THE GROTHENDIECK RING OF VARIETIES IS NOT A DOMAIN

BJORN POONEN

1. The Grothendieck ring of varieties

Let k be a field. By a k-variety we mean a geometrically reduced, separated scheme of finite type over k. Let V_{k} denote the category of k-varieties. Let K_{0}(V_{k}) denote the free abelian group generated by the isomorphism classes of k-varieties, modulo all relations of the form [X − Y] = [X] − [Y] where Y is a closed k-subvariety of a k-variety X. Here, and from now on, [X] denotes the class of X in K_{0}(V_{k}). The operation [X] · [Y] := [X ×_{k} Y] is well-defined, and makes K_{0}(V_{k}) a commutative ring with 1. It is known as the Grothendieck ring of k-varieties. A completed localization of K_{0}(V_{k}) is needed for the theory of motivic integration, which has many applications: see [Loo00] for a survey.

Our main result is the following. Theorem 1. Suppose that k is a field of characteristic zero. Then K_{0}(V_{k}) is not a domain. Remark. We conjecture that the result holds also for fields k of characteristic p. But we use a result whose proof relies on resolution of singularities and weak factorization of birational maps, which are known only in characteristic zero.

2. Abelian varieties of GL_{2}-type

If A is an abelian variety over a field k_{0}, and k is a field extension of k_{0}, then End_{k}(A) denotes the endomorphism ring of the base extension A_{k} := A ×_{k_{0}} k, that is, the ring of endomorphisms defined over k. Lemma 2. Let k be a field of characteristic zero, and let \overline{k} denote an algebraic closure. There exists an abelian variety A over k such that End_{k}(A) = End\overline{k}(A) ≃ O, where O is the ring of integers of a number field of class number 2.

Let us precede the proof of Lemma 2 with a few paragraphs of motivation. Our strategy will be to find a single abelian variety A over ℚ such that the base extension A_{k} works over k.

Suppose that A is a nonzero abelian variety over ℚ. Let Lie A be its Lie algebra, which is a ℚ-vector space of dimension dim A. If End_{ℚ}(A) is an order in a number field F, then the O-action makes Lie A a vector space over O ⊗ ℚ = F; hence [F : ℚ] ≤ dim_{ℚ} Lie A = dim A. If moreover equality holds, then A is said to be of GL_{2}-type. (The terminology is due to the following: If A is of GL_{2}-type, then the action of the Galois group Gal(\overline{ℚ}/ℚ) on a Tate module T_{ℓ}A can be viewed as a representation ρ_{ℓ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(O ⊗_{ℤ} ℤ_{ℓ}).)

Because ℚ has class number 1, we must take [F : ℚ] ≥ 2 to find an A over ℚ as in Lemma 2. The inequality [F : ℚ] ≤ dim A then forces dim A ≥ 2. Moreover, if we want dim A = 2, then A must be of GL_{2}-type.

Date: April 15, 2002.

This research was supported by NSF grant DMS-9801104, and a Packard Fellowship.

Abelian varieties of GL_{2}-type are closely connected to modular forms. For each N ≥ 1, let Γ_{1}(N) denote the classical modular group, let X_{1}(N) denote the corresponding modular curve over ℚ, and let J_{1}(N) be the Jacobian of X_{1}(N). G. Shimura [Shi71, Theorem 7.14],

[Shi73] attached to each weight-2 newform f on Γ_{1}(N) an abelian variety quotient A_{f} of J_{1}(N). It is known that dim A_{f} = [F : ℚ], where F is the number field generated over ℚ by the Fourier coefficients of f. These coefficients can also be identified with the endomorphisms of A_{f} induced by the Hecke correspondences on X_{1}(N); hence A_{f} is GL_{2}-type. Conversely, it is conjectured that each abelian variety of GL_{2}-type is ℚ-isogenous to some A_{f}. See [Rib92] for more details. The dim A = 1 case of this conjecture is the statement that elliptic curves over ℚ are modular, which is known [BCDT01].

Therefore we are led to consider A_{f} of dimension 2, where f is a newform as above.

Proof of Lemma 2. Tables [Ste] show that there exists a weight-2 newform f = ∑_{n=1}^{∞} a_{n}q^{n} on Γ_{0}(276) (hence also on Γ_{1}(276)) such that ℚ(\{a_{n} : n ≥ 1\}) = ℚ(\sqrt{10}), a_{17} = 4 − \sqrt{10}, and a_{19} = 2 + \sqrt{10}. Let A = A_{f} be the corresponding abelian variety over ℚ. Then dim A = [ℚ(\sqrt{10}) : ℚ] = 2. Also, End_{ℚ}(A) is an order of ℚ(\sqrt{10}) containing 4 − \sqrt{10}, so End_{ℚ}(A) is the maximal order ℤ[\sqrt{10}] of ℚ(\sqrt{10}). The class number of ℚ(\sqrt{10}) is 2.

It remains to show that End_{k}(A) = ℤ[\sqrt{10}] for any field extension k of ℚ. For any place of k at which A has good reduction, End_{k}(A) injects into the endomorphism ring of the reduction. We will use this to bound End_{k}(A). The abelian variety A has good reduction at all primes not dividing 276, so in particular it has good reduction at 17 and 19. Let A_{17} and A_{19} denote the resulting abelian varieties over 𝔽_{17} and 𝔽_{19}. The places 17 and 19 of ℚ extend to places of k taking values in \overline{𝔽}_{17} and \overline{𝔽}_{19}. Thus End_{k}(A) injects into End_{\overline{𝔽}_{17}}(A_{17}) and End_{\overline{𝔽}_{19}}(A_{19}).

By the work of Eichler and Shimura (see Theorem 4 in D. Rohrlich’s article in [CSS97]), the characteristic polynomial P_{17}(x) of Frobenius on A_{17} equals

N_{ℚ(\sqrt{10})/ℚ}(x^{2} − a_{17}x + 17) = x^{4} − 8x^{3} + 40x^{2} − 136x + 289.

This is irreducible over ℚ, and its middle coefficient is prime to 17, so A_{17} is a simple ordinary abelian surface. Checking the criterion in [HZ02] (see especially Theorem 6 and the last paragraph of Section 2), we find that End_{\overline{𝔽}_{17}}(A_{17}) ⊗ ℚ ≃ ℚ[x]/(P_{17}(x)). Similarly, End_{\overline{𝔽}_{19}}(A_{19}) ⊗ ℚ ≃ ℚ[x]/(P_{19}(x)). The ratio of the discriminants of P_{17}(x) and P_{19}(x) is not a square in ℚ, so ℚ[x]/(P_{17}(x)) and ℚ[x]/(P_{19}(x)) are distinct number fields of degree 4. But End_{k}(A) ⊗ ℚ embeds into both, so dim_{ℚ}(End_{k}(A) ⊗ ℚ) ≤ 2. On the other hand, ℤ[\sqrt{10}] ⊆ End_{k}(A), so End_{k}(A) = ℤ[\sqrt{10}]. □

Remark. The case k = ℂ of Lemma 2 has an easy proof: let A be an elliptic curve over ℂ with complex multiplication by ℤ[\sqrt{−5}].

3. Abelian varieties and projective modules

Let A be an abelian variety over a field k, and let O = End_{k}(A). Given a finite-rank projective right O-module M, we define an abelian variety M ⊗_{O} A as follows: choose a finite presentation O^{m} → O^{n} → M → 0, and let M ⊗_{O} A be the cokernel of the homomorphism A^{m} → A^{n} defined by the matrix that gives O^{m} → O^{n}. It is straightforward to check that this is independent of the presentation, and that M ↦ (M ⊗_{O} A) defines a fully faithful functor T from the category of finite-rank projective right O-modules to the category of abelian varieties over k. (Essentially the same construction is discussed in the appendix by J.-P. Serre in [Lau01].)

Lemma 3. Let k be a field of characteristic zero. There exist abelian varieties A and B over k such that A × A ≃ B × B but A\overline{k} ≄ B\overline{k}.

Proof. Let A and O be as in Lemma 2. Let I be a nonprincipal ideal of O. Since O is a Dedekind domain, the isomorphism type of a direct sum of fractional ideals I_{1} ⊕ … ⊕ I_{n} is determined exactly by the nonnegative integer n and the product of the classes of the I_{i} in the class group Pic(O). Since Pic(O) ≃ ℤ/2, we have O ⊕ O ≃ I ⊕ I as O-modules. Applying the functor T yields A × A ≃ B × B, where B := I ⊗_{O} A. Since End\overline{k}(A) also equals O, we have B\overline{k} = I ⊗_{O} A\overline{k}. Since T for \overline{k} is fully faithful, A\overline{k} ≄ B\overline{k}. □

4. Rings related to the Grothendieck ring of varieties

For any extension of fields k ⊆ k′, there is a ring homomorphism K_{0}(V_{k}) → K_{0}(V_{k′}) mapping [X] to [X_{k′}].

Let k be a field of characteristic zero. Smooth, projective, geometrically integral k-varieties X and Y are called stably birational if X × ℙ^{m} is birational to Y × ℙ^{n} for some integers m, n ≥ 0. The set SB_{k} of equivalence classes of this relation is a monoid under product of varieties over k. Let ℤ[SB_{k}] denote the corresponding monoid ring.

When k = ℂ, there is a unique ring homomorphism K_{0}(V_{k}) → ℤ[SB_{k}] mapping the class of any smooth projective integral variety to its stable birational class [LL01]. (In fact, this homomorphism is surjective, and its kernel is the ideal generated by 𝕃 := [𝔸^{1}].) The proof in [LL01] requires resolution of singularities and weak factorization of birational maps [AKMW00, Theorem 0.1.1], [Wło01, Conjecture 0.0.1]. The same proof works over any algebraically closed field of characteristic zero.

The set AV_{k} of isomorphism classes of abelian varieties over k is a monoid. The Albanese functor mapping a smooth, projective, geometrically integral variety to its Albanese variety induces a homomorphism of monoids SB_{k} → AV_{k}, since the Albanese variety is a birational invariant, since formation of the Albanese variety commutes with products, and since the Albanese variety of ℙ^{n} is trivial. Therefore we obtain a ring homomorphism ℤ[SB_{k}] → ℤ[AV_{k}].

5. Zerodivisors

Proof of Theorem 1. Let A and B be as in Lemma 3. Then ([A] + [B])([A] − [B]) = 0 in K_{0}(V_{k}). On the other hand, [A] + [B] and [A] − [B] are nonzero, because their images under the composition

K_{0}(V_{k}) → K_{0}(V\overline{k}) → ℤ[SB\overline{k}] → ℤ[AV\overline{k}]

are nonzero. (The Albanese variety of an abelian variety is itself.) □

Acknowledgements

I thank Eduard Looijenga and Arthur Ogus for discussions. The package GP-PARI was used to perform the calculations in the last paragraph of the proof of Lemma 2.

References

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[Shi71] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971, Kanô Memorial Lectures, No. 1.

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[Ste]  William Stein, The Modular Forms Database, http://modular.fas.harvard.edu/Tables/index.html.

[Wło01] Jarosław Włodarczyk, Toroidal varieties and the weak factorization theorem, 22 June 2001, arXiv:math.AG/9904076.

Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

E-mail address: poonen@math.berkeley.edu