebras over A/𝔞 for an open ideal 𝔞 of A, or equivalently all profinite A-algebras with finite A-length) and all ideals I in R with I^{2} = 0, the natural map Sp_{A}(B)(R) → Sp_{A}(B)(R/I) is surjective.

Remark We’ll see later, in Corollary 1.5.17, that if we replace ‘surjective’ by ‘bijective’ in the above definition, then we recover the notion of formally étale, as we’d expect. Also, it is clear by induction that if A → B is formally smooth, then for any profinite artinian A-algebra R and any nilpotent ideal I in R, the natural map Sp_{A}(B)(R) → Sp_{A}(B)(R/I) is surjective (intuitively, we can lift maps across infinitesimal thickenings).

In order to check formal smoothness, it clearly suffices to consider only local R, yet for local R and two profinite A-algebras B_{1} and B_{2}, we have as sets

Sp_{A}(B_{1} × B_{2})(R) = Sp_{A}(B_{1})(R) ∐ Sp_{A}(B_{2})(R).

Thus, B_{1} × B_{2} is formally smooth over A if and only if B_{1} and B_{2} are both formally smooth over A. Hence, we see that in general, if B is a profinite A-algebra, then B is formally smooth over A if and only if B_{𝔪} is formally smooth over A_{𝔪} for all open maximal ideals 𝔪 of A; when A is local, B is formally smooth over A if and only if the local factors of the pseudocompact ring B are formally smooth over A. Thus, it is clear that formally étale implies formally smooth (use [1, Prop 6, §2.2]).

Thus, the study of formally smooth profinite A-algebras essentially reduces to the study of formally smooth local profinite A-algebras, with A a local pseudocompact ring. We will now describe these local possibilities. This is the main result on formal smoothness in our setting and will take some time to prove rigorously. It should be emphasized that the definition of formal smoothness does not include topological flatness. Also, in contrast to formal étaleness, formal smoothness does not generally force separability on the residue field extensions. For example, if k is a non-perfect field and a ∈ k is not a pth power, then the completion of k[T]_{(T^{p}−a)} is formally smooth over k but has a residue field k(a^{1/p}) which is not separable over k. We omit the justification of this.

In view of the theorem below, formally smooth maps are non-trivial examples of topologically flat maps.

Theorem 1.5.15 Let A be a local pseudocompact ring and B a local profinite A-algebra. If B is formally smooth over A, then B is topologically flat over A. In this case, we can let C be the maximal formally étale A-subalgebra (so C is a local finite étale cover of A). If the finite residue field extension κ(A) → κ(B) is separable, then B ≃ C[[X_{i}]] over C for a suitable index set I. More generally, choose a local finite flat A-algebra A′ such that (κ(A′) ⊗_{κ(A)} κ(B))_{red} is a finite product of separable extensions of κ(A′) (in particular, A′ is a local profinite A-algebra). For any such A′, B′ = B ˆ⊗_{A}A′ is a finite product of formally smooth A′-algebras which satisfy the residual separability condition above (relative to A′).

Conversely, if B is a local profinite A-algebra and we choose A′ as above, then if the finitely many factor rings of B′ are formal power series rings over finite local étale A′-algebras, then B is formally smooth over A.

Finally, for any closed ideal 𝔟 in a profinite A-algebra B, any map of profinite A-algebras A[[X_{i}]] → B/𝔟 lifts to a map A[[X_{i}]] → B.

Remark. The existence of A′ as in the theorem is elementary. More precisely, basic field theory shows the existence of a finite extension k of κ(A) such that (k ⊗_{κ(A)} κ(B))_{red} is a finite product of separable extensions of k. We then wish to construct a finite flat local A-algebra with residue field k (as a κ(A)-algebra). It is enough to consider the case in which k has a primitive generator over κ(A). Lifting a minimal polynomial for such a generator then finishes the construction.

Proof. In order to check the final lifting property for formal power series rings, we simply use Theorem 1.3.9 to lift the image of X_{i} in B/𝔟 to an element b_{i} ∈ B with b_{i} → 0, in the sense that any open A-submodule of B (in particular, any open ideal) contains all but finitely many b_{i}’s. There is clearly a map A[[X_{i}]] → B of profinite A-algebras taking X_{i} to b_{i}.

We now prove that all local formally smooth B over A have the asserted form. Choosing A′ as in the theorem, we immediately reduce to the case in which κ(B) is a finite separable extension of κ(A). Suppose we have the result when A is a local artin ring. Let’s see how to finish the proof of the general case. Note that although we don’t yet claim that formal smoothness is preserved under arbitrary local (formal) base change (though this will be obvious once the theorem is proven), at the very least it is clear for a local base change A → A′ with A′ a pseudocompact ring which is profinite as an A-algebra (since then finitelength profinite A′-algebras are also finite-length profinite A-algebras). In particular, for an open ideal 𝔞 in A, B ˆ⊗_{A}(A/𝔞) is formally smooth over A/𝔞. Hence, by our assumption that the theorem is true for an artin local base, we see that B ˆ⊗_{A}(A/𝔞) is topologically flat over A/𝔞. By Theorem 1.3.13, B is topologically flat over A, so there is a maximal formally étale subalgebra C in B (and it is finite étale and local over A, thus finite and free as an A-module), with C/𝔞 ≃ C ˆ⊗_{A}(A/𝔞) the maximal formally étale subalgebra of B ˆ⊗_{A}(A/𝔞).

Now go back and (again by our assumption of the theorem over an artin local base) choose a C/𝔪_{A}-isomorphism (C/𝔪_{A})[[X_{i}]] ≃ B ˆ⊗_{A}(A/𝔪_{A}) of profinite C/𝔪_{A}-algebras (i.e., use the choice 𝔞 = 𝔪_{A} above). By the first part of the proof, this then lifts (in many ways) to a map of local profinite C-algebras f : C[[X_{i}]] → B. This lift is surjective by the formal Nakayama Lemma and is an isomorphism after applying ˆ⊗_{A}(A/𝔪_{A}). Hence, by the topological flatness of B over A, it follows that the map f is an isomorphism, as desired.

Thus, we are reduced to the case in which A is an artin local ring. Keep in mind that we do not yet know that B is topologically flat over A. From the henselian property of B, we construct a local finite étale closed subalgebra C in B over A which canonically lifts the separable closure of k = A/𝔪_{A} inside of the finite extension given by the residue field of B. By our separability hypothesis, this separable closure is the entire residue field of B. It is easy to see (check!) that B is formally smooth over C, so we can at least assume that A = C for the rest of the argument, so A and B have the ‘same’ residue field.

Since A is an artin local ring, by the formal smoothness of B over A we have that the residue field map B ↠ A/𝔪_{A} = k lifts to a section s : B → A to the A-algebra structure map A → B. Let J = ker(s), a closed ideal in B. Choose a topological k-basis \{\overline{x}_{i}\} for (J/\overline{J^{2}})ˆ⊗_{A}k ≃ J ˆ⊗_{B}k, so lifting to elements x_{i} ∈ J with x_{i} → 0 (by Theorem 1.3.9), we get a unique (necessarily local) map of profinite A-algebras

θ_{A} : A[[X_{i}]] → B

sending X_{i} ↦ x_{i}. Our goal is to show that this is an isomorphism.

The surjectivity of θ_{A} can be checked on the (formal) closed fiber over A, thanks to the formal Nakayama Lemma and the fact that θ_{A} ˆ⊗_{A}k = θ_{k} (using the choice \{\overline{x}_{i}\}). But on the closed fiber J is just the maximal ideal of B, so θ_{k} induces a surjection on the formal cotangent spaces at the maximal ideals. A ‘profinite’ analogue of [5, Lemma 7.4, Ch II], whose proof is essentially the same, except that it uses the formal Nakayama Lemma in place of the usual one, etc., shows that θ_{k} is surjective. Thus, θ_{A} is surjective.

Now we check that θ_{A} is injective, with A an artin local ring. The section s shows that any non-zero element u in ker(θ_{A}) (if such a thing exists) has the form

u = ∑_{r=r_{0}}^{∞} u_{r},

where u_{r} is a formal homogenous form of degree r in the X_{i}’s, with r_{0} ≥ 1 and u_{r_{0}} ≠ 0 (the point is that r_{0} ≠ 0). Choose such an element u. We will obtain a contradiction.

We know that u_{r_{0}} contains a monomial aX_{i_{1}}^{e_{1}} … X_{i_{m}}^{e_{m}} with a ∈ A non-zero and ∑ e_{i_{j}} = r_{0} ≥ 1. Since θ_{A} is a quotient map (!) of profinite A-modules, for an appropriate open ideal 𝔟 in B we have an isomorphism of A-algebras

\overline{θ_{A}} : A[X_{i_{1}}, …, X_{i_{m}}]/(X_{i_{j}}^{e_{j}+1}) = A[[X_{i}]]/\overline{(X_{i_{j}}^{e_{j}+1}, X_{i} (i ≠ i_{1}, …, i_{m}))} ≃ B/𝔟.

Choose some i_{n} ∈ I. The surjection of A-algebras

π : A[X_{i_{j}}]/(X_{i_{j}}^{e_{j}+1}X_{i_{n}}) ↠ A[X_{i_{j}}]/(X_{i_{j}}^{e_{j}+1})

has a nilpotent kernel, so by the formal smoothness the map

ψ_{1} : B ↠ B/𝔟 ≃ A[X_{i_{j}}]/(X_{i_{j}}^{e_{j}+1})

lifts to a continuous A-algebra map

ψ_{2} : B → A[X_{i_{j}}]/(X_{i_{j}}^{e_{j}+1}X_{i_{n}}).

In particular, ψ_{2}(x_{i_{j}}) ≡ X_{i_{j}} mod ker(π) and ψ_{2}(x_{i}) ∈ ker(π) for i ≠ i_{1}, …, i_{m}. Thus, ψ_{2}(x_{i_{j}}^{e}) = X_{i_{j}}^{e} for all e ≥ 2 and

ψ_{2}(x_{i}^{2}) = ψ_{2}(x_{i}x_{i_{j}}) = ψ_{2}(x_{i}x_{i′}) = 0.

Suppose now that r_{0} > 1 (we’ll handle r_{0} = 1 later). We easily see (for θ = θ_{A}) that

0 = ψ_{2}(θ(u)) = ψ_{2}(θ(u_{r_{0}}) + θ(u_{r_{0}+1}))

and if m > 1 then together with the above this yields

0 = ψ_{2}(ax_{i_{1}}^{e_{1}} … x_{i_{m}}^{e_{m}}) = aX_{i_{1}}^{e_{1}} … X_{i_{m}}^{e_{m}} + (ker(π) stuff).

Applying π gives a = 0, a contradiction. If m = 1, a similar argument with some care yields the same contradiction, provided that e_{i_{1}} = r_{0} is at least 2, which we’re assuming to be the case.

Now consider the remaining possibility r_{0} = 1. For this case we will use the topological basis property of the \overline{x}_{i}’s, which so far has not been used! We have 0 ≠ u_{1} = ∑ a_{i}X_{i}, yet θ(u) = 0 forces ∑ a_{i}x_{i} ∈ \overline{J^{2}}. Thus, ∑ a_{i}\overline{x}_{i} = 0, so all a_{i} ∈ 𝔪_{A}. If A is a field, we’d be done. Hence, we at least know that θ_{A} ˆ⊗_{A}(A/𝔪_{A}) is an isomorphism. Pick any i_{0} ∈ I. Since 𝔪_{A} is nilpotent and A → B is formally smooth, the continuous surjection

B ↠ B ˆ⊗_{A}(A/𝔪_{A}) ≃ (A/𝔪_{A})[[X_{i}]] ↠ (A/𝔪_{A})[[X_{i}]]/\overline{(X_{i}X_{i′}, X_{i} (i ≠ i_{0}))} ≃ (A/𝔪_{A})[X_{i_{0}}]/(X_{i_{0}}^{2}) lifts to a continuous map B → A[X_{i_{0}}]/(X_{i_{0}}^{2}), so the lift sends x_{i} to β_{i}X_{i_{0}}, with β_{i} ∈ 𝔪_{A} for i ≠ i_{0} (with almost all β_{i} = 0) and β_{i_{0}} ≡ 1 mod 𝔪_{A}. Since ∑ a_{i}x_{i} = 0, applying the map B → A[X_{i_{0}}]/(X_{i_{0}}^{2}) to this and looking at the coefficient of X_{i_{0}} then yields

a_{i_{0}} = −β_{i_{0}}^{−1} (∑_{i≠i_{0}} a_{i}β_{i}) ∈ 𝔪_{A}^{2}.

That is, all a_{i} ∈ 𝔪_{A}^{2}. Thus, the above formula now shows that all a_{i} ∈ 𝔪_{A}^{3}. Continuing in this way, all a_{i} ∈ ∩_{n≥1}𝔪_{A}^{n} = 0.

Now we prove the converse. Choose some profinite artinian A-algebra R and an ideal I in R with I^{2} = 0, and a map of profinite A-algebras f : B → R/I. This extends to a map

f′ : B ˆ⊗_{A}A′ → (R/I)ˆ⊗_{A}A′ ≃ (R ⊗_{A} A′)/(I ⊗_{A} A′),

so the final part of the theorem (already proven above) and the hypothesis on B ˆ⊗_{A}A′ imply that we can lift this to a map F′ : B ˆ⊗_{A}A′ → R ⊗_{A} A′. The resulting continuous A-algebra map F : B → R ⊗_{A} A′ has the property that its image in (R ⊗_{A} A′)/(I ⊗_{A} A′) ≃ (R/I) ⊗_{A} A′ lies inside of R/I. Thus, for b ∈ B we uniquely have F (b) = r(b) + i(b) with r(b) ∈ R and i(b) ∈ I ⊗_{A} X′, where X′ is a (fixed) A-module complement to A inside of A′. Since F is a map of profinite A-algebras and I^{2} = 0, it is easy to see that r : B → R is a lift of f of the desired type.

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Corollary 1.5.16 Let A and B be complete local noetherian rings and A → B a local map. Then B is profinite as an A-algebra if and only if the induced map on residue fields is a finite extension, and if the residue field extension is finite and separable, then B is formally smooth over A if and only if B ≃ A′[[X_{1}, …, X_{n}]] as A-algebras (for some n ≥ 0), with A′ a local finite étale cover of A.

This is clear.

Corollary 1.5.17 A profinite algebra B over a pseudocompact ring A is formally étale over A if and only if for all profinite artinian A-algebras R and all ideals I in R with I^{2} = 0, the natural map Sp_{A}(B)(R) → Sp_{A}(B)(R/I) is a bijection.

Proof. We reduce to the local case. By Theorem 1.5.9, it remains to prove sufficiency. If B satsifies the mapping property relative to A, then so does B ˆ⊗_{A}A′ relative to A′ for any profinite A-algebra A′. In particular, if we choose a finite local A-algebra A′ as in Theorem 1.5.15, then since B ⊗_{A} A′ ≃ B ˆ⊗_{A}A′ and the finite étale property can be checked after a faithfully flat base change, we may assume that B is a formal power series ring over a local finite étale cover of A. It is then obvious that the set of indeterminates must be empty.

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Corollary 1.5.18 Let A → A′ be a map of pseudocompact rings. If B is a formally smooth profinite A-algebra, then B′ = B ˆ⊗_{A}A′ is a formally smooth profinite A′-algebra.

Proof. By Corollary 1.3.8, B′ is a profinite A′-algebra. Formally ‘localizing’ at the open maximals of A′, we easily reduce ourselves to the case in which A → A′ is a local map of local pseudocompact rings. Moreover, Corollary 1.3.8 shows that it is sufficient to consider the local factor rings of B one at a time (note that we can’t use Theorem 1.3.1(2) because A′ was not assumed to be profinite over A). Thus, B is also local, so by Theorem 1.5.15 we choose a finite flat local A-algebra A_{0} such that (κ(A_{0})⊗_{κ(A)} κ(B))_{red} is étale over κ(A_{0}) and then B ˆ⊗_{A}A_{0} ≃ ∏ B_{j}, with each of the finitely many B_{j}’s a formal power series ring over a finite local étale cover of A_{0}. Define A′_{0} = A_{0} ˆ⊗_{A}A′ (which might not be local). Note that

B′ ˆ⊗_{A′}A′_{0} ≃ B′ ⊗_{A} A_{0} ≃ (B ⊗_{A} A_{0}) ⊗_{A_{0}} A′_{0}.

Also, (κ(A_{0}) ⊗_{κ(A)} κ(B))_{red} ⊗_{κ(A_{0})} (κ(A_{0}) ⊗_{κ(A)} κ(A′))_{red} is étale over (κ(A_{0}) ⊗_{κ(A)} κ(A′))_{red} and so is itself reduced. Thus, this large tensor product is isomorphic to the reduced ring underlying κ(B) ⊗_{κ(A)} κ(A′) ⊗_{κ(A)} κ(A_{0}). Since this reduced ring is exactly the product of the residue fields of the factors of B′ ˆ⊗_{A′}A′_{0}, we can take any local factor ring of A′_{0} and use it in the ‘converse’ part of Theorem 1.5.15 in order to show that B′ is formally étale over A′.

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The above result is of interest because it does not require A′ to be profinite over A. The main situation in which this is interesting is the case of the map from a field to its separable or algebraic closure. Such maps are typically not finite, yet in many arguments it is natural to want to reduce oneself to a situation over separably or algebraically closed fields. This is why it is important to have results like Corollary 1.5.18 and Theorem 1.5.7(3) without profiniteness conditions on base changes.

Let’s now briefly discuss a profinite analogue to the usual relation between Galois theory and finite étale covers of a field. Let k be a field, B a profinite k-algebra, k′/k an extension field, B′ = k′ ˆ⊗_{k}B (a profinite k′-algebra). Applying Corollary 1.3.8 to the profinite k-module Ω̂_{B/k}^{1} and recalling Theorem 1.3.6 for topological flatness over fields, we see that B is formally étale over k if and only if B′ is formally étale over k′. In particular, if k_{s} is a separable closure of k and B_{s} = k_{s} ˆ⊗_{k}B, then B is formally étale over k if and only if B_{s} is a product of copies of k_{s} as a k_{s}-algebra. Note that G = Gal(k_{s}/k) acts on B_{s} in an obvious way and that this induces a permutation action on the set M(B, k_{s}) of open maximals of B_{s} (which is in bijective correspondence with the set of continuous k-algebra maps from B to k_{s}). Moreover, every element of M(B, k_{s}) is fixed by an open subgroup of G, which is what it means to say that M(B, k_{s}) is a discrete G-set. Conversely, if X is a discrete G-set, then let B(X) be the fixed subset of ∏_{x∈X} k_{s}x under the obvious semilinear G-action. If we decompose X into a disjoint union of its finite G-orbits, then it is an easy consequence of the usual Galois-étale comparison for finite covers of fields that B(X) is a profinite k-algebra which is formally étale.

This yields

Theorem 1.5.19 Let A be a local pseudocompact ring. Choosing a separable closure k_{s} of the residue field k gives rise to an antiequivalence of categories between formally étale Aalgebras and the category of Gal(k_{s}/k)-sets. In particular, completed tensor products go over to products.

Proof. Since A is a henselian local ring, by Theorem 1.5.9 and [1, Lemma 7, §2.3] we see that we can consider the case in which A is a field without any loss of functoriality. But the case in which we are over a field is settled by the discussion above.

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Of course, in Theorem 1.5.19 we could just as well have chosen a strict Henselization of A in place of k_{s} and used the resulting étale fundamental group for A in place of Gal(k_{s}/k), but this amount to the same thing (except that k_{s} is a pseudocompact ring, whereas the strict Henselization of A generally is not).

We conclude this section with a proof that formal smoothness can be checked on (formal) closed fibers.

Theorem 1.5.20 Let A be a pseudocompact ring and B a profinite topologically flat Aalgebra. Then B is formally smooth over A if and only if for each open maximal 𝔪 of A and some extension field k′_{𝔪} of A/𝔪, the profinite k′_{𝔪}-algebra B ˆ⊗_{A}k′_{𝔪} is formally smooth over k′_{𝔪}.

Proof. We can clearly assume that B and A are local. Since ‘only if’ is clear, we now suppose that B ˆ⊗_{A}k′ is formally smooth over k′ for some extension field k′ of k = A/𝔪. Suppose we can show that B ˆ⊗_{A}k is formally smooth over k. Choose a finite extension k_{0}/k such that (k_{0} ⊗_{k} κ(B))_{red} is étale over k_{0}. Pick a local finite flat A-algebra A_{0} with residue field k_{0} (as a k-algebra). Thus, B_{0} = B ˆ⊗_{A}A_{0} has a formally smooth closed fiber with all residue fields étale over κ(A_{0}) = k_{0}. Since B_{0} is topologically flat over A, we may proceed as in the proof of Theorem 1.5.15 to show that each of the finitely many factor rings of B_{0} is a formal power series ring over a finite local étale over of A. By the ‘converse’ part of Theorem 1.5.15, B is formally smooth over A.

Since B ˆ⊗_{A}k′ ≃ (B ˆ⊗_{A}k)ˆ⊗_{k}k′, we may assume now that A = k is a field and B is a profinite k-algebra such that B ˆ⊗_{k}k′ is formally smooth over k′. The argument now proceeds essentially like the end of the proof of Theorem 1.5.15, except we note that since k′ is a vector space over k (of possibly infinite dimension), we can find a k-module complement X′ to k inside of k′.

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§2 Affine and Formal Group Schemes

Our goal in this section is to discuss the fundamental general notions in the theory of affine and formal group schemes. The essential issues are kernels and quotients, finite flat commutative group schemes over any base scheme, differentials on any group scheme, the connected-étale sequence over a henselian local base, and the analogous theory for formal group schemes over a pseudocompact base. Applications to the theory of p-divisible groups (including explanations of some of the essential missing details in [9]) are postponed until §3.

The general principle is that we carry out all constructions in a relative setting, but for many proofs we quickly reduce ourselves to the case in which the base is an algebraically closed (hence perfect) field.

We assume that the reader has seen the notion of a group scheme before, but not in any particular depth.

§2.1 Affine and Formal Group Schemes over Fields

We begin with some aspects of the theory over fields because in this setting, flatness (or topological flatness) over the base is always satisfied for all constructions we carry out. In later sections we will discuss a more satisfactory relative theory by incorporating flatness hypotheses. However, the situation over fields does enjoy some special properties (e.g., certain categories are abelian) and proofs in the relative case are often obtained by reducing to the case in which the base is a field. Thus, it is simplest if we first spend some time over fields, rather than periodically proving intermediate results over fields along the way.

Definition 2.1.1 Let k be a field. A group scheme over k is a group object in the category of k-schemes. A formal group scheme over k is a group object in the category of formal kschemes. The rings corresponding to such objects are called bialgebras over k (resp. formal bialgebras over k).

The condition that G = Sp_{k}(B) is a (commutative) formal group scheme over k is clearly equivalent to the existence of morphisms m : B → B ˆ⊗_{k}B, ϵ : B → A, and i : B → B of profinite k-algebras, satisfying certain compatibility conditions. It is clear that formal group schemes over k form a category, with formal commutative group schemes forming a full additive subcategory; these categories admit the existence of products and kernels, with the usual functorial descriptions (and the forgetful functor from the category of commutative formal group schemes over k to formal group schemes over k commutes with formation of products and kernels). Analogous statements hold for the categories of group schemes and affine group schemes over k.

Definition 2.1.2 For a group scheme G over k and a morphism H → G in the category, we say that H is a closed subgroup of G if the map of schemes is a closed immersion. In the formal case, we require the map on rings to be surjective.

If f : H → G is a morphism of formal group schemes or of group schemes over k (with G separated over k in the latter case), then ker(f) is clearly a closed subgroup of H. In the affine (or formal) case, on the ring level we just take the quotient of the ring 𝒪(H) by the (closure of the) ideal generated by the image of the augmentation ideal of 𝒪(G). It is fairly easy to check that a group scheme over k is separated if and only if the identity section is a closed immersion [1, Lemma 2, §7.1]. Cokernels for morphisms exist in the categories of affine and formal group schemes over k. These can be constructed ‘by hand’, as we now explain in the formal case (the non-formal affine case is similar).

Let f : G_{1} → G_{2} be a morphism of formal group schemes over k. Writing 𝒪(G_{i}) = k ⊕ I_{i}, with I_{j} the (closed) augmentation ideal, the map on rings has the form k ⊕ I_{2} → k ⊕ I_{1}, with the map compatible with the decompositions. For any formal group scheme G over k and a morphism G_{2} → G, the map G_{1} → G_{2} → G vanishes if and only if the ring map 𝒪(G) = k ⊕ I → k ⊕ I_{2} sends I into the kernel J of the map I_{2} → I_{1}. Thus, if k ⊕ J contains a maximal (necessarily closed) formal subbialgebra inside of 𝒪(G_{2}), then this is the ring corresponding to coker(f). Certainly k ⊕ (0) is a formal subbialgebra of 𝒪(G_{2}) inside k ⊕ J, so it suffices to check more generally that if R = k ⊕ I_{R} is a formal bialgebra over k and {R_{j}} is a collection of formal subbialgebras inside of R, then the closed k-subalgebra R′ generated by the R_{j}’s is again a formal subbialgebra inside of R.

Since we have compatible decompositions R_{j} = k ⊕ I_{j}, certainly there is a decomposition of the profinite k-algebra R′ = k ⊕ I compatible with that for R. It remains to check that

I ↪ I_{R} → (k ˆ⊗_{k}I_{R}) ⊕ (I_{R} ˆ⊗_{k}k) ⊕ (I_{R} ˆ⊗_{k}I_{R})

has its (closed!) image inside of the (necessarily closed!) profinite k-submodule (k ˆ⊗_{k}I) ⊕ (I ˆ⊗_{k}k) ⊕ (I ˆ⊗_{k}I). Note that we are using here quite critically the fact that we are working over a field, as this ensures that completed tensor products of injective maps are injective maps. Since I contains each I_{j} and R_{j} is a formal subbialgebra inside of R, an easy continuity argument finishes the proof that this is the ring for coker(f). Of course, we have not yet shown that coker(f) satisfies any nice properties; we will do this later under the presence of commutativity hypotheses.

When H is a closed ‘normal’ subgroup of G (i.e., H(T) is a normal subgroup of G(T) for all k-schemes T or all finite k-schemes T, depending on whether we are in the k-scheme or formal k-scheme setting), we let G/H denote the cokernel of H → G. The above discussion suggests that subbialgebras are the rings for ‘quotient groups’. From a geometric point of view, if H is a closed normal subgroup of G, then the map G → G/H looks like it should have ‘nicely varying’ fibers (just cosets of H) and so it should be flat. But the map should also be surjective, which is to say faithfully flat. This is indeed the case.

Theorem 2.1.3 Let B be a formal bialgbera over k and A a formal subbialgebra (so B is profinite as an A-algebra). Then the map A ↪ B of profinite k-algebras is topologically faithfully flat. Similarly, if A → B is an injection of bialgebras over k, then as a map of rings it is faithfully flat.

Proof. First consider the formal situation. Before showing that A → B is topologically faithfully flat, we prove the related result that B ˆ⊗_{A}B is topologically free over B as a left B-module. It is at this stage (and only this stage) that the ‘group theory’ will be used. Let G = Sp_{k}(B), H = Sp_{k}(A), so G ×_{H} G = Sp_{k}(B ˆ⊗_{A}B) is a formal k-scheme. Let K be the kernel of G → H, so K = Sp_{k}(B/𝔟) for some closed ideal 𝔟 in B. From the functor of points it is clear that there is an isomorphism of formal k-schemes G ×_{H} G ≃ G × K, via (g, g′) ↦ (g, g′g^{−1}), thereby giving an isomorphism of profinite (left) B-algebras B ˆ⊗_{A}B ≃ B ˆ⊗_{k}(B/𝔟). Since B/𝔟 is topologically free over k, the topological freeness of B ˆ⊗_{A}B over B (using the left B-module structure) now follows.

We will now show that if A is any pseudocompact ring and A → B is an injection of profinite k-algebras, then for any profinite A-module M, if the profinite B-module B ˆ⊗_{A}M is topologically free, then M is topologically flat over A. This will then prove that in the above group situation, B is topologically flat (resp. flat) over A. Decomposing along the local factors of the pseudocompact A, it suffices to consider the case in which A is a local pseudocompact, say with maximal ideal 𝔪 and residue field k′ = A/𝔪 which is a finite extension of k. Since A is an equicharacteristic complete local ring, there is a k-algebra section to the residue field map, so we can view k′ ⊆ A as a k-subalgebra. The induced topology on k′ is discrete, since any proper open ideal 𝔞 in A lives in the maximal ideal and k′ → A/𝔞 is injective. Thus, this section is continuous.

Choose a topological basis \{\overline{x}_{i}\} for k′ ˆ⊗_{A}M over k′. Since ∏ Ae_{i} is projective in P F_{A}, we can lift the map

∏ Ae_{i} ↠ ∏ k′e_{i} ≃ k′ ˆ⊗_{A}M ≃ M/\overline{𝔪M}

to a map π : ∏ Ae_{i} → M, say with x_{i} = π(e_{i}), and by the formal Nakayama’s Lemma this is surjective. Before showing that π is an isomorphism, we need to check that {1ˆ⊗x_{i}} is a pseudobasis of the topologically free profinite B-module B ˆ⊗_{A}M. Let \overline{B} = B ˆ⊗_{A}k′. Under \overline{B} ˆ⊗_{A}M ≃ \overline{B} ˆ⊗_{k′}(k′ ˆ⊗_{A}M) we see that the profinite \overline{B}-module \overline{B} ˆ⊗_{A}M is topologically free with a pseudobasis {1ˆ⊗x_{i}}. Since x_{i} → 0 in M (by the continuity of π), there exists a unique map of profinite B-modules ∏ Be_{i} → B ˆ⊗_{A}M with e_{i} ↦ 1ˆ⊗x_{i}. Both sides are topologically free over B, so to check this is an isomporphism, it is enough to check after applying (B/𝔟)ˆ⊗_{B}(·) for some closed ideal 𝔟 in the radical of B. For example, we can take 𝔟 = \overline{𝔪_{A}B}. Hence, \{1ˆ⊗x_{i}\} is a topological basis for B ˆ⊗_{A}M over B.

Since the surjection π becomes an isomorphism after applying B ˆ⊗_{A}(·), if (a_{i}e_{i}) lies in the kernel, then it vanishes under the map ∏ Ae_{i} → B ˆ⊗_{A}(∏ Ae_{i}) ≃ ∏ Be_{i}, where this composite is the natural one arising from the map A → B. But this map is injective, so we conclude that ker(π) = 0 and so M is topologically free over A (which is equivalent to being topologically flat).

Back to the formal group schemes. We now know that the injection A → B is topologically flat, so for each open maximal ideal 𝔪 of A, A_{𝔪} → B_{𝔪} is topologically flat and injective. Thus, B_{𝔪} is non-zero and so B is topologically faithfully flat over A.

Now consider the algebraic situation. By [10, §3.3], A and B are unions of k-subalgebras of finite type which are stable under the group operations. Thus, it is easy to see that we are reduced to the case in which A and B are of finite type over k and we can also assume that k is algebraically closed. Since A → B is injective, Spec(B) → Spec(A) has a dense image. However, this image is constructible by [5, Exercise 3.19, Ch II] and so contains a dense open set. Since k is algebraically closed, an easy group-theoretic/topological argument shows that the map of group schemes is surjective on closed points, and so it is also surjective on the entire spaces of the group schemes, as is well-known. Hence, by a translation argument (using the group structure!) together with [8, Thm 7.1], faithful flatness will follow if we can show that the induced map on local rings at the identity is flat.

In fact, by a translation argument we see that it is enough to produce a single closed point x ∈ Spec(B) with image point y ∈ Spec(A) (necessarily closed) so that the induced map between local rings at x and y is flat. But by [8, Thm 24.3] and the denseness of closed points in Spec(B), it is enough to prove this for one single point x ∈ Spec(B), possibly not closed. Since k is algebraically closed, it is clear that the connected components at the identity form closed (and open) subgroups and they map to each other, so we can certainly assume our group schemes are connected. A reduced group scheme is clearly smooth (by a translation argument, together with the denseness of smooth points for reduced schemes of finite type over an algebraically closed field) and thus integral if it is connected. Hence, if A is reduced (e.g., if B is reduced) and therefore integral we can take x to be any point on Spec(B) over the generic point of Spec(A) (such an x exists, since we already noted that the map on spectra is surjective).

There remains the case in which B is possibly non-reduced. By a theorem of Cartier [10, Thm 11.4], this can only happen when k has characteristic p > 0. The intuitive reason that characteristic 0 affine group schemes over fields must be reduced is that group schemes of finite type over a field have a plentiful supply of linearly independent invariant differentials (see [1, Prop 1, §4.2] for a truly natural proof, which simplifies a little in the case of affine group schemes over a field), so after reducing to the finite type setting over an algebraically closed field of characteristic 0, one can dualize and get many linearly independent derivations on the local ring at the identity. In particular, we can use these, together with reciprocated factorials (characteristic 0!) in order to construct ‘Taylor expansions’ and thereby obtain an explicit isomorphism of the completion of the local ring at the identity with a formal power series ring. This yields smoothness of the group scheme (and so it is reduced). Now consider the case of characteristic p. In this latter case, we give a special argument to handle the case in which the schemes might not be reduced.

First, suppose that Spec(A) is a single point; that is, the augmentation ideal I of A is nilpotent. In this case, A is finite over k and the completion B̂_{0} of B at its identity point is a profinite k-algebra, with A → B̂_{0} injective. Since B̂_{0} naturally has the structure of a formal group scheme over k (see §2.3), compatible with the injection from A, our previous results show that B̂_{0} is topologically flat over A, so B̂_{0} is flat over A by Theorem 1.3.10. Since B̂_{0} is faithfully flat over the local ring for Spec(B) at the identity, it follows that this local ring at the identity is flat over A. Hence, we can take x above to be the identity point on Spec(B).

More generally, since the nilradical of A is finitely generated, for suitably large n the ring A′ = {a^{p^{n}}|a ∈ A} is a reduced finite-type k-subalgebra of A (note that k = k^{p^{n}} since k is algebraically closed, so this really is a k-subalgebra). It now follows by the ‘reduced case’ results (valid in any characteristic) that A′ → B is faithfully flat, so it suffices to check that A ⊗_{A′} B → B ⊗_{A′} B is faithfully flat. At the very least, we know this is injective. Consider the scheme-theoretic kernels of the group scheme maps Spec(B) → Spec(A′) and Spec(A) → Spec(A′), represented by k-bialgebras K_{B} and K_{A} respectively. If one considers grouptheoretic arguments analogous to the proof in the formal case that B ˆ⊗_{A}B ≃ B ˆ⊗_{k}(B/𝔟), one arrives naturally at isomorphisms of (left) B-algebras B ⊗_{A} B ≃ B ⊗_{k} K_{B} and B ⊗_{A′} A ≃ B⊗_{k}K_{A}. Thus, it follows that the natural map K_{A}⊗_{k}B → K_{B}⊗_{k}B is injective, so K_{A} → K_{B} is injective. But Spec(K_{A}) is constructed to be a scheme whose underlying space is a single point (i.e., the augmentation ideal is nilpotent, and even killed by raising to the p^{n}th power), so by the case considered above we know that K_{A} → K_{B} is faithfully flat. Applying ⊗_{k}B and using the above isomorphisms, we arrive at the faithful flatness of A ⊗_{A′} B → B ⊗_{A} B, which we saw suffices.

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Theorem 2.1.4 The categories of affine and formal commutative group schemes over a field k are abelian. A map f : H → G is a monomorphism if and only if it is surjective on rings and is an epimorphism if and only if the map on rings is faithfully flat (resp. topologically faithfully flat).

Proof. The essential role of commutativity in our proof is that it allows us to exploit duality between affine commutative group schemes over k and formal ones. Let’s first check the description of monomorphisms and epimorphisms, as this will be needed in the rest of the proof. Since (topologically) faithfully flat maps on rings for the group objects are exactly the injective maps (by Theorem 2.1.3, Corollary 1.3.7, and [8, Thm 7.5(i)]), by duality it suffices to check in both the formal and affine cases that monomorphisms G → H must induce surjections on the rings (as it is obvious that surjections on rings give rise to monomorphisms). We give the argument in the formal case, with the affine case argument being essentially the same. Since the image of 𝒪(H) in 𝒪(G) is a closed k-subalgebra, it is clearly the ring for a formal commutative group scheme G′ over k. Moreover, G → G′ is a monomorphism since G → H is a monomorphism. Thus, we may assume that 𝒪(H) → 𝒪(G) is injective (and thus topologically faithfully flat, by Theorem 2.1.3) and we wish to conclude that this is an isomorphism. Since G → H is a monomorphism of formal commutative group schemes over k, so is G ×_{H} G → G (projection to the second factor). However, this map has an obvious section (the diagonal) and so by Yoneda’s Lemma we see that this is an isomorphism. That is, f : G → H becomes an isomorphism after applying the topologically faithfully flat base extension G → H. Hence, f was an isomorphism to begin with, as desired.

One consequence of our description of epimorphisms and monomorphisms is that if f : G → H is a map of formal or affine commutative group schemes over k such that both f and its dual are monic (i.e., f is monic and epic), then f is an isomorphism. This will be essential shortly.

If f : G → G′ is a morphism in either the formal or affine setting and we let K denote the kernel of f and π : G′ → C denote the cokernel of f, the there is a naturally induced map G/K → ker(π) between ‘image’ and ‘coimage’; the abelian category assertion is equivalent to the claim that this final map is an isomorphism. Note that our construction behaves well with respect to dualization, so it is enough to show that in both the formal and affine cases, the map G/K → ker(π) is monic. Since ker(π) → G′ is monic (it is even a closed subgroup), it is therefore sufficient to show that the natural map f′ : G/K → G′ induced by f is monic. This is what we will now prove. We will use the more suggestive language of functors, though of course everything below could be written in terms of rings. Also, we’ll give the argument in the formal case; the argument in the affine case is essentially the same.

Choose a finite k-algebra R and x ∈ (G/K)(R) = Hom(Sp_{k}(R), G/K) which has trivial image in G′(R). We want to show x = 1. Since G → G/K is topologically faithfully flat (as i : 𝒪(G/K) → 𝒪(G) is injective, by the construction of G/K as a cokernel), the map of formal k-schemes π_{1} : Sp_{k}(R)×_{G/K} G → Sp_{k}(R) is topologically faithfully flat (Corollary 1.3.8). We will show that x∘π_{1} = 1 in (G/K)(Sp_{k}(R)×_{G/K}G), for then xˆ⊗i : 𝒪(G/K) → Rˆ⊗_{𝒪(G/K)} 𝒪(G) kills the augmentation ideal, yet this map is the composite of x with the topologically faithfully flat (and therefore injective!) map R → Rˆ⊗_{𝒪(G/K)} 𝒪(G), so we could conclude that x kills the augmentation ideal, as desired.

In order to compute x ∘ π_{1}, we note that it is equal to q ∘ π_{2}, where q : G → G/K is the canonical map. Our given condition is that f′ ∘ x ∘ π_{1} is the trivial map of group functors, yet this is precisely f ∘ q ∘ π_{2} = f ∘ π_{2}. By definition of K = ker(f), we see that π_{2} factors through K ↪ G, so q ∘ π_{2} is the trivial map, as desired.

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The study of commutative affine and formal group schemes over fields has some additional special features when the base field has positive characteristic p, particularly in the analysis of multiplication by p. First, we state the fundamental result for connected formal commutative group schemes over fields:

Theorem 2.1.5 Let k be a field and G a commutative connected formal group scheme over k. If k has characteristic 0, then G = Sp_{k}(k[[X_{j}]]) as a formal k-scheme. If k has characteristic p > 0, then there is a set J and a function h : J → Z^{+} ∪ {∞} and a formal k-scheme isomorphism G ≃ Sp_{k}(k[[X_{j}]]/I_{h}), with I_{h} the closed ideal generated by the monomials X_{j}^{p^{h(j)}} for all j ∈ J with h(j) ≠ ∞.

Proof. The full details are lengthy to write out, so see [3, §9, Ch I]. The rough idea is that if G ≃ Sp_{k}(A), with 𝔪 the maximal ideal of A, first choose a pseudobasis for the topologically free profinite A/𝔪-module 𝔪/\overline{𝔪^{2}}. This then gives rise (non-canonically) to a surjection k[[X_{j}]] → A whose kernel must be determined. A study of invariant differentials on formal commutative group schemes shows that the profinite A-module Ω̂_{A/k}^{1} is topologically free with {dX_{j}} as a pseudobasis (essentially since the X_{j}’s give a topological basis for the ‘formal cotangent space’ 𝔪/\overline{𝔪^{2}} over k). From this one concludes that the kernel is trivial if k has characteristic 0, since df = 0 implies f = 0 for f ∈ k[[X_{j}]]. Meanwhile, if k has characteristic p, all elements f of the kernel have df = 0 and so the desired description of the kernel looks plausible. To make this intuition precise, one reduces to the case in which 𝔪^{p^{r}} = 0 for some r ≥ 0 and then one uses induction on r. At one critical step an auxiliary formal group scheme is constructed and it is necessary to know the topological (faithful) flatness result Theorem 2.1.3 (and the fact that topological flatness amounts to topological freeness over local pseudocompact rings). This is the essential input from the theory of formal commutative group schemes over fields. The rest is tedious algebra.

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When k has characteristic 0, one can be even more precise: G is isomorphic to a product of copies of the formal affine line. That is, we can choose the X_{j}’s so that m(X_{j}) = 1ˆ⊗X_{j} + X_{j} ˆ⊗1, ϵ(X_{j}) = 0, and i(X_{j}) = −X_{j} for all j ∈ J. This requires somewhat more specialized methods which we won’t enter into here; see [3, §10, Ch I] for the full details. In characteristic p > 0, a special case of the above is a classification of finite commuative connected group schemes over k: they must have the form Spec(k[x_{1}, …, x_{n}]/(x_{i}^{p^{m_{i}}})). This result is actually true for non-commutative finite connected group schemes over k also, but we won’t need this.

Since the map x ↦ x^{p} on F_{p}-algebras commutes with all rings maps, we can make the following useful definitions:

Definition 2.1.6 If G is any affine or formal commutative group scheme over a field k with characteristic p > 0, let G^{∗} denote the corresponding ‘dual’ commutative group scheme (formal or affine). Let G^{f} denote the base extension of G by the absolute frobenius morphism f : k ↪ k (so clearly there is a natural isomorphism (G^{∗})^{f} ≃ (G^{f})^{∗}). Let F_{G} : G → G^{f} denote the map corresponding to x ↦ x^{p} on the ring level. This is clearly a map of group objects. The induced map G^{f} → G arising from F_{G^{∗}} via duality is denoted V_{G} (for ‘Verschiebung’).

Via duality, it is clear that F_{G} and V_{G} are suitably functorial in G. Though V_{G} may seem somewhat mysterious, it is possible to give a somewhat explicit formula for it when k is perfect. Also, it turns out that F_{G} and V_{G} encode a lot of information about G. For example, the maps V_{G} ∘ F_{G} and F_{G} ∘ V_{G} are ‘canonical’ group object endomorphisms of G and G^{f} respectively. What are they? This is answered by:

Theorem 2.1.7 Let G be a formal or affine commutative group scheme over a field k with characteristic p > 0. Then V_{G} ∘ F_{G} and F_{G} ∘ V_{G} are multiplication by p on G and G^{f} respectively.

Proof. Clearly we can base extend to an algebraically closed field, so we may assume that k is perfect. We’ll give the proof in the formal case; the affine case is then either settled by duality or else can be proven directly by essentially the same method. The idea is to find an explicit formula for V_{G} for a formal k-bialgebra A with G ≃ Sp_{k}(A) and then to observe from the formula that the theorem is true (here, we abuse notation and write V_{G} for the corresponding ring map as well). Notice that because G is commutative, the p-fold comultiplication map A → A^{ˆ⊗p} has its image inside of the subset consisting of symmetric tensors (i.e., those invariant under S_{p}). For this reason, we first need to prove a lemma about the structure of symmetric tensors.

Let W be a vector space over k and let T S^{j}(W) denote the space of symmetric j-fold tensors; that is, the fixed space of S_{j} acting on W^{⊗j}, with T S^{0}(W) = k. Define the map Σ_{j} : W^{⊗j} → T S^{j}(W) via

w_{1} ⊗ ⋯ ⊗ w_{j} ↦ ∑_{σ∈S_{j}} w_{σ(1)} ⊗ ⋯ ⊗ w_{σ(j)}

(with Σ_{0} : k → k the identity map). This easily satisfies

Σ_{j}((W_{1} ⊕ W_{2})^{⊗j}) ≃ ⨁_{i=0}^{j} (Σ_{i}(W_{1}^{⊗i}) ⊗ Σ_{j−i}(W_{2}^{⊗(j−i)}))

compatibly with

T S^{j}(W_{1} ⊕ W_{2}) ≃ ⨁_{i=0}^{j}(T S^{i}(W_{1}) ⊗ T S^{j−i}(W_{2})).

But for 0 ≤ j ≤ p − 1 it is obvious that Σ_{j}(W^{⊗j}) = T S^{j}(W), so we obtain

(T S^{p}(W_{1})/Σ_{p}(W_{1}^{⊗p})) ⊕ (T S^{p}(W_{2})/Σ_{p}(W_{2}^{⊗p})) ≃ T S^{p}(W_{1} ⊕ W_{2})/Σ_{p} ((W_{1} ⊕ W_{2})^{⊗p}) .

From this, it follows easily by induction on dimension (with the 0 and 1 dimensional cases being easy) that if dim_{k}(W) is finite and W^{(p)} denotes W ⊗_{k} k (using f : k → k), then the k-linear map

W^{(p)} → T S^{p}(W)/Σ_{p}(W^{⊗p})

given by

a ⊗ λ ↦ λ(a ⊗ ⋯ ⊗ a)

is an isomorphism. Since k is perfect, then we can remove the ambiguity of λ and so can conclude that every symmetric tensor can be uniquely written in the form a^{⊗p} + t, where a ∈ W and t ∈ Σ_{p}(W^{⊗p}).

Passing to the direct limit gives the same assertion without restriction on the dimension of W and passing to the inverse limit (with some care) gives the same assertion for profinite k-modules W (using ˆ⊗ in place of ⊗).

Now consider the map p : G → G, which translates into the composite ring map [p]^{∗} : A → A^{ˆ⊗p} → A, with the first map the p-fold comultiplication m_{p} and the second map induced by the profinite k-algebra structure on A. Thus, if we write

m_{p}(a) = V (a)^{ˆ⊗p} + t,

with t a symmetric tensor and V (a) ∈ A, then the association a ↦ V (a) is f^{−1}-semilinear and clearly [p]^{∗}(a) = V (a)^{p} (we used the fact that A has characteristic p in order to see that t does not contribute here). Via the canonical semi-linear k-algebra identification ι_{A} of A with its base extension A^{f} by the Frobenius automorphism of k, it makes sense to ask if V_{G}(a) = ι_{A}(V (a)). If this were true, then we would be able to conclude that the map V_{G} ∘ F_{G} : G → G is what we expect. Also, since the pth power map commutes with all maps between rings of characteristic p, then it would be immediate that F_{G} ∘ V_{G} is also what we expect (one just has to keep track of k-linearity issues).

So it remains to prove that if

⟨, ⟩ : A × A^{∗} → k

is the canonical pairing and ⟨, ⟩_{f} is the analogous pairing for A^{f} (via the identification (A^{f})^{∗} ≃ (A^{∗})^{f}), then ⟨ι_{A}(V (a)), ι_{A^{∗}}(a′)⟩_{f} = ⟨V_{G}(a), ι_{A^{∗}}(a′)⟩_{f} for all a′ ∈ A^{∗}. By definition of V_{G} and the ring structure on A^{∗},

⟨V_{G}(a), ι_{A^{∗}}(a′)⟩_{f} = ⟨a, F_{G^{∗}}(ι_{A^{∗}}(a′))⟩ = ⟨m_{p}(a), a′^{⊗p}⟩,

where the last pairing is really the one between A^{ˆ⊗p} and (A^{∗})^{⊗p} ≃ (A^{⊗p})^{∗}. Since any symmetric tensor in A^{ˆ⊗p} pairs to 0 with a′^{⊗p} (as k has characteristic p), this pairing is equal to

⟨V (a)^{ˆ⊗p}, a′^{⊗p}⟩ = (⟨V (a), a′⟩)^{p} = ⟨ι_{A}(V (a)), ι_{A^{∗}}(a′)⟩_{f},

as desired.

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As one example, note that if we consider the formal affine line Ĝ_{a} over a perfect field of characteristic p, m_{p}(X) = Σ_{p}((1/(p − 1)!)X ˆ⊗1ˆ⊗ … ˆ⊗1), so V (X) = 0. Clearly by base extension it follows that the result VĜ_{a} = 0 holds even without the perfectness assumption. This is certainly what we expect, since the multiplication by p map on the formal affine line in characteristic p is the zero map.

We conclude our discussion over fields with an observation that will come up in the study of p-divisible groups.

Theorem 2.1.8 Let G be a connected formal commuative group scheme over a field k with characteristic p. Then the Ab-valued covariant functor \underrightarrow{lim} G[p^{n}] on P F_{k} is a formal kscheme and its natural map to G is an isomorphism.

Proof. Note that the maps G[p^{n}] → G are monomorphisms and so the ring maps are surjective. Thus, the ring maps for G[p^{n}] → G[p^{m}] (for m ≥ n) are surjective. By Lemma 1.4.4(3), the Ab-valued covariant functor \underrightarrow{lim} G[p^{n}] on P F_{k} is a priori a formal k-scheme and thus is a formal commutative group scheme over k which is a subgroup of G. It remains to check that for any finite k-algebra R, any x ∈ G(R) is killed by some power of p. For this, it is certainly enough to base extend to an algebraic closure of k, so we may assume that k is perfect. Clearly G(R_{1} × R_{2}) is the disjoint union of G(R_{1}) and G(R_{2}), so we can assume that R is local. Also, the connectedness of G implies that the augmentation ideal of 𝒪(G) consists of topologically nilpotent elements, so the map x : 𝒪(G) → R sends the augmentation ideal into the nilpotent maximal ideal of R. Hence, for sufficiently large n, the semilinear k-algebra map x^{p^{n}} kills the augmentation ideal of G. Combining this with Theorem 2.1.7, which shows that p^{n} on G can be factored into a product of n Frobnieus maps and n Verscheibung maps, we’re done (i.e., x ∈ G(R)[p^{n}]).

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§2.2 Finite Flat Group Schemes

Definition 2.2.1 Let S be a scheme. A group scheme over S is a group object G in the category of S-schemes. These form a category gp_{S}. We say that H → G is a closed subgroup if the map is a closed immersion. A finite flat group scheme over S is a commutative (!) group object G in the category of S-schemes such that π : G → S is finite, flat, and locally of finite presentation (i.e., π is affine and π_{∗}(𝒪_{G}) is a locally free sheaf of 𝒪_{S}-modules of finite rank). The order of such a G is the locally constant function on S whose value at s ∈ S is the (finite) dimension over k(s) of the ring of G ×_{S} k(s). We let F F_{S} denote the category of finite flat group schemes over S.

Standard examples of objects in gp_{S} which are not in F F_{S} are G_{m}_{/S} and, more generally, GL(n)_{/S}. We could of course have omitted the commutativity condition in the definition of F F_{S}, but the notion is not at all useful for us without commutativity.

Definition 2.2.2 Let A be a pseudocompact ring. A formal group scheme over A is a (possibly non-commutative) group object G in the category of formal A-schemes which is also topologically flat over A.

It is clear that all of the above notions behave well with respect to the appropriate notion of base change and that products exist. When S = Spec(A) with A a local pseudocompact ring, we can view objects in F F_{A} as formal group schemes over A also. More generally, finite flat A-algebras are just (Zariski-)locally free and so usually can’t be expected to have a profinite A-module structure. Hence, we can view formal commutative group schemes G over A for which 𝒪(G) is a finite A-module as objects in F F_{A}, but not conversely (in general). The connection with formal groups will be discussed in §2.3.

For G in gp_{S} and S′ → S an S-scheme, we let G_{S′} or G_{/S′} denote the base extension. The existence of kernels in F F_{S} and in the category of formal groups is more subtle, due to the flatness conditions; in general, the scheme-theoretic kernels will not be flat. However, since G in gp_{S} is separated over S if and only if the identity section is a closed immerion [1, Lemma 2, §7.1], we see that G → H in gp_{S} has a scheme-theoretic kernel which is a closed subscheme of G provided that H is separated over S. It is the failure of this kernel to be flat in general that makes the relative theory technically complicated (since over fields, flatness properties are automatically satisfied).

We know from Theorem 2.1.3 and Theorem 2.1.4 that when S = Spec(k) with k a field, F F_{S} is an abelian category and the monomorphisms correspond to ring surjections and the epimorphisms correspond to ring injections, always necessarily faithfully flat. Moreover, the duality in §1.3 gives rise to an exact contravariant Cartier duality functor on F F_{k}. Our first task is to relativize this.

Theorem 2.2.3 Let G be in F F_{S}. The functor from S-schemes to Ab given by

T ⇝ Hom_{gp_{T}} (G_{T} , G_{m}_{/T})

is represented by an object G^{∗}, called the Cartier dual of G. Explicitly,

G^{∗} = Spec(ℋom_{𝒪_{S}} (π_{∗}(𝒪_{G}), 𝒪_{S})),

with π : G → S the structure morphism. The functor G ⇝ G^{∗} is an order-preserving antiequivalence of the category F F_{S} with itself and there is a natural isomorphism G ≃ (G^{∗})^{∗}.

Proof. The sheafified bialgebra structure on π_{∗}(𝒪_{G}) can be dualized, as this sheaf is locally free of finite rank. It is clear that this is functorial in G. We omit the details, as they are routine.

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Of course, the intuitive example to keep in mind is when S = Spec(C), so G is the constant group scheme attached to a finite abelian group A. Then G^{∗} is the constant group scheme attached to Hom_{Ab}(A, C^{∗}), the classical ‘dual’ group of A. It is obvious that Cartier duality commutes with base change in a natural manner.

We will now use Cartier duality in order to construct quotients by closed subgroups in F F_{S}.

Theorem 2.2.4 Let f : H → G be a closed subgroup in F F_{S}. Then G/H = coker(f) exists in the category F F_{S} and H → G is the kernel of G → G/H in gp_{S}. The quotient G/H is of formation compatible with base change and (G/H)^{∗} → G^{∗} is the kernel of G^{∗} → H^{∗} in gp_{S}. The map G^{∗}/((G/H)^{∗}) → H^{∗} is an isomorphism and the map G → G/H is faithfully flat.

Proof. We wish to define the group scheme (G/H)^{∗} to be the kernel of G^{∗} → H^{∗} in gp_{S}. If we check that G^{∗} → H^{∗} is faithfully flat, then the desired definition does yield an object in F F_{S} and we can then define G/H to be its Cartier dual. Since scheme-theoretic kernels commute with arbitrary base change, the finite presentation condition lets use reduce to the case in which S is affine and noetherian. Thus, we can use the fiber-by-fiber criteria for flatness and our results over fields to see that G^{∗} → H^{∗} is indeed faithfully flat.

Everything else can also now be checked by passing to geometric fibers. Over a field, everything has been checked in §2.1.

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Corollary 2.2.5 The order functions satisfy order(G) = order(H)order(G/H) as locally constant functions on S.

Proof. This can be checked on geometric fibers, so we suppose S is an algebraically closed (and hence perfect) field. The theory of the connected-étale sequence (see Definition 2.2.9ff below), together with the obvious fact that orders multiply when forming product groups, reduces us to the case in which G and H are either both étale or both connected. The étale case is trivial (classical group theory) and the connected case follows from the fact that G → G/H is faithfully flat and that a finite flat module over a local ring (such as 𝒪(G) over 𝒪(G/H) when G is connected) is free, with rank unaffected by any base change (such as the identity section k → 𝒪(G/H)).

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Observe that the above theorem also shows implicitly that Cartier duality behaves well with respect to the sequence H ↪ G → G/H.

Theorem 2.2.6 Let f : G → G′ be a map in F F_{S}. Assume that H = ker(f) is in F F_{S}. Then the natural map G/H → G′ is a closed immersion and is an isomorphism if and only if order(G) = order(H)order(G′). In general, there exists a closed subgroup H ↪ G with H → G → G′ vanishing and G/H → G an isomorphism if and only if G → G′ is faithfully flat. This is true if and only if G′^{∗} → G^{∗} is a closed immersion.

Proof. Since the property of being a closed immersion, faithfully flat, or an isomorphism in our setting can be checked on fibers, everything is clear in view of our previous results, once we note that a closed subgroup G_{1} ↪ G_{2} in F F_{k} is the full group if and only if G_{1} and G_{2} have the same order.

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We now connect up our ad hoc construction of quotients with the more high-brow point of view (this will never be used).

Corollary 2.2.7 For an fppf cover X → S, let X̃ denote the fppf sheaf Hom_{S}(·, X). For a closed subgroup H ↪ G in F F_{S}, the sequence of fppf sheaves

0 → H̃ → G̃ → { \atop G/H} → 0

is exact. The same is true on the fpqc site over S.

Proof. The only point to check is that G̃ → { \atop G/H} is fppf (or fpqc) surjective. The main reason for this is that G → G/H is faithfully flat (and certainly quasi-compact and locally of finite presentation too). Thus, G → G/H is an fppf (and fpqc) cover for which the base extension G×_{G/H} G → G (by projection to second factor) has a section. Thus, for T → G/H in either site over S and T′ = T ×_{G/H} G a cover of T, the map G̃(T′) → { \atop G/H}(T) is surjective.

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Although F F_{S} is generally not an abelian category, we can still work with short exact sequences, provided we exercise some care.

Definition 2.2.8 We say that a diagram

0 → G′ → G → G′′ → 0

in F F_{S} is a short exact sequence if G′ → G is a closed subgroup which dies in G′′ and the induced map G/G′ → G′′ is an isomorphism (so G′ is the precise kernel of G → G′′).

By using the fiber-by-fiber criteria for (faithful) flatness and Nakayama’s Lemma, it is not hard to check that a diagram G′ → G → G′′ in F F_{S} yields a short exact sequence if and only if the composite map G′ → G′′ vanishes and we get a short exact sequence on geometric fibers. Also, clearly the property of being a short exact sequence is compatible with base change. A check on geometric fibers shows that in a short exact sequence such as the above, G′ = 0 if and only if G ≃ G′′ and G′′ = 0 if and only if G′ ≃ G.

The fundamental short exact sequence is the connected-étale sequence, which we will now construct. This construction works over any henselian local base ring, so we operate in such generality for conceptual clarity. This will make essential use of our previous results concerning maximal étale subalgebras.

Let A be a henselian local ring, G = Spec(R) a finite flat group scheme over A. Let G^{ét} = Spec(R^{ét}), with R^{ét} the maximal étale subalgebra of R over A. Since we know by Theorem 1.5.3 that R^{ét} ⊗_{A} R^{ét} is the maximal étale subalgebra of R ⊗_{A} R, it follows from Corollary 1.5.5 that the group law morphisms on G uniquely induce group law morphisms on G^{ét} such that G → G^{ét} is a map of finite flat group schemes over A. Since G → G^{ét} is faithfully flat (by Theorem 1.5.2), we can form the kernel G^{0} in gp_{A} and this is again in F F_{A}.

Definition 2.2.9 We call

0 → G^{0} → G → G^{ét} → 0

the connected-étale sequence of G. G^{ét} is called the maximal étale quotient of G and G^{0} is called the connected component of G.

It is clear that the connected-étale sequence of G in F F_{A} really is a short exact sequence. Moreover, by Theorem 1.5.4 we see that it is compatible with any local base change to another henselian local ring (e.g., passage to the closed fiber). Of course, since a map such as Z_{p} → Q_{p} is not local, base changing from Z_{p} to Q_{p} would not behave well with respect to connected-étale sequences. This is to be expected, since base change preserves the property of being a short exact sequence, yet all objects in F F_{Q_{p}} are étale (of course, what goes wrong is that base change to Q_{p} does not preserve the property of being connected, due to the fact that Z_{p} → Q_{p} is not local). By Theorem 1.5.3, it is also clear that formation of G^{ét} (and thus G^{0}) is compatible with taking products in F F_{A}.

Before we study the connected-étale sequence more closely, we note that we really have used nothing about commutativity, except to justify the existence of quotients. Thus, if we had a more general theory of quotients of (possibly non-commutative) finite flat group schemes over A by finite flat ‘normal’ closed subgroups, we could actually study a connectedétale sequence in that situation. We won’t require this, though, and so do not pursue this point.

When A = k is a perfect field, the connected-ètale sequence splits canonically (in the noncommutative case, we would get a semi-direct product). We should remark that this sort of behavior is generally not true if A is merely a henselian local ring with a perfect residue field. For example, elliptic curves over Z_{p} with ordinary reduction give rise to examples of non-split extensions of étale objects in F F_{Z_{p}} of order p by connected objects of order p. The precise statement concerning splitting is:

Theorem 2.2.10 Let k be a perfect field and G a finite flat group scheme over k, possibly noncommutative. The connected-étale sequence of G canonically splits, in a manner compatible with formation of products of group schemes and base change k → k′ with k′ a perfect field.

Proof. Let R denote the affine k-algebra of G and 𝔯 the nilradical of R, so the canonical map R^{ét} → R → R/𝔯 is an isomorphism. Indeed, to check this we can reduce to the case in which R is local and then it is just the statement that the residue field of R is a finite extension of k which is separable. But this holds because we assumed k is perfect. Thus, we get a canonical map R → R^{ét} which corresponds to a section to G → G^{ét}. This gives the canonical (semi-direct product) splitting, via a group theory and Yoneda Lemma argument.

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The above theorem is the main technical reason for reducing to the study of an algebraically closed (and hence perfect) base field when one considers finite flat group schemes.

In order to justify the terminology ‘maximal étale quotient’ and ‘connected component’, we prove the following lemma:

Lemma 2.2.11 For any G in F F_{A}, G^{0} is connected. Also, if G in F F_{A} is connected and G′ in F F_{A} is étale, then there are no non-trivial maps from G to G′ in F F_{A}. In particular, H in F F_{A} is connected if and only if H^{ét} = 0 and is étale if and only if H^{0} = 0, so if H → G is a morphism in F F_{A}, then this map factors through G^{0} ↪ G if H is connected and factors through H → H^{ét} if G is étale.

Proof. An object in F F_{A} is connected if and only if it is so over the closed fiber, so we can assume for the first part that A = k is a field. Since a finite k-algebra R with a section to the structure map k → R is local if and only if its geometric fiber is local, we can even assume for simplicity that k is an algebraically closed field. Thus, 𝒪(G) is the product of finitely many local k-algebras with residue field k, thereby giving canonical idempotents {e_{i}} in the ring, so 𝒪(G)^{ét} = ⊕ke_{i}. All but one of these idempotents is killed by the identity section of G, and so it is clear that 𝒪(G^{0}) is exactly the local component of 𝒪(G) corresponding to the idempotent not killed by the identity section of G.

Now consider connected G = Spec(R) in F F_{A} and étale G′ = Spec(R′) in F F_{A} (with A no longer assumed to be a field). We wish to show that Hom_{gp_{A}}(G, G′) = 0. Since G has an identity section and is local, we see that the residue field of the closed fiber is just the residue field of A. Thus, R has maximal étale subalgebra A, which is complemented by the augmentation ideal of R. We also can write R′ = A ⊕ I, where I is the direct sum of the other factor rings of the finite étale cover R′ over A. By Corollary 1.5.5, any map from R′ to R over A must have image inside of A ↪ R. This implies that the augmentation ideal of R′ is killed (as its image lies inside the augmentation ideal of R), so G → G′ is the trivial map of group schemes over A.

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The above lemma enables us to check certain functoriality properties of the connected-étale sequence.

Theorem 2.2.12 Let G′ → G be a morphism in F F_{A}, for a henselian local ring A. There are unique maps between connected components and maximal étale quotients such that the diagram

0 → G′^{0} → G′ → G′^{ét} → 0
↓ ↓ ↓
0 → G^{0} → G → G^{ét} → 0

commutes. Moreover, if

0 → G′ → G → G′′ → 0

is a short exact sequence in F F_{A}, then the resulting 3 by 3 commutative diagram has all rows and columns exact.

Proof. The first part is clear. Because the connected-étale sequence is compatible with local base change of henselian local rings, the second part can be checked on geometric fibers, over which the connected-étale sequence canonically splits and so the result is obvious (since finite flat group schemes over a field form an abelian category).

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§2.3 Formal Group Schemes

Before delving into any grand theory, let’s first see how formal group schemes naturally arise in the study of (possibly non-commutative) separated group schemes G which are flat and of finite type over a noetherian local base S = Spec(A). This is the main case of interest in applications, as it’s usually easy to reduce oneself to cases in which the base in local and noetherian. Consider the category ℱ_{A} of artinian A-algebras with finite A-length (i.e., finite algebras over some A/𝔪^{n}). This is naturally isomorphic to the category of artinian profinite algebras over the (pseudocompact) completion Â of A.

There is a covariant functor Ĝ : ℱ_{A} → Gp which associates to each R the subgroup (!) in G(R) consisting of those maps Spec(R) → G which have the entire image at the closed point on the identity section subscheme (recall that G is separated over A, so the identity section is a closed immerion). By Theorem 1.4.7, this is a formal scheme over Â! Of course, it is straightfoward to check ‘by hand’ that this is nothing other than the formal completion of G at the closed point on the identity section (it is not generally the formal completion of G along the entire identity section, unless of course A is artinian). In particular, we see that this is profinite over A and that the ring of Ĝ is faithfully flat over A and thus is faithfully flat over Â (here we use flatness properties of completions of noetherian local rings, particularly [8, Thm 22.4]). By Theorem 1.3.10 and the fact that topologically flat injections between psuedocompact rings are topologically faithfully flat, we see that Ĝ is topologically faithfully flat over Â. Thus, Ĝ is a formal group scheme over Â (recall we require topological flatness in the definition).

If G were separated and of finite type over a field, then Ĝ is just the completion of the local ring at the origin point and the result is the usual ‘formal group’ of the group scheme, which one can explicitly ‘see’ does represent the functor Ĝ (and it is not too hard to see that the group law on G does extend to the completion of this local ring, even though it does not extend to any of its artinian quotients; the idea is to see that comultiplication fails to be well-defined on artinian quotients only up to an amount that becomes negligible when we pass to the inverse limit).

From the definition, it is clear that Ĝ is compatible with local base change in an appropriate manner (i.e., using completed tensor products). Of course, a formal group scheme Ĝ constructed as above is always connected. Nevertheless, there certainly are many formal group schemes that are étale and not connected. The extent to which formal group schemes fail to be connected is measured by a formal analogue of the connected-étale sequence. Let’s give the precise formulation of the connected-étale sequence in the context of formal group schemes.

Theorem 2.3.1 Let A be a pseudocompact local ring, G = Sp_{A}(R) a topologically flat formal group scheme over A. Let R^{ét} be the maximal formally étale subalgebra for R over A. Then G^{ét} = Sp_{A}(R^{ét}) has a unique structure of formal group scheme over A such that G → G^{ét} is a map of formal group schemes. This map is topologically faithfully flat and the kernel G^{0} = Sp_{A}(R^{0}) is a connected formal group scheme over A (i.e., R^{0} is local). The construction of G^{ét} and G^{0} is compatible with taking products and with local base change A → A′ with A′ another pseudocompact local ring.

Proof. The proof is just like that given in the case of finite flat group schemes, except we use the analogous results we proved in the case of pseudocompact rings. We omit the details.

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Note that if A = k is a perfect field, then G^{ét} can also be defined as the functor R ⇝ G(R/𝔯_{R}), with 𝔯_{R} the radical of the artin ring R. Of course, if G is a finite flat group scheme over a local pseudocompact (and thus henselian) ring A, then the above notions coincide with those discussed in §2.2. Before pursuing the connected-étale sequence much further, we should first properly formulate the notion of a quotient by a closed subgroup and a short exact sequence in the category of formal group schemes over A. Thus, we now will work with just commutative formal group schemes over A (as usual, one can get results in the non-commutative case by imposing normality hypotheses, but we won’t need such results and therefore stick with commutativity conditions).

Over a field, we have seen that commutative and affine formal group schemes form an abelian category. In order to extend this to bases other than fields, the essential technical fact we need is a fiber-by-fiber criteria for (topological) flatness.

Theorem 2.3.2 Let A be a pseudocompact ring and B_{1} → B_{2} a map between topologically flat profinite A-algebras. This map is topologically flat if and only if it is topologically flat on the formal fiber over all open maximals of A (i.e., after applying ˆ⊗_{A}(A/𝔪) for open maximals 𝔪 of A).

If A is an artin local ring, then all flat A-algebras are free as A-modules and a map between flat A-algebras is flat if and only if it is flat on the closed fiber. The same holds for faithful flatness.

Proof. First consider the formal setting. We obviously reduce to the case in which all the rings and maps are local (there may be some cases that arise involving maps to the zero ring, but these can be safely ignored since such maps are trivially flat). Since ‘only if’ is clear, we can assume that B_{1} ˆ⊗_{A}(A/𝔪_{A}) → B_{2} ˆ⊗_{A}(A/𝔪_{A}) is topologically flat and we wish to conclude that B_{1} → B_{2} is topologically flat. By Theorem 1.3.6, there is an isomorphism of B_{1} ˆ⊗_{A}(A/𝔪_{A})-modules

∏(B_{1} ˆ⊗_{A}(A/𝔪_{A}))e_{i} ≃ B_{2} ˆ⊗_{A}(A/𝔪_{A}).

By the projectivity of ∏ B_{1}e_{i} in P F_{B_{1}} and the formal Nakayama’s Lemma, this lifts to a surjection π : ∏ B_{1}e_{i} ↠ B_{2}. Since B_{2} is topologically flat and thus projective over the local pseudocompact ring A, π splits in P F_{A} and so the kernel has trivial ‘closed fiber’. By the formal Nakayama’s Lemma, the kernel is 0, so we have proven that B_{2} is topologically free over B_{1}.

When A is an artin local ring, then by [8, Thm 7.10] all flat A-modules are free. Since 𝔪_{A} is nilpotent, it is easy to apply an analogue of the above argument to get the desired result in the flat case. Passing to the closed fiber over A has no effect on the spaces of the spectra (since 𝔪_{A} is nilpotent), so faithful flatness on the closed fiber implies faithful flatness for the map over A.

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We can now construct a quotient for a closed formal subgroup H → G of a formal commutative group scheme over a (possibly non-local) pseudocompact ring A (here, recall that H and G are both required to be topologically flat over A). Instead of giving all of the details and side results, we will content ourselves with sketching the method of the construction and stating a few of its properties. The details that are then required amount to nothing more than a modification of the methods used in the discussion of finite flat group schemes. The essential point is that we have the above fiber-by-fiber criteria for topological flatness to allow ourselves to reduce proofs to the case in which the base is a field (where everything is an abelian category and so is easy). It should be emphasized that our methods are somewhat ad hoc, but they enable us to avoid a huge digression into a theory of general quotients of formal schemes by equivalence relations. This more sophisticated point of view is needed for a correct treatment of the non-commutative case and is the point of view in SGA3.

We will explicitly construct G/H when A is an artin local ring. For A any local pseudocompact ring, we then will define 𝒪(G/H) = \underleftarrow{lim} 𝒪((G ×_{A} (A/𝔞))/(H ×_{A} (A/𝔞))) and if A is any pseudocompact ring we will define (in obvious notation) 𝒪(G/H) = ∏ 𝒪((G_{𝔪}/H_{𝔪})), with the product taken over all open maximals of A. As for the construction in the case where A is an artin local ring, we can use the duality from §1.3. More precisely, since 𝒪(H^{∗}) → 𝒪(G^{∗}) is an injection of flat A-modules which stays injective on the closed fiber (check using the comments follows Corollary 1.3.8), by Theorem 2.3.2 and Theorem 2.1.3 we see that this map is faithfully flat. Thus, if we base change by the identity section we get a faithfully flat affine commutative group scheme (G/H)^{∗} over A. By duality, this gives rise to a topologically flat commutative formal group scheme G/H over A. It is straightfoward to check (using closed fibers!) that G → G/H is topologically faithfully flat and has formal scheme-theortic kernel H. Moreover, this serves as a cokernel to H → G and the construction of G/H is compatible with arbitrary base change A → A′ of pseudocompact rings. Again looking on closed fibers along A, it is easy to see that if H ↪ G is a closed (topologically flat!) subgroup and G → G′ has kernel H, then the induced map G/H → G′ is a closed subgroup.

Definition 2.3.3 We say that a diagram of formal commutative group schemes over A

0 → G′ → G → G′′ → 0

is a short exact sequence if G′ → G is a closed subgroup which dies in G′′ and the natural map G/G′ → G′′ is an isomorphism (so G′ is the precise kernel of G → G′′).

It is clear that the notion of a short exact sequence is compatible with taking products and base change and that the connected-étale sequence is a short exact sequence. Moreover, the analogue of Theorem 2.2.10 is easily seen to hold in our present setting, so it is now clear that the analogues of Lemma 2.2.11 and Theorem 2.2.12 hold for formal commutative groups as well. Of course, it is obvious that these notions agree with the ones previously defined on the full subcategory of commutative formal group schemes over A which lie in F F_{A} (which coincides with all of F F_{A} if A is local).

The following result will be useful in the study of p-divisible groups:

Theorem 2.3.4 If G′, G, and G′′ are formal commutative group schemes over a pseudocompact ring A, with G′ and G′′ in F F_{A}, then if there exists a short exact sequence

0 → G′ → G → G′′ → 0,

G is also in F F_{A}.

Proof. Since there is an upper bound on the orders of G′ and G′′ along the local factors of A, it suffices to work locally and to show that G is in F F_{A} with order equal to the product of the orders of G′ and G′′. However, once A is local we can check everything on the closed fiber, and so are reduced to the case in which A = k is a field, even algebraically closed. The splitting of the connected-étale sequences then reduces us to the case in which all three formal group schemes are formally étale or else all three are connected. The formally étale case is trivial, and in the connected case we know from the topological faithful flatness of the (necessarily local) map 𝒪(G′′) → 𝒪(G) that 𝒪(G) is topologically free over 𝒪(G′′) with a pseudobasis of cardinality equal to that for 𝒪(G′) over k. As the latter is finite, and 𝒪(G′′) is finite over k, we’re done.

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One final application of the connected-étale sequence is that it can detect smoothness through the connected component. More precisely, we have:

Theorem 2.3.5 Let A be a local pseudocompact ring with maximal ideal 𝔪 and residue field k, and G a formal commutative group scheme over A. Then G is formally smooth over A if and only if the closed fiber of G^{0} is formally smooth over the residue field of k. That is, G is formally smooth over A if and only if there is an isomorphism G^{0} ×_{A} k ≃ Sp_{k}(k[[X_{i}]]) of formal k-schemes. In this case, G^{0} ≃ Sp_{A}(A[[X_{i}]]) as formal A-schemes.

Proof. By Theorem 1.5.20, we may assume that A = k is an algebraically closed field, in which case the connected-étale sequence splits. Since formally étale k-schemes are always formally smooth over k, it is necessary and sufficient to check the other factor, as claimed. By Theorem 1.5.15, we’re done.

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Definition 2.3.6 For G a formally smooth commutative group scheme over a local pseudocompact ring A, with G^{0} given by a power series ring over A in finitely many variables, we say that G is finite-dimensional, with dimension equal to the number of variables (this is just the rank of the finite free A-module Ω̂_{G^{0}/A}^{1}). Otherwise, we say that G is infinite-dimensional.

We close by noting that if A is a local pseudocompact ring and B is a formally smooth local profinite A-algebra, then from the non-canonical isomorphism B ≃ B^{ét}[[X_{i}]] we see that for any profinite artinian A-algebra R, the canonical map Hom(B, R) → Hom(B^{ét}, R) is surjective. This then easily extends to the case in which B is not local, so if G is a formal commutative group scheme over A which is formally smooth over A, then for all profinite artinian A-algebras R, the sequence of groups

0 → G^{0}(R) → G(R) → G^{ét}(R) → 0

is exact.

§3 p-divisible Groups

We are now in a position to define p-divisible groups and to prove some of their central properties. We also will briefly study isogenies between p-divisible groups, as well as duality. There are actually two different notions one can consider, so for conceptual clarity we call them by different names. The notion of a p-Barsotti-Tate group will make sense over any base scheme at all and will be a collection of finite flat group schemes with certain properties. The notion of a p-divisible group will make sense over a pseudocompact base and is a ‘gluing’ of the notion of a p-Barsotti-Tate group into a single group object, namely a certain type of formal commutative group scheme over the base. We’ll then check that over a pseudocompact local base, these two notions in fact coincide in an appropriate manner. Each notion has its own advantages, but it is best to be familiar with both points of view.

§3.1 Generalities

Definition 3.1.1 Let S be a scheme, p a prime. A p-Barsotti-Tate group over S of height h is an inductive system G = {G_{n}, i_{n}} of objects in F F_{S} (with n ≥ 0) such that G_{n} has order p^{nh} and i_{n} : G_{n} → G_{n+1} is a closed immersion which identifies G_{n} with the scheme-theoretic kernel of p^{n} on G_{n+1}. For fixed p, these objects form a category denoted BT_{S} = BT (p)_{S}.

Clearly the notion of a p-Barsotti-Tate group behaves well under base extension (this does not affect the height) and admits there is an obvious notion of a product of p-Barsotti-Tate groups (in which the heights add).

Let G be a p-Barsotti-Tate group over S. For n, m ≥ 0, there is a canonical closed immersion i_{n,m} : G_{n} ↪ G_{n+m} which identifies G_{n} with the scheme-theoretic kernel of p^{n} on G_{n+m}. Since the map p^{n} : G_{n+m} → G_{n+m} is killed by p^{m}, it factors uniquely through i_{m,n} : G_{m} ↪ G_{n+m}, say via j_{n,m} : G_{n+m} → G_{m}. Since G_{n} is the scheme-theoretic kernel of j_{n,m} and

order(G_{n+m}) = p^{(n+m)h} = p^{nh}p^{mh} = order(G_{n})order(G_{m}),

it follows that the diagram in F F_{S}

0 → G_{n} →^{i_{n,m}} G_{n+m} →^{j_{n,m}} G_{m} → 0

is a short exact sequence. It is not hard to check that this construction behaves well with respect to base extension and products and is functorial is G. Moreover, if we apply Cartier duality, then it is easy to see that G^{∗} = {G_{n}^{∗}, j_{1,n}^{∗}} is a p-Barsotti-Tate group over S.

Definition 3.1.2 If G is in BT_{S}, then G^{∗} in BT_{S} is called the dual of G.

From the properties of Cartier duality in F F_{S}, it is obvious that G ⇝ G^{∗} is a heightpreserving antiequivalence from BT_{S} to itself which commutes with products and there is a natural isomorphism G ≃ (G^{∗})^{∗}. Also, dualizing commutes with base extension. When S = Spec(A) with A a henselian local ring, we can construct a connected-étale sequence in BT_{S}:

Theorem 3.1.3 Let G be in BT_{A} with height h, for A a henselian local ring. Consider the natural inductive systems G^{ét} and G^{0} formed by the étale and connected parts of the G_{n}’s. These are objects in BT_{A}, with heights h^{ét} and h^{0} which add up to h. Moreover, G → G^{ét} has kernel G^{0} in the additive category BT_{S} and is a cokernel for G^{0} → G. The map G^{0} → G is a categorical monomorphism and the map G → G^{ét} is a categorical epimorphism. These constructions are functorial in G and are compatible with local base change to another henselian local ring.

Proof. If A is a field, it is clearly enough to check the result over an algebraic closure. But over an algebraically closed field, the connected-étale sequences for each G_{n} canonically split, from which the result follows. Hence, we have proven the theorem when A is a field. Consider the diagrams

0 → G_{n}^{ét} →^{i_{n}^{ét}} G_{n+1}^{ét} →^{j_{n,1}^{ét}} G_{1}^{ét} → 0

and

0 → G_{n}^{0} →^{i_{n}^{0}} G_{n+1}^{0} →^{j_{n,1}^{0}} G_{1}^{0} → 0

in F F_{A}. By functoriality, the composite maps are certainly 0 (since G_{n} → G_{n+1} → G_{1} vanishes). By the result over fields, we can therefore conclude that the above sequences are short exact. Since (again by functoriality) p^{n} : G_{n+1}^{ét} → G_{n+1}^{ét} factors as the composite of j_{n,1}^{ét} and the closed immersion (!) G_{1}^{ét} ↪ G_{n+1}^{ét}, and likewise for the connected parts (here we apply the final part of Theorem 2.2.12 to the short exact sequence

0 → G_{1} → G_{n+1} → G_{n} → 0),

the result follows via a fiber-by-fiber check.

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We now define the notion of a p-divisible group, with the base a pseudocompact ring A. Before giving the definition, we make a simple observation. If G is a commutative formal p-group over A for which p : G → G is topologically flat, G[p^{n}] is a formal commutative group scheme over A and G[p^{n}] → G[p^{n+1}] is a closed subgroup, as can be checked on the closed fibers along A. Thus, by Lemma 1.4.4(3), we see that the covariant functor \underrightarrow{lim} G[p^{n}] is a formal commuative group scheme over A and is even a closed subgroup of G. We say that G is a formal p-group over A if the natural map \underrightarrow{lim} G[p^{n}] → G is an isomorphism. Note that we require in the definition that p : G → G is topologically flat. This notion obviously behaves well under base change and formation of products. Also, Theorem 2.1.8 shows that if A is a field of characteritic p, then all connected commuative formal group schemes over A are formal p-groups.

Definition 3.1.4 Let A be a pseudocompact ring. A p-divisible group of height h over A is a formal p-group G over A for which p : G → G is topologically faithfully flat and the formal commutative group scheme G[p] lies in F F_{A}, with order p^{h}.

The notion of a p-divisible group over A certainly behaves well under base change (which does not affect the height) and formation of products. Such objects form a full subcategory of the category of commutative formal group schemes over A. Before we show how the notion of a p-divisible group of height h over A coincides with the notion of a p-Barsotti-Tate group of height h over A when A is local, we should point out some advantages of each point of view. The definition of duality is certainly only possible via the Barsotti-Tate perspective. On the other hand, when discussing connected-étale sequences, it is more natural to work with the notion of a p-divisible group as defined above. Let’s now show that our two definitions coincide.

Theorem 3.1.5 Let A be a pseudocompact local ring. There is height-preserving equivalence of categories between p-Barsotti-Tate groups over A and p-divisible groups over A. More precisely, a p-Barsotti-Tate group {G_{n}} over A gives rise to a commuative formal group scheme \underrightarrow{lim} G_{n} over A which is in fact a p-divisible group; on the other hand, a p-divisible group G gives rise to an inductive system {G[p^{n}]} of commutative formal group schemes over A which is actually an inductive system in F F_{A} and forms a p-Barsotti-Tate group.

This equivalence takes a connected-étale sequence of p-Barsotti-Tate groups over to a connected-étale sequence in the category of commutative formal group schemes over A. The equivalence is also compatible with base change.

Proof. Let G be a p-divisible group of height h over A. In order to show that {G[p^{n}]} is in BT_{A} with height h, we just need to show that the formal commutative group scheme G[p^{n}] is finite over A, with order p^{nh}. When n = 1, this is given to us. In general, p : G[p^{n+1}] → G[p^{n+1}] factors through G[p^{n}], so we have a diagram of commuative formal group schemes over A

0 → G[p] → G[p^{n+1}] → G[p^{n}] → 0

in which the composite map is 0 and the first term is the formal scheme-theoretic kernel of the second map. Also, the second map is the base extension of p : G → G by G[p^{n}] ↪ G and so this second map in the diagram is topologically faithfully flat. Hence, the sequence is short exact. By induction on n, we conclude that G[p^{n}] lies in F F_{A} and has order p^{nh}.

Now let \{G_{n}\} be a p-Barsotti-Tate group over A with height h. Define G = \underrightarrow{lim} G_{n}, which is a formal A-scheme by Lemma 1.4.4(3) and which is topologicaly flat over A since each G_{n} is topologically flat over A. Thus, G is a formal commutative group scheme over A, and trivially G[p^{n}] ≃ G_{n}, so the only thing left to check is that p : G → G is topologically faithfully flat. The faithfully flat maps j_{1,n} : G_{n+1} → G_{n} are topologically faithfully flat, as can be checked on the closed fiber. Passing to the limit, it is not hard to check that the resulting map \underrightarrow{lim} G_{n+1} → \underrightarrow{lim} G_{n} is topologically faithfully flat (exercise!). But this is exactly p : G → G.

If {G_{n}} is a p-divisible group, with G the resulting p-Barsotti-Tate group, the natural isomorphism between {G_{n}} and {G[p^{n}]} is clear. Conversely, if G is a p-divisible group, there is a natural map \underrightarrow{lim} G[p^{n}] → G of p-divisible groups and this is an isomorphism precisely because G is a formal p-group!

The rest of the assertion is straightfoward.

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Of course, if we dropped the assumption that A is local for the first part of the above theorem, then we would simply get a height-preserving fully faithful exact functor from pdivisible groups over A to p-Barsotti-Tate groups over A, with the essential image being those p-Barsotti-Tate groups whose constituent pieces admit a profinite A-module structure.

Theorem 3.1.6 Let A be a pseudocompact ring and G a commutative formal group scheme over A. Then G is a p-divisible group if and only if for all over maximals 𝔪 of A, G×_{A}(A/𝔪) is a p-divisible group over A/𝔪. This latter criteria can be checked after base extension from A/𝔪 to an extension field.

Proof. Since G is topologically flat over A, by the standard fiber-by-fiber criteria and formal Nakayama’s Lemma arguments, etc., the first part is clear. For the second part, we have a commutative formal group scheme G over a field k and are given an extension field k′ of k for which G×_{k} k′ is a p-divisible group over k′ and we wish to conclude that G is a p-divisible group over k. This is clear, thanks to the wonder of fields.

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§3.2 Aspects of the Theory of p-divisible groups

We now wish to supply some important fundamental results on p-divisible groups.

Theorem 3.2.1 (Serre-Tate) Let A be a pseudocompact local ring and G a p-divisible group over A. Then G is formally smooth and finite-dimensional over A. A connected p-divisible group over A is trivial if A has residue characteristic not equal to p and is the same thing as a commutative formal Lie group Γ over A with p : Γ → Γ topologically faithfully flat if A has residue characteristic equal to p.

Proof. Since we can detect whether a formal commutative group scheme over A is p-divisible or trivial by looking on the closed fiber, the second assertion follows from the first in the case of residue characteristic p. In the case of residue characteristic not equal to p, the second assertion follows from the fact that for a field k and a finite commutative group scheme G with order m which is prime to the characteristic of k, G is necessarily étale (indeed, we’ve seen earlier than connected finite group schemes over a field are trivial in characteristic 0 and have order a power of the characteristic otherwise).

Thus, it suffices to prove the first part, for which we may assume that G is connected and A = k is an algebraically closed field of characteristic p (for in any other characteristic, we just explained that G must be trivial). In [9], an outline of a proof is given that such a formal group scheme is smooth on finitely many parameters. This proof involves some delicate algebraic analysis. A somewhat more high-brow, and also more conceptual, explanation (or so it seems to me) can be supplied by the theory of Dieudonne modules as developed in [3, Ch III], which was developed precisely to understand the structure of formal p-groups over perfect fields of characteristic p. We will give the essential steps in the argument; what we omit are statements of the standard results in the theory of Dieudonne modules, which are all given in [3, Ch III].

Consider the Dieudonne modules M of G and M_{n} of G[p^{n}]. The axioms for a p-divisible group show that M_{n} ≃ (W (k)/p^{n})^{⊕h} as profinite W (k)-modules for all n and so by exactness of the contravairant Dieudonne functor, M = \underleftarrow{lim} M_{n} ≃ W (k)^{⊕h} as a profinite W (k)-module. In particular, since p = V F as endomorphisms of M, we see that the Frobenius-semilinear endomorphism F : M → M is injective. This implies that G is formally smooth over k (compare with Theorem 2.1.5). On the other hand, M/F M is surjected onto by M/pM ≃ M(G[p]) = M_{1}, which is finite-dimensional over k, so M/F M has finite dimension over k. This implies that G is finite-dimensional over k (with dimension equal to dim_{k}(M/F M); again, compare with Theorem 2.1.5).

Note how the argument with Dieudonne modules allows us to carry over to characteristic p the intuition of working with Q_{p}/Z_{p} as a p-divisible group.

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Now we wish to discuss the important notion of an isogeny between p-divisible groups. This can be formulated in the context of p-Barsotti-Tate groups, though our ad hoc method of constructing quotients will make this formulation somewhat awkward. Thus, in order to streamline things, we’ll first discuss isogenies over pseudocompact rings, in the context of p-divisible groups, as this makes the definitions and proofs more natural. At the end we will see how this connects up with a formulation for p-Barsotti-Tate groups.

We should mention that because a p-Barsotti-Tate group involves an ‘infinite’ amount of information, it is not always possible to ‘reduce to the (locally) noetherian case’ when studying general p-Barsotti-Tate groups over a base which is not assumed to be locally noetherian. In our study below, though, this will not be an obstacle. Of course, if the p-Barsotti-Tate group comes from an abelian scheme, this is not a problem (since an abelian scheme always descends to a noetherian situation locally on the base).

Definition 3.2.2 Let A be a pseudocompact ring, G_{1} and G_{2} two p-divisible groups over A. An isogeny from G_{1} to G_{2} is a map f : G_{1} → G_{2} of commutative formal group schemes which is topologically faithfully flat and has a kernel which is in F F_{A}.

From our discussion of quotients of p-divisible groups, we see that isogenies arise from taking the quotient of a p-divisible group G by a closed subgroup N which lies in F F_{A}. Conversely, if N is a closed subgroup of a p-divisible group such that N also lies in F F_{A}, then we claim that the commutative formal group scheme G/N is a p-divisible group with the same height as G and that G → G/N is an isogeny with kernel N.

In order to check that G/N is p-divisible with height h, we immediately reduce to the case in which A = k is an algebraically closed field, so we may assume that G and N are either both étale or both connected. When they are both étale, the assertion is trivial (a quotient of (Q_{p}/Z_{p})^{⊕h} by a finite subgroup is again isomorphic to (Q_{p}/Z_{p})^{⊕h}). Now consider the connected case. As this is a triviality if k has characteristic distinct from p, we may assume that k has characteristic p, and of course we note that k is perfect.

Since G/N is clearly connected, it is a p-formal group (Theorem 2.1.8). From the theory of Dieudonne modules, the Dieudonne module M(G/N) injects as a W (k)-submodule inside of M(G) ≃ W (k)^{⊕h}, with M(G)/M(G/N) ≃ M(N), which has finite W (k)-length. Thus, as a profinite W (k)-module we have M(G/N) ≃ W (k)^{⊕h}, from which it follows that G/N is a p-divisible group with the same height h as G [3, Prop 6.1, Ch III].

In order to be a little more explicit, note that since Spec(A) is quasi-compact and N is killed by its order (by a theorem of Deligne), we have N ↪ G[p^{t}] for some t ≥ 0. By an examination of closed fibers, it is then not difficult to check that we have natural maps (G[p^{n+t}]/N)[p^{n}] ≃ H[p^{n}] for all n ≥ 0 (in particular, the left side is flat over A), and also that the natural map (G[p^{n+t}]/N)[p^{n}] → (G[p^{m+t}]/N)[p^{n}] is an isomorphism for all m ≥ n (in particular, the right side is flat over A).

By passing to closed fibers, it is clear that if N_{1} ↪ N_{2} is an inclusion of closed subgroups inside of a p-divisible group G, then the natural map N_{2}/N_{1} → G/N_{1} is a closed subgroup and the quotient is G/N_{2}, with G/N_{1} → G/N_{2} the natural map. Of course, the usual fiberby-fiber checking shows that a map f : G → H of p-divisible groups over A is an isogeny if and only if for each open maximal 𝔪 of A, there exists an extension k′_{𝔪} of A/𝔪 such the the induced map on fibers over k′_{𝔪} is an isogeny. This is sometimes a convenient criteria for detecting isogenies. A more useful criteria for detecting isogenies is:

Theorem 3.2.3 A map f : G → H between p-divisible groups over A is an isogeny if and only if there exists a map f′ : H → G such that f′ ∘ f = p^{t} for some t ≥ 0. In this case, f′ is also an isogeny.

Proof. Suppose f is an isogeny, say with kernel N. Since N ↪ G[p^{t}] for some t ≥ 0, we can consider the map

f′ : H ≃ G/N → (G/N)/(G[p^{t}]/N) ≃ G/G[p^{t}] ≃ G.

It is clear that f′ is topologically faithfully flat with kernel G[p^{t}]/N ↪ G/N in F F_{A}, so f′ is an isogeny. It is equally clear that f′ ∘ f = p^{t}. Moreover, f is a categorical epimorphism (as the induced map on rings is topologically faithfully flat and hence injective), so such an f′ is unique. This shows that even as we vary t, the choices for f′ are composites of iterates of p with a fixed ‘minimal’ f′, so all possible f′ are isogenies.

Conversely, suppose there exists f′ : H → G such that f′ ∘ f = p^{t} for some t ≥ 0. We need to check that f is an isogeny. We reduce to the case in which A = k is a field. Clearly f has finite kernel (a closed subgroup of G[p^{t}]). Also, f′ is an epimorphism, since p^{t} : G → G is an epimorphism, so f′ is topologically faithfully flat. Combining this with the topological faithful flatness of p^{t} on G, we see that f is topologically faithfully flat, as desired, so f is an isogeny.

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Another interesting issue is the relation between duality and isogenies. More precisely, since any map f : G → H between p-divisible groups over A gives rise to a dual map f^{∗} : H^{∗} → G^{∗} (with f = (f^{∗})^{∗}), it is reasonable to ask if f^{∗} is an isogeny if and only if f is an isogeny, and when this is the case, whether ker(f) and ker(f^{∗}) are naturally Cartier duals. The answer to these questions is (of course) yes:

Theorem 3.2.4 For f : G → H a map between p-divisible groups over A, if f is an isogeny then f^{∗} is an isogeny and the two kernels are naturally Cartier duals.

Proof. Since f is an isogeny, there exists an isogeny f′ : H → G such that f′ ∘ f = p^{t}. Thus, f^{∗} ∘ (f′)^{∗} = p^{t}, so we see that (f′)^{∗} is an isogeny and then also that f^{∗} is an isogeny.

The heights of G and H are the same, say h, so we have a map f_{n} : G[p^{n}] → H[p^{n}] in F F_{A} between group schemes of the same order p^{nh}, and for large n, ker(f_{n}) = ker(f) (by a consideration of orders) while ker(f_{n}^{∗}) = ker(f^{∗}) (again by a consideration of orders). For large n, f_{n} induces a closed immersion G[p^{n}]/ ker(f) ↪ H[p^{n}], with quotient denoted Q_{n}. Naturally we have Q_{n}^{∗} ≃ ker(f^{∗}) for all large n, compatibly with change in n. Thus, we want to naturally construct an isomorphism Q_{n} ≃ ker(f) for all large n, compatible with change in n. Well, by construction Q_{n} = ((G[p^{n+t}]/ ker(f))[p^{n}])/(G[p^{n}]/ ker(f)) is a closed subgroup of (G[p^{n+t}]/ ker(f))/(G[p^{n}]/ ker(f)), yet this is isomorphic to G[p^{n+t}]/G[p^{n}] ≃ G[p^{t}], so it suffices to check that this factors through ker(f) ↪ G[p^{t}], for then by a comparison of orders it will follow that Q_{n} ↪ ker(f) is an isomorphism of the desired type.

It is enough to check that for all large n, the composite map

(G[p^{n+t}]/ ker(f))[p^{n}] → G[p^{n+t}]/ ker(f) → G[p^{n+t}]/G[p^{n}] ≃ G[p^{t}] → G[p^{t}]/ ker(f)

is 0. But the left side is the kernel of G[p^{n+t}]/ ker(f) → G[p^{t}]/ ker(f), as we noted above, so the observation that

G[p^{n+t}]/ ker(f) → G[p^{n+t}]/G[p^{n}]

factors naturally through (G[p^{n+t}]/ ker(f))/(G[p^{n}]/ ker(f)) finishes the proof.

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The construction in the above proof also shows that the isomorphism ker(f^{∗}) ≃ ker(f)^{∗} is symmetric in a suitable sense.

We conclude by briefly indicating how to carry out the above in the setting of p-Barsotti-Tate groups over a base scheme S (possibly not locally noetherian). If G lies in BT_{S}, we say that N in F F_{S} is a finite subgroup of G if there exists Zariski-locally on S a closed immersion of group schemes N ↪ G[p^{t}], where t is allowed to vary (it can be taken so that p^{t} bounds the order of N in the local region on S, again by Deligne’s theorem that finite flat (commutative!) group schemes are annihilated by their orders). We claim that the locally-on-S group scheme (G[p^{n+t}]/N)[p^{n}] (where t varies as above) is locally a finite flat group scheme on S and the natural transition maps (G[p^{n+t}]/N)[p^{n}] → (G[p^{m+t}]/N)[p^{n}] for m ≥ n are isomorphisms, so we can glue to get an inductive system H = {H_{n}} in F F_{S}. The flatness of (G[p^{n+t}]/N)[p^{n}] is fairly easy, since G[p^{t}]/N → G[p^{n+t}]/N is a closed immersion and G[p^{n+t}]/N → G[p^{n+t}]/N is faithfully flat, with kernel exactly (G[p^{n+t}]/N)[p^{n}]. The rest is now easy, as it can be checked along fibers, where we already have verified everything (since fields are pseudocompact).

Moreover, we claim that H is a p-Barsotti-Tate group, denoted G/N. This can also be checked on the closed fiber, where it is clear in view of the above results. A map such as G → G/N in the context of p-Barsotti-Tate groups is what we call an isogeny. It is clear that if the base S is a pseudocompact ring, then this notion of G/N and an isogeny is compatible (via the identification of p-divisible and p-Barsotti-Tate groups) with what has already been defined. Everything we said above for isogenies of p-divisible groups (preserve height, etc.) still holds in this more general setting. Most proofs quickly reduce to checking statements over fields, yet we know everything here because fields are pseudocompact. The statement concerning duality is simply checked using the same method as in the proof of Theorem 3.2.4.

§4 Invariant Differentials

As we have remarked earlier, invariant differentials are an important fundamental tool in the study of group schemes. The primordial example to keep in mind is that of complex Lie groups, where one knows classically that every 1-form at the identity uniquely extends to a global 1-form which is left-invariant under the action of the group on differentials. A similar result is true in the setting of group schemes and formal group schemes, but we require a much more functorial point of view in order to construct global invariant differentials on a group scheme. In particular, the results we get provide an algebraic analogue to the analytic construction of invariant differentials via propogation from the identity by translations.

Everything we do will be in the context of formal group schemes over a pseudocompact ring A, except we now drop the hypothesis of topological flatness in the definition of formal group schemes. Our arguments will be done almost completely in terms of functors and so will carry over essentially without change to the setting of group schemes over an arbitrary base scheme (with no finite type or flatness assumptions, etc). However, in the non-affine case there are a few minor places where one needs to be a little more careful. With these changes, the methods below can be used to show that an abelian scheme A of relative dimension d over a local ring R has the property that the sheaf Ω_{A/R}^{1} is globally free of rank d. When R = C, this is well-known from the theory of complex Lie groups. In fact, the methods below apply to complex Lie groups!

Our methods are based on the lucid exposition in [1, §4.2], except we connect this up with some alternate perspectives from [3] which are not explicitly commented on in [1]. In [1, §4.2], one can find the minor adaptations needed for non-affine situations (the only genuine extra twist is that one needs to choose an auxiliary open set U, something which is unnecessary in the affine setting).

§4.1 Different Definitions

Let S = Sp_{A}(A) for a fixed pseudocompact ring A and let G = Sp_{A}(B) be a formal group scheme over S (again, recall that we do not assume G is topologically flat over S). There are two definitions of the meaning of invariant differential inside Ω_{G/S}^{1} = Ω̂_{B/A}^{1}, one which applies in general (and is the one used in [1]) and one which applies in the commutative case (and is the one used in [3]). It is important (and useful!) to know that in the commutative case, both definitions coincide.

Let p_{i} : G ×_{S} G → G be the projection maps, ϵ_{G} : S → G the identity section, and m : G ×_{S} G → G the multiplication. By Theorem 1.5.7(3),(4), we have the fundamental natural isomorphism of profinite B ˆ⊗_{A}B-modules

Ω_{G×G /S}^{1} ≃ p_{1}^{∗}Ω_{G/S}^{1} ⊕ p_{2}^{∗}Ω_{G/S}^{1}.

Here, the isomorphism goes from right to left and p_{i}^{∗}Ω_{G/S}^{1} = (B ˆ⊗_{A}B)ˆ⊗_{B}Ω_{G/S}^{1}, where B → B ˆ⊗_{A}B maps to the ith factor. Also, the above isomorphism is compatible with base change. For any ω ∈ Ω_{G/S}^{1}, m^{∗}ω makes sense on the left side and p_{1}^{∗}ω + p_{2}^{∗}ω makes sense on the right side.

For any formal S-scheme T, we let G_{T} = G ×_{S} T denote the base-extended formal group scheme over T and we let G(T) = G_{T} (T) denote the group of S-maps from T to G. The left translation

t_{g} : G_{T} ≃ T ×_{T} G_{T} →^{g×id} G_{T} ×_{T} G_{T} →^{m_{T}} G_{T}

is given functorially by x ↦ gx on G(T′) = G_{T} (T′) for all formal T -schemes T′ (where g ∈ G(T′) via the natural map G(T) → G(T′)). Of course, we can similarly define the formal T -scheme isomorphism given by the right translation t′_{g} : G_{T} ≃ G_{T}. Since the automorphism t_{g} over T induces an isomorphism t_{g}^{∗}Ω_{G_{T} /T}^{1} ≃ Ω_{G_{T} /T}^{1} and any ω ∈ Ω_{G/S}^{1} gives rise to an element ω_{T} ∈ Ω_{G_{T} /T}^{1} via pullback, it makes sense to ask if t_{g}^{∗}ω_{T} = ω_{T} inside of Ω_{G_{T} /T}^{1} . Note that ω_{T} depends not only on T → S but also on the choice of T → G over S, so the notation is a little sloppy. This should not cause too much confusion. Of course, t_{g} and t′_{g} are compatible with base change in an evident manner.

Definition 4.1.1 We say that ω ∈ Ω_{G/S}^{1} is left-invariant if t_{g}^{∗}ω_{T} = ω_{T} for all formal Sschemes T and all g ∈ G(T). Right-invariant differentials are defined similarly, using t′_{g}. The module of left-invariant differentials is denoted Ω_{G/S}^{1,l}.

Note that left- and right-invariance are properties which are preserved under base change. Also, in order to check such invariance, it actually suffices to check invariance under translation by a very special point, namely the ‘universal point’ id_{G} ∈ G(G) = G_{G}(G), where G_{G} = G × G is viewed as a formal group scheme over G via p_{2} (and in terms of points, for all T → G the map G_{G}(T) ×_{G} (T)G_{G}(T) → G_{G}(T) is just (x, y) · (x′, y) = (xx′, y)). The left translation

t_{id_{G}} : G_{G} = G × G → G × G

is the ‘universal left translation’ (x, y) → (yx, y) on points, and this is certainly compatible with base change. It is also clear that pullback by inversion interchanges left- and rightinvariant differentials.

Now suppose that ω ∈ Ω_{G/S}^{1} is left-invariant under the universal left translation. In order to check that ω is left-invariant, by base change it suffices to check that t_{g}^{∗}ω = ω for all g ∈ G(S). We are given t_{id_{G}}^{∗}ω_{G} = ω_{G}, with ω_{G} = p_{1}^{∗}ω. Pulling back along the map

id_{G} × g : G ≃ G ×_{S} S → G × G,

we get

(id_{G} × g)^{∗}t_{id_{G}}^{∗}ω_{G} = (id_{G} × g)^{∗}ω_{G}.

But it is easy to compute that the left side is t_{g}^{∗}ω and the right side is ω. One consequence of this ‘universal point’ criteria is that it is clear that inside of Ω_{G/S}^{1} = Ω̂_{B/A}^{1}, the A-submodule Ω_{G/S}^{1,l} of left-invariant differentials is closed and so is naturally profinite as an A-module (and we have likewise for the right-invariant submodule). Of course, this could also be seen directly without the ‘universal point’ criteria.

In the case of reduced (and therefore smooth) group schemes G of finite type over an algebraically closed field k, there is a more classical definition which we wish to show is consistent with the above definition (suitably formulated in terms of k-schemes rather than formal schemes). More generally, consider any finite type group scheme G over a field k such that G is smooth over k. We claim that to show ω ∈ Γ(G, Ω_{G/k}^{1}) is left-invariant in the above functorial sense, it is necessary and sufficient to check left-invariance under all closed points of G. Necessity is clear, and for sufficiency we can (by the Nullstellensatz) obviously reduce to the case in which k is algebraically closed. In this case, we are given that t_{g}^{∗}ω = ω for all g ∈ G(k), where t_{g} : G → G corresponds to nothing other than the classical notion of translation on a smooth, possibly disconnected, group variety over an algebraically closed field. We want to conclude that ω is invariant under left translation by the universal point g ∈ G(G).

By smoothness of G → k, we see that G and G × G are both regular and smooth over k and p_{2} : G × G → G is also smooth. Thus, Ω_{G×G /G}^{1} ≃ p_{2}^{∗}Ω_{G/k}^{1} is locally free on G × G and so from the fact that finite type algebras over a field are Jacobson rings, we see that a global section of Ω_{G×G /G}^{1} vanishes if and only if it vanishes at all closed points. Thus, in order to show that p_{1}^{∗}ω is left-invariant under translation by the universal point, it is enough to check t_{id_{G}}^{∗}p_{1}^{∗}ω and p_{1}^{∗}ω pull back to the same thing at each closed point (x, y) ∈ (G × G)(k). In fact, it suffices to check that for all y ∈ G(k), the two forms pull back to the same thing in Γ(G, Ω_{G/k}^{1}) under id_{G} × y : G → G × G. The two pullbacks are t_{y}^{∗}ω and ω, which are equal by hypothesis.

Classically on a group variety, the abstract differential of m : G ×_{k} G → G induces a tangent space map

T_{0}(G) × T_{0}(G) ≃ T_{(0,0)}(G ×_{k} G) → T_{0}(G)

which is nothing other than addition of the underlying vector space This is best seen by passing to the formal group at the identity (which still ‘contains’ the cotangent space) and using the fact that a formal group law on k[[T_{1}, …, T_{n}]] necessarily has linear terms in the comultiplication which coincide with those on the n-fold product of the formal additive group scheme. This fact suggest the following definition:

Definition 4.1.2 We say that ω ∈ Ω_{G/S}^{1} is bivariant if

m^{∗}ω = p_{1}^{∗}ω + p_{2}^{∗}ω.

Clearly this definition is compatible with base change. In order to bring things down to earth a little, let’s now consider an example. Say we have a formal group scheme G underlying Sp_{A}(A[[T]]), so we have an A-algebra map m : A[[T]] → A[[U, V]] satisying the usual conditions (where we can clearly ‘change coordinates’ so that the augmentation ideal is (T)). Since Ω̂_{A[[T]]/A}^{1} ≃ A[[T]]dT, pick some ω = f(T)dT in here. Bivariance of ω requires that the differential

m^{∗}ω = f(m(T))d(m(T)) = f(m(T))(∂_{U}(m(T))dU + ∂_{V} (m(T))dV)

is equal to f(U)dU + f(V)dV. That is,

f(m(T))∂_{U}(m(T)) = f(U), f(m(T))∂_{V} (m(T)) = f(V).

Meanwhile, to say that ω is left-invariant, we require that f(U)dU = p_{1}^{∗}ω ∈ Ω̂_{A[[U,V]]/A[[V]]}^{1} is unaffected by the universal left translation, so using the ‘flip’ map φ : G × G → G × G (interchanging U and V), this becomes

f(U)dU = f(φ(m(T)))d_{A[[U,V]]/A[[V]]}(φ(m(T))) = f(φ(m(T)))∂_{U}(φ(m(T)))dU.

Applying φ^{∗}, this is equivalent to

f(V)dV = f(m(T))∂_{V} (m(T))dV.

In a similar manner, invariance under right translation is equivalent to

f(U)dU = f(m(T))∂_{U}(m(T)dU.

That is, bivariance in this case is equivalent to being both left- and right-invariant.

When the example is the formal affine line, we see that the bivariant differentials are exactly those f(T)dT for which f(U + V) = f(U). Setting U = 0, we see that these are exactly the differentials a · dT with a ∈ A, so in this case, the natural map of profinite B-modules

B ˆ⊗_{A}Ω_{G/S}^{1,l} → Ω_{G/S}^{1}

is an isomorphism. We will prove this more generally in §4.2. Let us now show that the equivalence of bivariance with simultaneous left- and right-invariance is not a quirky feature of the above example.

Theorem 4.1.3 For any formal group scheme G_{/S}, bivariant differentials are the same thing as differentials which are simultaneously left- and right-invariant.

Proof. Let ω ∈ Ω_{G/S}^{1} be a bivariant differential. In order to show that it is left-invariant, by base extension we reduce ourselves to checking ω = t_{g}^{∗}ω for g ∈ G(S). Let g̃ : G → G × G

denote the morphism given on points by x ↦ (g, x) (that is, g̃ : G ≃ S ×_{S} G →^{g×id_{G}} G × G), so t_{g} = m ∘ g̃. Thus,

t_{g}^{∗}ω = g̃^{∗}m^{∗}ω = g̃^{∗}(p_{1}^{∗}ω + p_{2}^{∗}ω) = (p_{1} ∘ g̃)^{∗}ω + ω,

since p_{2} ∘ g̃ = id_{G}. If we can show that the natural map (p_{1} ∘ g̃)^{∗}Ω_{G/S}^{1} → Ω_{G/S}^{1} of profinite B-modules is 0, then it follows that ω is left-invariant. But on the affine level this is clear, since if i : A → B is the structure map and g : B → A is the section, then

(i ∘ g)^{∗}d_{B/A}(b) = d_{B/A}(i(g(b))) = i(g(b))d_{B/A}(1) = 0

(and now use the denseness remark following Definition 1.5.6). A similar argument works to establish right-invariance by using g̃′(x) = (x, g).

Choose a left- and right-invariant differential ω ∈ Ω_{G/S}^{1} and write

m^{∗}ω = ω_{1} + ω_{2}

compatibly with Ω_{G×G/S}^{1} ≃ p_{1}^{∗}Ω_{G/S}^{1} ⊕ p_{2}^{∗}Ω_{G/S}^{1}. We wish to show that ω_{j} = p_{j}^{∗}ω. The rightinvariance under the universal right-translation t′ says that p_{1}^{∗}ω ∈ p_{1}^{∗}Ω_{G/S}^{1} = Ω_{G×G /G}^{1} is unaffected by t′^{∗}. Thus,

p_{1}^{∗}ω = t′^{∗}p_{1}^{∗}ω = (p_{1} ∘ t′)^{∗}ω = m^{∗}ω

inside of Ω_{G×G /G}^{1} (viewed as the first factor of Ω_{G×G /S}^{1}). Note that in the above calculation we used the fact that p_{1} ∘ t′ = m for right translation maps. Anyway, this says that m^{∗}ω projects onto p_{1}^{∗}ω as its ‘first coordinate’, so ω_{1} = p_{1}^{∗}ω.

If we let i : G → G denote inversion, then i^{∗}ω is also right-invariant (as ω is left-invariant) and so m^{∗}i^{∗}ω has ‘first coordinate’ p_{1}^{∗}i^{∗}ω = (i × i)^{∗}p_{1}^{∗}ω. But i ∘ m = m ∘ φ ∘ (i × i), where φ : G × G → G × G is the ‘flip’ morphism that changes coordinates, so

m^{∗}i^{∗}ω = (i × i)^{∗}(φ^{∗}m^{∗}ω).

Since i×i : G×G → G×G is of the form of a ‘product’ of two maps G → G, it is compatible with the decomposition of Ω_{G×G /S}^{1} and so applying (i × i)^{∗} throughout, we get that φ^{∗}m^{∗}ω has ‘first coordinate’ p_{1}^{∗}ω.

Since

φ^{∗}m^{∗}ω = φ^{∗}ω_{2} ⊕ φ^{∗}ω_{1}

(note the order of the factors), it follows that φ^{∗}ω_{2} = p_{1}^{∗}ω. Applying φ^{∗} to both sides, we get ω_{2} = p_{2}^{∗}ω, as desired.

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We now abandon the notion of a bivariant differential, except to remind the reader that this is the definition used in [3] (in which all group objects considered are commutative). From now on, we will work with left-invariant differentials (though right-invariant ones could just as well be used; the only difference this makes is to slightly alter some morphisms).

§4.2 Existence of Enough Invariant Differentials

We begin by indicating the relation between the formal cotangent space t_{G}^{∗} = I_{G}/\overline{I_{G}^{2}} of G = Sp_{A}(B) (with I_{G} the augmentation ideal of B) and differentials. First, we claim there is a natural identification of profinite A-modules t_{G}^{∗} ≃ ϵ_{G}^{∗}Ω_{G/S}^{1} = Ω̂_{B/A}^{1} ˆ⊗_{B}A, where B → A is the identity section. This is easily seen by noting that both objects represent the same functor on the category of profinite A-modules (recall the functor represented by the module of continuous differentials). Of course, this is somewhat similar to the global proof that for any separated scheme X over a base S and any section s : S → X (necessarily a closed immersion), the ideal sheaf I_{s} determined by this closed subscheme has the property that s^{∗}Ω_{X/S}^{1} ≃ I_{s}/I_{s}^{2} as sheaves of quasi-coherent 𝒪_{S} ≃ 𝒪_{X}/I_{s}-modules (via the identification of 𝒪_{S} with s_{∗}𝒪_{S}).

We will now related t_{G}^{∗} to left-invariant differentials in a somewhat ‘explicit’ manner, and then we will give a more general functorial argument (which carries over to non-affine settings also) that relates ϵ_{G}^{∗}Ω_{G/S}^{1} to left-invariant differentials. The explicit formulation is given because that is what Fontaine does in his book [3]; however, the more general functorial version (taken from [1, §4.2]) is more conceptually satisfying. We leave it as an exercise to the reader to check carefully that the two points of view are consistent with each other under the above identification of t_{G}^{∗} and ϵ_{G}^{∗}Ω_{G/S}^{1} (e.g., one wants to make sure that things aren’t off by a minus sign).

Consider the commutative diagram

G  →^{id} G
Δ_{G/S} ↓  ↓
G × G →^{j} G × G

with the right column id_{G} × 1 and j(x, y) = (x, y^{−1}x). In terms of profinite B-algebras, this corresponds to

B →^{id}  B
↑  ↑ idˆ⊗ϵ_{B}
B ˆ⊗_{A}B ←^{∼}  B ˆ⊗_{A}B
↑  ↑
J_{G}  B ˆ⊗_{A}I_{G}

where the bottom of each column in the kernel in that column, since

0 → I_{G} → B → A → 0

is split in the category P F_{A} of profinite A-modules. Thus, we get an isomorphism of the kernels and so modding out each by the closure of its square, we get an isomorphism of profinite B-modules

ι_{G} : B ˆ⊗_{A}(I_{G}/\overline{I_{G}^{2}}) ≃ (B ˆ⊗_{A}I_{G})/\overline{(B ˆ⊗_{A}I_{G})^{2}} ≃ J_{G}/\overline{J_{G}^{2}} = Ω̂_{B/A}^{1}.

Note that the first map is an isomorphism because the inverse can be written down.

The above procedure gives an injection of profinite A-modules from t_{G}^{∗} into Ω_{G/S}^{1}. This is injective because of the identity section. We would like to show that this is an isomorphism from t_{G}^{∗} onto the closed submodule Ω_{G/S}^{1,l} of left-invariant differentials. Let’s first check that the image is inside of Ω_{G/S}^{1,l}. By base change, it is enough to check left-invariance under t_{g} for g ∈ G(S). This is an easy consequence of the fact that we have a commutative diagram

G × G  →^{j}  G × G
t_{g} × t_{g} ↑  ↑ t_{g} × id
G × G  →^{j}  G × G

with the identity on the right factor of the second column (and recall that I_{G} is mapped to J_{G} via the right factor).

Now consider the natural commutative diagram

B ˆ⊗_{A}Ω_{G/S}^{1,l} → Ω_{G/S}^{1}
↑  ↑∼
B ˆ⊗_{A}t_{G}^{∗}  →^{id} B ˆ⊗_{A}t_{G}^{∗}

in which the right column is ι_{G} and the left column is an injection. If we could show that the top row is an isomorphism, then we could conclude that t_{G}^{∗} ↪ Ω_{G/S}^{1,l} becomes an isomorphism after the base extension B ˆ⊗_{A}(·), so applying the identity section shows that this map was an isomorphism to begin with. That is, this would give an algebraic method of using the formal cotangent space at the identity to generate the left-invariant differentials (and would also show that all differentials are generated by the left-invariant ones). When A = k is a field, this would yield the non-obvious fact that Ω_{G/k}^{1} is topologically free over B (by choosing a pseudobasis for Ω_{G/k}^{1,l} over k).

Thus, for the issue of connecting the formal cotangent space and left-invariant differentials, we can ignore t_{G}^{∗} and instead can concentrate on trying to prove that the natural map

τ_{G} : B ˆ⊗_{A}Ω_{G/S}^{1,l} → Ω_{G/S}^{1}

of profinite B-modules is an isomorphism. Actually, we will prove a slightly different statement.

Theorem 4.2.1 Let G → S be a formal group scheme (possibly not topologically flat). For each choice of

ω_{0} ∈ ϵ_{G}^{∗}Ω_{G/S}^{1} ≃ Ω_{G/S}^{1}/\overline{I_{G}Ω_{G/S}^{1}},

there exists a unique ω ∈ Ω_{G/S}^{1,l} ⊆ Ω_{G/S}^{1} such that ϵ_{G}^{∗}ω = ω_{0}. Thus,

Ω_{G/S}^{1,l} ≃ ϵ_{G}^{∗}Ω_{G/S}^{1}

as profinite A-modules.

Let’s grant this theorem for a moment and we will show how to use it to show that the map

τ_{G} : B ˆ⊗_{A}Ω_{G/S}^{1,l} → Ω_{G/S}^{1}

(for S = Sp_{A}(A), G = Sp_{A}(B)) is an isomorphism. The theorem says that the map

ϵ_{G}^{∗}(τ_{G}) : Ω_{G/S}^{1,l} → ϵ_{G}^{∗}(Ω_{G/S}^{1})

is a continuous bijection of profinite A-modules and so it is an isomorphism. The inverse therefore gives

ϵ_{G}^{∗}(Ω_{G/S}^{1}) ≃ Ω_{G/S}^{1,l}

as profinite A-modules (by ‘lifting’). However, using the structure map π : G → S, we get from this a map

π^{∗}ϵ_{G}^{∗}(Ω_{G/S}^{1}) = B ˆ⊗_{A}ϵ_{G}^{∗}(Ω_{G/S}^{1}) ≃ B ˆ⊗_{A}Ω_{G/S}^{1,l} →^{τ_{G}} Ω_{G/S}^{1}

that is an isomorphism (even the ‘identity’) after applying ϵ_{G}^{∗}. What we want to prove using Theorem 4.2.1 is that this map

α_{G} : π^{∗}ϵ_{G}^{∗}(Ω_{G/S}^{1}) → Ω_{G/S}^{1}

is an isomorphism. This is the formulation that generalizes to group schemes over arbitrary base schemes. Of course, we note that α_{G} is functorial in G and is of formation compatible with base change.

Though we know that α_{G} becomes an isomorphism after pullback by ϵ_{G} ∈ G(S), we claim that it becomes an isomorphism after pullback by all g ∈ G(S). By base extension, it then follows that α_{G} becomes an isomorphism after pullback by all g ∈ G(T) for all formal Sschemes T. Choosing T = G and g = id_{G} ∈ G(G) the ‘universal point’, we indeed get that α_{G} is an isomorphism. Now pick g ∈ G(S). We will prove that g^{∗}(α_{G}) is an isomorphism.

Note that g^{∗} = ϵ_{G}^{∗}t_{g}^{∗}. Also, the above construction of α_{G} involves lifting to left-invariant differentials. This guarantees that the diagram

t_{g}^{∗}π^{∗}ϵ_{G}^{∗}Ω_{G/S}^{1} →^{t_{g}^{∗}(α_{G})} t_{g}^{∗}Ω_{G/S}^{1}
∼↓  ↓∼
π^{∗}ϵ_{G}^{∗}Ω_{G/S}^{1}  →^{α_{G}} Ω_{G/S}^{1}

commutes. Here, the left column uses π ∘ t_{g} = π and the right column is as usual.

In order to check commutativity, pick a typical element bˆ⊗ω_{0} in the upper left. It goes down to t_{g}(b)ˆ⊗ω_{0} in the lower left and then over to t_{g}(b)ω in the lower right, with ω the left-invariant lift of ω_{0}. Meanwhile, going across the top row, bˆ⊗ω_{0} goes over to bˆ⊗ω. The right column isomorphism takes this down to t_{g}(b)t_{g}^{∗}ω = t_{g}(b)ω, since ω is left-invariant. By continuity, it follows that the diagram commutes. Now applying ϵ_{G}^{∗} throughout, the bottom row is the known isomorphism ϵ_{G}^{∗}(α_{G}), while the top row is g^{∗}(α_{G}), so this latter map must be an isomorphism also. Modulo proving Theorem 4.2.1, this then completes the proof that

B ˆ⊗_{A}Ω_{G/S}^{1,l} ≃ Ω_{G/S}^{1}

is an isomorphism for (possibly not topologically flat) formal group schemes G = Sp_{A}(B) over A. We remind the reader to now check as an exercise that the profinite A-module isomorphism t_{G}^{∗} ≃ Ω_{G/S}^{1,l} obtained from Theorem 4.2.1 and the initial ‘representable functor’ proof that t_{G}^{∗} ≃ ϵ_{G}^{∗}Ω_{G/S}^{1} is the same as the isomorphism obtained from the injection t_{G}^{∗} ↪ Ω_{G/S}^{1,l} in the first part of this section.

Let us now prove Theorem 4.2.1.

Proof. The uniqueness is easy. If ω_{0} = ϵ_{G}^{∗}ω = ϵ_{G}^{∗}ω̂ for two left-invariant differentials ω and ω̂, the relation g = t_{g} ∘ ϵ_{G} for g ∈ G(S) yields

g^{∗}ω = g^{∗}ω̂

inside of g^{∗}Ω_{G/S}^{1}. Using the usual base chage arguments, this holds for all g ∈ G(T) for all formal S-schemes T. The only point to check is that under h : T → S, ϵ_{G_{T}}^{∗} (ω_{T}) ∈ ϵ_{G_{T}}^{∗} Ω_{G_{T} /T}^{1} is equal to h^{∗}ω_{0} ∈ h^{∗}ϵ_{G}^{∗}Ω_{G/S}^{1} (via h^{∗}ϵ_{G}^{∗} = ϵ_{G_{T}}^{∗}), and likewise for ω̂. Taking the universal point g = id_{G} ∈ G(G) then gives ω = ω̂, as desired.

The main mystery is to see how one would ever figure out how to construct the right ω. At the very least, we can pick some ω′ ∈ Ω_{G/S}^{1} reducing to ω_{0}. It is this step that requires extra care in the setting of non-affine group schemes, as a surjection of quasi-coherent sheaves on a non-affine scheme admits lifts of sections only locally and not globally. In order to circumvent this in the non-affine setting, it is important to have established uniqueness first, as we did above, so that existence can be checked locally. We won’t say anything more about the non-affine case (see [1, §4.2] for further details). The choice of ω′ is very likely not the one we seek (i.e., it won’t be left-invariant), so we must somehow alter it.

Let us for a moment indulge in an intuitive and non-rigorous discussion. The result we want is certainly made geometrically plausible based on one’s experience with complex Lie groups, so let’s pretend we knew the result, and so all differentials can be generated by the left-invariant ones. Roughly speaking, this means that there should be left-invariant differentials ω_{i} and functions a_{i} so that ω′(x) = ∑ a_{i}(x)ω_{i}(x) for all points x. The lift we really seek is the differential for which

ω(x) = t_{x}^{∗}(ω_{0}) = t_{x}^{∗}(ω′(0)) = t_{x}^{∗} (∑ a_{i}(0)ω_{i}(0)) = ∑ a_{i}(0)t_{x}^{∗}(ω_{i}(0)).

But since ω_{i} is left-invariant, this is exactly ∑ a_{i}(0)ω_{i}(x). Can we somehow recover this sum intrinsically from ω′? Well, since ω_{i} is left-invariant, the method of proof of Theorem 4.1.3 shows that

ω_{i}(xy) = (m^{∗}ω_{i})(x, y) = η_{i}(x, y) + ω_{i}(y),

where η_{i} is a complete mystery. Thus, we at least have

(m^{∗}ω′)(x, y) = ω′(xy) = η′(x, y) + ∑ a_{i}(xy)ω_{i}(y).

More precisely, under the decomposition of m^{∗}ω′ into its two ‘coordinate’ differentials, the second one is ∑ a_{i}(xy)ω_{i}(y). Thus, under the canonical decomposition m^{∗}ω′ = ω_{1} ⊕ ω_{2}, we see that pulling back ω_{2} along the map y ↦ (y^{−1}, y) gives the desired ∑ a_{i}(0)ω_{i}(y). Note that if we were considering right-invariant differentials, this discussion suggests we should pullback ω_{1} along the map x ↦ (x, x^{−1}). We will leave it to the reader to carry out the easy modifications to the argument below so in order to adapt it to the right-invariant case.

Now to be rigorous, consider the twisted diagonal map

δ : G → G × G

given by

x ↦ (x^{−1}, x)

on points. Define ω = δ^{∗}ω_{2} ∈ Ω_{G/S}^{1}. We will show that this works.

Let’s first check that ϵ_{G}^{∗}ω = ω_{0}. Well, define

ϵ̂_{G} : G → G × G

via x ↦ (1, x) on points, so inside of Ω_{G/S}^{1} we have ϵ̂_{G}^{∗}ω_{2} = ϵ̂_{G}^{∗}(ω_{1} + ω_{2}), since the natural map ϵ̂_{G}^{∗}p_{1}^{∗}Ω_{G/S}^{1} → Ω_{G/S}^{1} vanishes (this is proven by a similar argument to that used in the proof of Theorem 4.1.3). Hence,

ϵ̂_{G}^{∗}ω_{2} = ϵ̂_{G}^{∗}m^{∗}ω′ = (m ∘ ϵ̂_{G})^{∗}ω′ = ω′

inside of Ω_{G/S}^{1}. Thus, we have ϵ_{G}^{∗}ω = ϵ_{G}^{∗}δ^{∗}ω_{2} by definition of ω and ω_{0} = ϵ_{G}^{∗}ω′ = ϵ_{G}^{∗}ϵ̂_{G}^{∗}ω_{2} by the above computation, so we want to check that

ϵ_{G}^{∗}δ^{∗}ω_{2} = ϵ_{G}^{∗}ϵ̂_{G}^{∗}ω_{2}

inside of ϵ_{G}^{∗}Ω_{G/S}^{1}. But p_{2} ∘ ϵ̂_{G} = id_{G} = p_{2} ∘ δ and δ ∘ ϵ_{G} = ϵ̂_{G} ∘ ϵ_{G}, so we get canonical isomorphisms

ϵ_{G}^{∗}(δ^{∗}p_{2}^{∗}Ω_{G/S}^{1}) ≃ ϵ_{G}^{∗}Ω_{G/S}^{1} ≃ ϵ_{G}^{∗}(ϵ̂_{G}^{∗}p_{2}^{∗}Ω_{G/S}^{1}).

Under these isomorphisms, we get the desired equality

ϵ_{G}^{∗}δ^{∗}ω_{2} = ϵ_{G}^{∗}ϵ̂_{G}^{∗}ω_{2}.

Now it remains to check that ω = δ^{∗}ω_{2} is left-invariant. By the usual base change arguments, it is enough to check left-invariance against t_{g} for g ∈ G(S). Consider the commutative diagram

G →^{t_{g}} G
δ ↓  ↓ δ
G × G ⟶^{t′_{g^{−1}}×t_{g}} G × G

(this is where we use that inverse in the definition of δ). For the differential ω_{2} ∈ p_{2}^{∗}Ω_{G/S}^{1} on the lower right G × G, it certainly suffices to check that going across the bottom row pulls ω_{2} back to itself. But ω_{2} is intrinsically described as the ‘second coordinate’ of m^{∗}ω′, and the bottom row morphism is clearly compatible with the decomposition

Ω_{G×G/ S}^{1} ≃ p_{1}^{∗}Ω_{G/S}^{1} ⊕ p_{2}^{∗}Ω_{G/S}^{1},

as t′_{g^{−1}} × t_{g} is a ‘product’ of two maps. It now suffices to observe that m ∘ (t′_{g^{−1}} × t_{g}) = m, so the bottom row morphism pulls m^{∗}ω′ back to itself.

■

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