THE GROTHENDIECK RING OF VARIETIES IS NOT A DOMAIN

BJORN POONEN

Abstract. If k is a field, the ring K_{0}(V_{k}) is defined as the free abelian group generated by the isomorphism classes of geometrically reduced k-varieties modulo the set of relations of the form [X − Y] = [X] − [Y] whenever Y is a closed subvariety of X. The multiplication is defined using the product operation on varieties. We prove that if the characteristic of k is zero, then K_{0}(V_{k}) is not a domain.

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.

Date: May 27, 2002.

1991 Mathematics Subject Classification. Primary 14A10; Secondary 14G35.

Key words and phrases. Grothendieck ring of varieties, modular abelian variety, stable birational equivalence, Albanese variety.

This research was supported by NSF grant DMS-9801104, and a Packard Fellowship. This paper has appeared in Math. Res. Letters 9 (2002), no. 4, 493–498.

Let A be a simple abelian variety over ℚ. Let E = End_{ℚ}(A) ⊗ ℚ. Since A is simple, E is a division algebra. The Lie algebra Lie A is a nonzero left E-vector space, so [E : ℚ] ≤ dim_{ℚ} Lie A = dim A. If equality holds and E is commutative (hence a number field), 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 V_{ℓ}A can be viewed as a representation ρ_{ℓ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(E ⊗ ℚ_{ℓ}).)

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

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, in Theorem 1 of [Shi73], attached to each weight-2 newform f on Γ_{1}(N) an abelian variety quotient A_{f} of J_{1}(N). (Previously, in Theorem 7.14 of [Shi71], he had attached to f an abelian subvariety of J_{1}(N).) Let E_{f} be the number field generated over ℚ by the Fourier coefficients of f. Theorem 1 of [Shi73] shows also that dim A_{f} = [E_{f} : ℚ], and that there is an injective ℚ-algebra homomorphism θ : E_{f} ↪ E := End_{ℚ}(A_{f}) ⊗ ℚ mapping each Fourier coefficient to the endomorphism of A_{f} induced by the associated Hecke correspondence on X_{1}(N). Corollary 4.2 of [Rib80] proves that θ is an isomorphism. It follows that A_{f} is of GL_{2}-type.

Conversely, it is conjectured that each simple abelian variety over ℚ of GL_{2}-type is ℚisogenous 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}(590) (hence also on Γ_{1}(590)) such that E_{f} = ℚ(\sqrt{10}) and a_{3} = \sqrt{10}. Let A = A_{f} be the corresponding abelian variety over ℚ. Then dim A = [E_{f} : ℚ] = 2. But End_{ℚ}(A) is an order of E = E_{f} containing a_{3} = \sqrt{10}, so End_{ℚ}(A) is the maximal order ℤ[\sqrt{10}] of E. Since 590 is squarefree, A is semistable over ℚ by Theorem 6.9 of [DR73], and then Corollary 1.4(a) of [Rib75] shows that all endomorphisms of A over any field extension k of ℚ are defined over ℚ. Finally, the class number of ℤ[\sqrt{10}] is 2. □

Remarks.

(1) After one knows that End_{ℚ}(A) = ℤ[\sqrt{10}], another way to prove End\overline{ℚ}(A) = ℤ[\sqrt{10}] is to use the fact that End\overline{ℚ}(A) injects into the endomorphism ring of the reduction A_{p} over \overline{𝔽}_{p} for any prime p not dividing 590. The latter endomorphism rings can be computed using Eichler-Shimura theory and Honda-Tate theory. Combining the information from a few primes p yields the result.

(2) The smallest N for which there exists a newform f on Γ_{0}(N) with E_{f} of class number 2 is 276. The advantage of 590 is that it is squarefree. (In fact, our original proof applied the technique in the previous remark at level 276.)

(3) 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. Stable birational classes and Albanese 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 Ken Ribet for several comments regarding Section 2, and in particular for suggesting a less computational proof of Lemma 2. I thank also Eduard Looijenga and Arthur Ogus for discussions. The tables [Ste] were developed using MAGMA, C++, LiDIA, and GP-PARI. The package GP-PARI was used also to search the data from [Ste] for a newform f suitable for the proof of Lemma 2.

References

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

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