CHAPTER 11 Lecture 11

1. L/K Forms

1.1. Definitions and Examples. Let L/K be a finite extension of fields. We assume that L/K is Galois, although that will not really be necessary until later. Let V be an algebraic variety defined over K. We will say that another algebraic variety W, defined over K, is an L/K form of V if

W ×_{Spec K} Spec L ≅ V ×_{Spec K} Spec L.

(Note that the specific isomorphism is not part of the data attached to an L/K form; we merely assume that one exists.) An L/K form W of V, then, is simply an algebraic variety which becomes isomorphic to K after base change to L; this, of course, does not imply that V and W are isomorphic over K. Such a W is often called a twist of V. We define E(L/K, V) to be the set of K-isomorphism classes of varieties W, defined over K, which are L/K forms of V. In fact, it is a pointed set, with the class of V itself being the distinguished element.

This notion of L/K form is not in any way restricted to the case of algebraic varieties. Grothendieck was able to extend the key ideas to a much more general situation, replacing L/K with any faithfully flat quasi-compact map of schemes Y → X, and replacing the above twists with certain fibered categories.

We will content ourselves with a few more examples. First, suppose that instead of simply considering a variety V over K, we consider a base-pointed variety V over K; that is, we consider pairs (V, v), where v is a K-valued point of V. L/K forms of V are now pairs (W, w) of a variety W defined over K and a K-valued point of W. There is an obvious notion of isomorphism of base-pointed schemes, and we can define E(L/K, (V, v)) to be the set of K-isomorphism classes of base-pointed varieties, defined over K, which are L/K forms of (V, v). In the exact same way one can define L/K forms for algebraic groups defined over K, isomorphisms now required to be isomorphism as algebraic groups, or even L/K forms for algebraic varieties with chosen groups of automorphisms.

Moving away from varieties, we note that a central simple algebra over K of rank n^{2} which splits over L is nothing more than an L/K form for the central simple K-algebra M_{n}(K). Thus E(L/K, M_{n}(K)) is simply the set of central simples Kalgebras which split over L. As another example, let F be a number field and recall the notion of locally free rank 1 O_{F,S}-module introduced in Lecture 10, Section 1.3. If V is such a module, then for some N we have V^{N} ≅ O_{F,S} since Pic(O_{F,S}) is finite. Interpreting V as a fractional ideal, we have V^{N} = αO_{F,S}. It follows easily that V becomes trivial over F (α^{1/N}), so that it is an F (α^{1/N})/F form for O_{F,S}.

0Last modified March 11, 1998

1.2. The Automorphism Group. We return now to the case of algebraic varieties over K; almost everything we will say remains true in the other situations we mentioned, but this case is the simplest to explain. So let V be a fixed varieity over K. If L is an extension of K, we define the automorphism group for V over L, Aut_{L}(V), to be the group of automorphisms of V ×_{Spec K} Spec L as an algebraic variety over L. Thus Aut_{L}(V) is just the set of isomorphisms

φ : V ×_{Spec K} Spec L → V ×_{Spec K} Spec L.

(Viewing Aut(V) as a function of L one obtains a presheaf for the étale topology on Spec K; in fact, it is actually a sheaf. See [Mi-EC, Chapter 3, Section 4, pp. 134-135]. We will not really need this interpretation, however.)

The (not necessarily abelian) group Aut_{L}(V) has a natural right Gal(L/K)- action. Specifically, given φ ∈ Aut_{L}(V) and g ∈ Gal(L/K), we define φ^{g} ∈ Aut_{L}(V) by the following commutative diagram of isomorphisms:

\xymatrix{V ×_{Spec K} Spec L \ar[d]^{1×g} \ar[r]^{φ} & V ×_{Spec K} Spec L \ar[d]^{1×g} \\ V ×_{Spec K} Spec L \ar@{-->}[r]^{φ^{g}} & V ×_{Spec K} Spec L}

That is, φ^{g} is the inverse of the isomorphism 1 × g followed by φ followed by 1 × g. Here by 1 × g we mean the map on V ×_{Spec K} Spec L making the following diagram commute:

\xymatrix{ & V \ar[r]^{1} & V \ar[dr] \\ V ×_{Spec K} Spec L \ar@{-->}[r]^{1×g} \ar[ur]^{π_{V}} \ar[dr]^{π_{L}} & V ×_{Spec K} Spec L \ar[ur]^{π_{V}} \ar[dr]^{π_{L}} &  & Spec K \\  & Spec L \ar[r]^{g} & Spec L \ar[ur]}

It is easy to check that this is really a group action, so that we have realized Gal(L/K) as a group of automorphisms of Aut_{L}(V). An important fact is that the subgroup of Aut_{L}(V) of automorphisms fixed by every g ∈ Gal(L/K) can be canonically identified with Aut_{K}(V). If one is willing to work simply with equations this is just standard Galois theory, but the full statement is really that the étale presheaf defined above is actually a sheaf.

If Aut_{L}(V) were abelian it would thus be a Gal(L/K)-module in our usual sense. For the next few sections we will amend the definition of Gal(L/K)-module to include modules whose underlying groups are non-abelian, so that any Aut_{L}(V) is a Gal(L/K)-module. However, to make the context in which Aut_{L}(V) is being used clear, we will write \underline{Aut}_{L}(V) for it when we want to regard it as a Gal(L/K)- module.

As an example, note that \underline{Aut}_{L}(Spec K) is canonically just Gal(L/K), the Gal(L/K)-action being conjugation.

Let us now generalize the above situation somewhat. We continue to let V be an algebraic variety over K, and we now fix a finite extension L/K and an L/K form W of V. We define Isom_{L}(W, V) to be the set of isomorphisms from W to V defined over L; that is, the set of isomorphisms

W ×_{Spec K} Spec L → V ×_{Spec K} Spec L.

First note that Isom_{L}(W, V) is non-empty, since W is an L/K form for V. It has a natural Aut_{L}(V) action given by postcomposition: if ψ ∈ Isom_{L}(W, V) and φ ∈ Aut_{L}(V), then we define φψ to simply be the composition

W ×_{Spec K} Spec L →^{ψ} V ×_{Spec K} Spec L →^{φ} V ×_{Spec K} Spec L.

It is easy to see that this action exhibits Isom_{L}(W, V) as a principal homogeneous space for Aut_{L}(V); that is, it is an Aut_{L}(V)-torsor.

Isom_{L}(W, V) in fact has even more structure. It has a natural Gal(L/K)- action: if φ ∈ Isom_{L}(W, V) and g ∈ Gal(L/K), we define φ^{g} ∈ Isom_{L}(W, V) to be the composition

\xymatrix{W ×_{Spec K} Spec L \ar[d]^{1×g} \ar[r]^{φ} & V ×_{Spec K} Spec L \ar[d]^{1×g} \\ W ×_{Spec K} Spec L \ar@{-->}[r]^{φ^{g}} & V ×_{Spec K} Spec L}

We will write \underline{Isom}_{L}(W, V) when we want to emphasize that Isom_{L}(W, V) should be thought of with its Gal(L/K)-action. In fact, the Gal(L/K)-actions on Isom_{L}(W, V) and \underline{Aut}_{L}(V) are compatible with the action of \underline{Aut}_{L}(V) on \underline{Isom}_{L}(W, V). We will make this notion precise in the next section.

2. Cohomological Interpretations

2.1. G-action Torsors. In this section we wish to formalize the sort of situation studied in the previous section. Let G be a group and let A be a finite group on which G acts. Let X be a set on which both G and A act. We will say that X is a G-action torsor for A if it is a torsor for A in the usual sense and if the A-action is compatible with the G-structures. That is, there is a map

A × X ⟶^{α} X

exhibiting X as a principal homogeneous space for A and such that

α(a^{g}, x^{g}) = α(a, x)^{g}

for any a ∈ A, x ∈ X and g ∈ G.

As a first example, it is easy to see that the constructions of the previous section exhibit \underline{Isom}_{L}(W, V) as a Gal(L/K)-action torsor for \underline{Aut}_{L}(V), where V is an algebraic variety over K and W is an L/K form for V.

If G is a group and A is another group on which G acts, we define T (G, A) to be the set of isomorphism classes of G-action torsors for A. T (G, A) is actually a pointed set, with the trivial torsor A as the distinguished element.

The fundamental classification theorem for G-action torsors is the following cohomological expression, which we have essentially seen before.

Theorem 2.1. If G is a group and A is a finite group with a G-action, then there is a canonical isomorphism of pointed sets

T (G, A) ≅ H^{1}(G, A).

We state this for non-abelian A, even though we have not yet defined H^{1}(G, A) when A is non-abelian, because it seems unnecessarily restrictive to assume that A is abelian. In fact, the definition of H^{1} for non-abelian modules is essentially exactly what makes the proof of the theorem work. We recall it briefly here; for more details on non-abelian cohomology, see [Se-LF, Appendix to Chapter 7] or [Se-GC, Chapter 1, Section 5] (although the reader should be forewarned that in the second source Serre appears to change his conventions halfway through).

To define non-abelian cohomology we work with cocycles and we must be careful as to what order operations occur in. All actions are left actions, and we will write all G-actions as exponentials on the left. We define the set CrossHom(G, A) to be the pointed set of maps

φ : G → A

such that

φ(g_{1}g_{2}) = φ(g_{1}) ^{g_{1}}φ(g_{2})

for all g_{1}, g_{2} ∈ G. (We write A multiplicatively, of course.) The distinguished element is the trivial map sending all of G to 1 ∈ A.

CrossHom(G, A) has a natural left A-action: for any a ∈ A and φ ∈ CrossHom(G, A) we define a ∗ φ ∈ CrossHom(G, A) by

a ∗ φ(g) = aφ(g) ^{g}(a^{−1}).

(Of course one must check that a ∗ φ is still a crossed homomorphism; this is an easy computation. Note also that this action does not preserve the base point of CrossHom(G, A).) We define H^{1}(G, A) to be the set of orbits of CrossHom(G, A) under this action; it is a pointed set with the distinguished element given by the class of the trivial crossed homomorphism. One checks immediately that if A is abelian this yields the usual definition of H^{1}(G, A).

We now give the proof of Theorem 2.1.

Proof. Let (X, α) be a G-action torsor for A. Choose any x ∈ X. We will use x to define a cocycle

c_{x} : G → A.

Specifically, we define c_{x}(g) to be the unique element of A such that

α(c_{x}(g),^{g}x) = x.

c_{x}(g) exists and is unique since α is a principal homogeneous action.

We check that c_{x} really is a cocycle. For g_{1}, g_{2} ∈ G,

α(c_{x}(g_{2}),^{g_{2}}x) = x
g_{1}α(c_{x}(g_{2}),^{g_{2}}x) = ^{g_{1}}x
α(^{g_{1}}c_{x}(g_{2}),^{g_{1}g_{2}}x) = ^{g_{2}}x
α(c_{x}(g_{1}), α(^{g_{1}}c_{x}(g_{2}),^{g_{1}g_{2}}x)) = α(c_{x}(g_{1}),^{g_{2}}x)
α(c_{x}(g_{1}) ^{g_{1}}c_{x}(g_{2}),^{g_{1}g_{2}}x) = x.

On the other hand, by definition

α(c_{x}(g_{1}g_{2}),^{g_{1}g_{2}}x) = x,

so

c_{x}(g_{1}g_{2}) = c_{x}(g_{1}) ^{g_{1}}c_{x}(g_{2})

which shows that c_{x} is a cocycle.

We now must check that the cohomology class of c_{x} is independent of the choice of x ∈ X. So choose any other y ∈ X. Then there is a unique a ∈ A such that

y = α(a, x).

We have

α(c_{y}(g),^{g}y) = y
α(c_{y}(g), α(^{g}a,^{g}x)) = α(a, x)
α(c_{y}(g) ^{g}a,^{g}x) = α(a, x)
α(a^{−1}c_{y}(g) ^{g}a,^{g}x) = x.

Thus

c_{x}(g) = a^{−1}c_{y}(g) ^{g}a

which shows that c_{x} and c_{y} differ by a coboundary. This shows that we have a well-defined map

T (L/K, Aut_{L}(V)) → H^{1}(Gal(L/K), Aut_{L}(V)).

We now define the inverse map. Given any cocycle φ : G → A, we will define a corresponding G-action torsor. In order to get the notation straight we define a new set P_{φ} to be identical to A as a set; we let

p : A → P_{φ}

be the natural map. We let b ∈ A act on p(a) ∈ P_{φ} by

bp(a) = p(ba).

This clearly makes P into a principal homogeneous space for A. We put a twisted G-action on P by

gp(a) = p(^{g}aφ(g)^{−1}).

We leave it to the reader to check that this really is a G-action torsor; it is not difficult. To show that the isomorphism class of P_{φ} depends only on the cohomology class of φ, one shows that if ψ(g) = bφ(g) ^{g}(b^{−1}), then the map

P_{ψ} → P_{φ}

sending p_{φ}(a) to p_{φ}(ab) is an isomorphism of G-action torsors. We also leave it to the reader to check that the maps are inverses.

2.2. Descent. We now apply the formalism of the previous section to the particular case of interest. We again let V be an algebraic variety defined over K and let W be an L/K form for V. Recall that \underline{Isom}_{L}(W, V) is a Gal(L/K)-action torsor for \underline{Aut}_{L}(V).

This fact can be summarized by saying that we have defined a map

E(L/K, V) ⟶^{ι} T (Gal(L/K), \underline{Aut}_{L}(V))

given by

ι(W) = \underline{Isom}_{L}(W, V).

We claim that ι is actually injective. To show this, suppose that we have an L/K form W for V and that ι(W) is isomorphic to \underline{Aut}_{L}(V) as an \underline{Aut}_{L}(V) torsor. Let φ be the image in ι(W) of 1 ∈ \underline{Aut}_{L}(V). A priori φ is a map W → V defined over L. In fact, since the Gal(L/K)-actions are compatible and Gal(L/K) acts trivially on 1, it also acts trivially on φ. This implies that φ is defined over K, and thus that W is the trivial L/K form. Of course, since neither the domain nor the range of ι need be a group this does not suffice to prove injectivity; however, the general case is quite similar.

The map ι is sometimes but not always surjective. In any case, we have the following fact.

Proposition 2.2. There is a canonical injection

E(L/K, V) ↪ H^{1}(Gal(L/K), \underline{Aut}_{L}(V))

and it is an isomorphism if and only if the natural injection

E(L/K, V) ↪ T (L/K, \underline{Aut}_{L}(V))

is an isomorphism.

Proof. This follows immediately from Theorem 2.1 and the above discussion.

If the injection of Proposition 2.2 is actually an isomorphism we will say that we have descent for V. A fundamental theorem due to Weil is that if V is a projective variety then we do have descent.

Theorem 2.3 (Weil). Let V be a projective variety. Then there is a canonical isomorphism of sets

E(L/K, V) ⟶^{≅} H^{1}(Gal(L/K), Aut_{L}(V)).

Proof. We sketch the main ideas. For the complete proof in the case of curves, see [Si-AEC, Chapter 10, Section 2, Theorem 2.2]. The general case is considerably harder.

Given any cocycle c : Gal(L/K) → Aut_{L}(V), we must define a twist V^{c} of V such that V^{c} becomes isomorphic to V over L and such that V^{c} gives the correct cohomology class.

To do this, consider V ×_{Spec K} Spec L. We define a Gal(L/K) action on this by first acting on Spec L via g in the usual way, and then acting on the “new” V ×_{Spec K} Spec L by c(g). V^{c} is then the quotient of V ×_{Spec K} Spec L by this Gal(L/K)-action. The hard part, of course, is to show that this quotient can be given the structure of an algebraic variety.

There is a rather peculiar situation that arises when one has descent, in that E(L/K, V) depends only on Aut_{L}(V) and not at all on the actual structures involved. This can be exploited in some circumstances. For example, the octonians and G_{2} have the same automorphism groups, so given any twist for one structure we ought to be able to obtain a corresponding twist for the other. Another example is central simple algebras and Brauer-Severi varieties; see [Se-LF, Chapter 10, Section 6].

As a final example, suppose that Aut_{L}(V) is cyclic of order 2. (For example, V could be a general hyperelliptic curve.) A group of order 2 must have trivial Galois action, so E(L/K, V) is independent of the variety V! In particular, it is simply

H^{1}(Gal(L/K), ±1) = Hom(Gal(L/K), ±1).