ZERO-CYCLES ON ALGEBRAIC TORI

ALEXANDER S. MERKURJEV

1. The map φ_{T}

Let T be an algebraic torus over a field F and X a smooth compactification of T, i.e., a geometrically irreducible smooth complete variety containing T as an open set. The Chow group CH_{0}(X) of classes of zero dimensional cycles on X does not depend (up to canonical isomorphism) on the choice of X (cf. [7, 16.1.11], [4, Prop. 6.3], [8]).

Recall that two F -points t, t′ ∈ T (F) are called R-equivalent if there is rational morphism f : 𝔸^{1} −−→ T defined at 0 and 1 satisfying f(0) = t and f(1) = t′ (cf. [2, §4]). We write T (F)/R for the group of R-equivalence classes in T (F).

For an F -point t ∈ T (F) let [t] denote its class in CH_{0}(X). Consider the map from T (F) to CH_{0}(X) taking a point t to the class [t] − [1]. This map does not depend on the choice of X (up to canonical isomorphism) and it factors through R-equivalence. Indeed, a map f as above extends to a morphism g : ℙ^{1} → X and [t] = g_{∗}([0]) = g_{∗}([1]) = [t′], where g_{∗} : CH_{0}(ℙ^{1}) → CH_{0}(X) is the push-forward homomorphism (cf. [7, 1.4]).

We denote the resulting map by

φ_{T} : T (F)/R → CH_{0}(X).

Note that there is a homomorphism ψ_{T} : A_{0}(X) → T (F)/R such that ψ_{T} ∘ φ_{T} is the identity (cf. [2, Prop. 12]). It follows that the map φ_{T} is injective.

One can ask whether φ_{T} is a homomorphism. It is known that φ_{T} is a homomorphism for all tori T of dimension at most 3 (cf. [10]). In this note we shall give an example of a torus T such that φ_{T} is not a homomorphism although it has left inverse map ψ_{T} that is a homomorphism. It follows that φ_{T} is not surjective.

The map φ_{T} is a homomorphism if and only if for any two points t_{1} and t_{2} in T (F) one has

(1)  [t_{1}t_{2}] − [t_{1}] − [t_{2}] + [1] = 0

in CH_{0}(X).

Let T′ be another torus with a compactification X′. Then X ×X′ is a compactification of T × T′. Let t ∈ T (F) and t′ ∈ T′(F). The condition (1) for the elements t_{1} = (t, 1) and t_{2} = (1, t′) of (T × T′)(F) amounts to

(2)  ([t] − [1]) × ([t′] − [1]) = 0

in CH_{0}(X × X′), where × denotes the external product for Chow groups (cf. [7, 1.10]). In the next section we shall give examples of tori T and T′ such that the condition (2) fails for some t and t′. It would follows that φ_{T ×T′} is not a homomorphism.

Key words and phrases. 2000 Mathematical Subject Classifications:

The work has been supported by the NSF grant DMS #0652316.

2. The tori R_{L/F}^{1} (G_{m})

Let F be a field with char F ≠ 2. For an element a ∈ F^{×}, let F_{a} denote the quadratic (étale) F -algebra F [t]/(t^{2} − a).

Let a, b ∈ F^{×}. Consider the biquadratic F -algebra L = F_{a} ⊗ F_{b} and let G be the Galois group Gal(L/F). Write σ ∈ G for the generator of Gal(L/F_{a}) and τ ∈ G for the generator of Gal(L/F_{b}).

Let T be the torus R_{L/F}^{1} (G_{m}) of norm 1 elements of the extension L/F. For a field extension K/F, a point t of T (K) is an element t ∈ (K ⊗ L)^{×} satisfying N_{(K⊗L)/K}(t) := t · σ(t) · τ(t) · στ(t) = 1, where N_{(K⊗L)/K} : (K ⊗ L)^{×} → K^{×} is the norm homomorphism. The element N_{(K⊗L)/(K⊗F_{a})} = t · σ(t) in K ⊗ F_{a} has norm 1 in K. By Hilbert Theorem 90, applied to the quadratic extension (K ⊗ F_{a})/K, there is an element z ∈ (K ⊗F_{a})^{×} with t·σ(t) = z ·τ(z)^{−1}. Note that z is unique up to a multiple from K^{×}. Hence the norm N_{(K⊗F_{a})/K}(z) = z · τ(z) is unique up to a multiple from K^{×2}. It follows that the class q_{K}(t) quaternion algebra (z · τ(z), b)_{K} in the Brauer group Br(K) is well defined. Thus, we get a group homomorphism

q_{K} : T (K) → Br(K), t ↦ q_{K}(t).

The collection of the homomorphisms q_{K} over all field extensions K of F form a morphism q of functors T and Br from the category of all field extensions of F to the category of groups. In other words, q is an invariant of the algebraic torus T with values in the Brauer group (cf. [9]).

Remark 2.1. It is shown in [11, p. 427] that q_{F} induces an isomorphism between T (F)/R and the subgroup of Br(F) consisting of classes of algebras that are split over all three quadratic subalgebras of L.

Example 2.2. Assume that F contains a square root i of −1. Then we can view i as an element of T (F). We have i·σ(i) = −1 = z ·τ(z)^{−1} with z = \sqrt{a} in F_{a}. Hence q_{F} (i) is the class of the quaternion algebra (z · τ(z), b)_{F} ≃ (−a, b)_{F} ≃ (a, b)_{F}.

Let F (T) be the function field of T over F and let v be a discrete valuation on F (T) over F. The residue field F (v) is a field extension of F. By [5, §5], there is the residue homomorphism

∂_{v} : Br(L(T)/F (T)) → G^{∗},

where G^{∗} is the character group of G. An element α in Br(L(T)/F (T)) is called unramified with respect to v if ∂_{v}(α) = 0 and (totally) unramified if α is unramified with respect to every discrete valuation of F (T) over F.

Proposition 2.3. For any t ∈ T (F (T)), the element q_{F (T)}(t) in Br(L(T)/F (T)) is unramified.

Proof. Write K for F (T), so L(T) = K ⊗ L = KL. As the character group G^{∗} is of exponent 2, it suffices to show that q_{K}(t) is divisible by 2 in Br(KL/K). By Hilbert Theorem 90, there are elements z ∈ K_{a}^{×} and w ∈ K_{b}^{×} such that t · σ(t) = z · τ(z)^{−1} and t · τ(t) = w^{−1} · σ(w). Consider the cross product central simple K-algebra (cf. [6, §12]):

A = KL1 ⊕ KLu_{σ} ⊕ KLu_{τ} ⊕ KLu_{τ} u_{σ}

with multiplication table:

u_{σ}^{2} = z, u_{τ}^{2} = w, u_{σ}u_{τ} = tu_{τ} u_{σ}.

As KL is a maximal subalgebra of A, the Brauer class of A belongs to Br(KL/K).

The centralizer C of the quadratic subalgebra K_{a} ⊂ KL ⊂ A in A is generated by KL and u_{σ} and hence is isomorphic to the quaternion algebra (z, b)_{K_{a}}. It follows from [6, §7] that

[A ⊗_{K} K_{a}] = [(z, b)_{K_{a}}] in Br(KL/K_{a}),

hence

q_{K}(t) = [(z · σ(z), b)_{K}] = cor_{K_{a}/K}[(z, b)_{K_{a}}] = cor_{K_{a}/K}[A ⊗_{K} K_{a}] = 2[A]. □

We write α_{T} for the element q_{F (T)}(t) in Br(L(T)/F (T)), where t is the generic point in T (F (T)). As 2α_{T} = 0, we can view α_{T} as an element of the group H^{2}(F (T), ℤ/2ℤ) = Br_{2}(F (T)). By Proposition 2.3, α_{T} is an unramified element of H^{2}(F (T), ℤ/2ℤ) in the sense of [1] (cf. [10, 2.2]).

Remark 2.4. If L/F is a field extension, by [3, Prop. 9.5], the factor group of the group of unramified elements in Br(F (T)) modulo Br(F) is canonically isomorphic to H^{2}(G, T̂) ≃ H^{3}(G, ℤ) ≃ ℤ/2ℤ, where T̂ is the Galois module of characters of T. The class α_{T} corresponds to the only nontrivial element of the group H^{2}(G, T̂).

Choose a smooth compactifications X of T, so we can view α as an unramified element of H^{2}(F (X), ℤ/2ℤ). Let x ∈ X(F) be any point over F. We write α(x) ∈ H^{2}(F, ℤ/2ℤ) for the value of α at x (cf. [10, 2.1]). If x ∈ T (F), then α(x) = q_{F} (x). In particular, we have α(1) = 0 and α(i) = (a) ∪ (b) by Example 2.2 if F contains a square root i of −1.

Let L′ = F_{a′} ⊗ F_{b′} be another biquadratic F -algebra and T′ := R_{L′/F}^{1} (G_{m}) and let α_{T′} ∈ H^{2}(F (T′), ℤ/2ℤ) be the element as above. Choose also a smooth compactification X′ of T′. Restricting α and α′ to F (X × X′) and taking the cup-product, we get the unramified element

β = α × α′ ∈ H^{4}(F (X × X′), ℤ/2ℤ).

Let Z_{0}(X × X′) be the group of zero-dimensional cycles on X × X′. The map Z_{0}(X × X′) → H^{4}(F, ℤ/2ℤ) taking the class of a closed point z ∈ X × X′ to N_{F (z)/F} (β(z)) factors through a homomorphism

ρ : CH_{0}(X × X′) → H^{4}(F, ℤ/2ℤ)

(cf. [10, 2.4]). Note that for every t ∈ T (F) and t′ ∈ T′(F) we have

ρ([t] × [t′]) = β(t, t′) = α(t) ∪ α′(t′) ∈ H^{4}(F, ℤ/2ℤ).

It follows that

ρ(([t] − [1]) × ([t′] − [1])) = (α(t) − α(1)) ∪ (α′(t′) − α′(1)) = α(t) ∪ α′(t′)

in H^{4}(F, ℤ/2ℤ).

Assume that F contains a square root i of −1, so i ∈ T (F). We then have

ρ(([i] − [1]) × ([i] − [1])) = (a) ∪ (b) ∪ (a′) ∪ (b′) ∈ H^{4}(F, ℤ/2ℤ).

One can easily find a field F and elements a, b, a′, b′ with (a) ∪ (b) ∪ (a′) ∪ (b′) ≠ 0 in H^{4}(F, ℤ/2ℤ). For example, one can take F = k(a, b, a′, b′), where a, b, a′, b′ are variables over a field k. This contradicts (2). Hence φ_{T ×T′} is not a homomorphism.

References

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Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

   E-mail address: merkurev@math.ucla.edu