RATIONALITY OF ALGEBRAIC CYCLES OVER FUNCTION FIELD OF SL_{1}(A)-TORSORS

RAPHAËL FINO

Abstract. In this note we prove a result comparing rationality of algebraic cycles over the function field of a SL_{1}(A)-torsor for a central simple algebra A and over the base field. Keywords: Chow groups, central simple algebras, principal homogeneous spaces.

Contents

1. Introduction 1 2. Preliminaries 2 3. Proof of the result 4 4. Exceptional projective homogeneous varieties 5 References 6

1. Introduction

Let A be a central simple algebra over a field F and let Nrd : A^{×} → F^{×} be the reduced norm homomorphism . We recall that the homomorphism F^{×} → H^{1}(F, SL_{1}(A)), associating to c ∈ F^{×} the SL_{1}(A)-torsor X_{c} given by the equation Nrd = c, is surjective (with kernel Nrd(A^{×})) – see [7, Proposition 2.7.3] for instance.

The main purpose of this note is to prove the following theorem dealing with rationality of algebraic cycles over function field of SL_{1}(A)-torsors.

Theorem 1.1. Let A be a central simple algebra of prime degree p over a field F and let X be a SL_{1}(A)-torsor. Then

(i) for any equidimensional F -variety Y, the change of field homomorphism

CH(Y) → CH(Y_{F (X)}),

where CH is the integral Chow group, is surjective in codimension < p + 1. (ii) it is also surjective in codimension p+1 for a given Y provided that the variety X_{F (ζ)}

does not have any closed point of prime to p degree for each generic point ζ ∈ Y.

The method of proof mainly relies on the following statement. This proposition is a version of the result [3, Lemma 88.5] slightly altered to fit our situation (see also the proof of [8, Proposition 2.8]).

Date: 3 June 2015.

2010 Mathematics Subject Classification. 14C25; 20G15.

Proposition 1.2 (Karpenko, Merkurjev). Let X be a smooth variety, and Y an equidimensional variety. Given an integer m such that for any nonnegative integer i and any point y ∈ Y of codimension i the change of field homomorphism

CH^{m−i}(X) ⟶ CH^{m−i}(X_{F (y)})

is surjective, the change of field homomorphism

CH^{m}(Y) ⟶ CH^{m}(Y_{F (X)})

is also surjective.

The proof of Theorem 1.1 is given in Section 3. In Section 4, we describe how this theorem can be related to a similar result dealing with rationality of algebraic cycles over function field of projective homogeneous varieties under some groups of exceptional type.

Acknowledgements. This note has been conceived while I was visiting the University of Alberta and I would like to thank the Department of Mathematical and Statistical Sciences for the hospitality. I am very gratefull to Nikita Karpenko for suggestions having improved this note and to Philippe Gille for Remark 4.2. This work has been supported by the Université Pierre et Marie Curie - Paris VI.

2. Preliminaries

2.1. Topological filtration and Chow groups. For any smooth variety X over a field F (in this paper, an F -variety is a separated scheme of finite type over F), one can consider the topological filtration on the Grothendieck ring K_{0}(X), whose term of codimension i is given by

τ^{i}(X) = ⟨[O_{Z}] | Z ↪ X and codim(Z) ≥ i⟩,

where [O_{Z}] is the class in K_{0}(X) of the structure sheaf of a closed subvariety Z. We write τ^{i/i+1}(X) for the successive quotients. We denote by pr^{i} the canonical surjection

\begin{matrix}CH^{i}(X) & −↠ & τ^{i/i+1}(X) \\ [Z] & ⟼ & [O_{Z}] \end{matrix} ,

where CH is the integral Chow group. By the Riemann-Roch Theorem without denominators the i-th Chern class induces an homomorphism in the opposite way c_{i} : τ^{i/i+1}(X) → CH^{i}(X) such that the composition c_{i} ∘ pr is the multiplication by (−1)^{i−1}(i − 1)!.

Note that for any prime p, one can also consider the topological filtration τ_{p} on the ring K_{0}(X)/pK_{0}(X) by replacing K_{0}(X) by K_{0}(X)/pK_{0}(X) in the previous definition. In particular, we get that for any 0 ≤ i ≤ p, the map pr_{p}^{i} : Ch^{i}(X) ↠ τ_{p}^{i/i+1}(X), where Ch is the Chow group modulo p, is an isomorphism.

RATIONALITY OF CYCLES OVER FUNCTION FIELD OF SL_{1}(A)-TORSORS 3

Remark 2.1. Assume that X is a SL_{1}(A)-torsor and let p be a prime. One has K_{0}(X) = ℤ by the result [14, Theorem A] of I. Panin and consequently, for i ≥ 1, the term τ^{i}(X) is equal to zero. Therefore, for any 1 ≤ i ≤ p, one has Ch^{i}(X) = 0. Moreover, by the result [17, Theorem 2.7] of A. Suslin, one has CH^{i}(SL_{p}) = 0 for any i ≥ 1. Hence, for A of degree p (then there exists a splitting field of A of degree p), it follows by transfert argument that p · CH^{i}(X) = 0 for any i ≥ 1. Therefore, for X a SL_{1}(A)-torsor, with A of prime degree p, one has CH^{i}(X) = 0 for any 1 ≤ i ≤ p. Note that, by Proposition 1.2, this gives Theorem 1.1(i) already.

2.2. Brown-Gersten-Quillen spectral sequence. For any smooth variety X and any i ≥ 1, the epimorphism pr^{i} coincides with the edge homomorphism of the spectral Brown-Gersten-Quillen structure E_{2}^{i,−i}(X) ⇒ K_{0}(X) (see [16, §7]), that is to say

pr^{i} : CH^{i}(X) ≃ E_{2}^{i,−i}(X) ↠ ⋯ ↠ E_{i+1}^{i,−i}(X) = τ^{i/i+1}(X).

Assume that X is a SL_{1}(A)-torsor, with A of prime degree p. Then it follows from Remark 2.1 that E_{i−1}^{i,−i}(X) = 0 for 3 ≤ i ≤ p. Consequently, one has A^{1}(X, K_{2}) = E_{p}^{1,−2}(X).

Moreover, by the result [11, Theorem 3.4] of A. Merkurjev, for any smooth variety X, every prime divisor l of the order of the differential δ_{r} ending in E_{r}^{p+1,−p−1}(X) is such that l − 1 divides r − 1. Therefore, for any prime p and 2 ≤ r ≤ p − 1, the differential δ_{r} is of prime to p order. Assume furthermore that X is a SL_{1}(A)-torsor, with A of prime degree p. Since p · CH^{p+1}(X) = 0 (see Remark 2.1), one deduce that, for 2 ≤ r ≤ p − 1, the differential δ_{r} is trivial. Consequently, one has CH^{p+1}(X) = E_{p}^{p+1,−p−1}(X).

Therefore, for X a SL_{1}(A)-torsor, with A of prime degree p, the differential δ_{p} in the BGQ-structure is an homomorphism

δ : A^{1}(X, K_{2}) → CH^{p+1}(X).

Remark 2.2. Let X be a principal homogeneous space for a semisimple group G. By [6, Part II, Example 4.3.3 and Corollary 5.4], one has E_{2}^{0,−1}(X) = A^{0}(X, K_{1}) = F^{×} and the composition F^{×} = K_{1}(F) → K_{1}(X) → A^{0}(X, K_{1}) of the pullback of the structural morphism with the inclusions

K_{1}^{(0/1)}(X) = E_{∞}^{0,−1}(X) ⊂ ⋯ ⊂ E_{3}^{0,−1}(X) ⊂ E_{2}^{0,−1}(X)

given by the BGQ spectral sequence, is the identity. Therefore, for any i ≥ 2, the differential starting from E_{i}^{0,−1}(X) is zero, i.e for any i ≥ 2, one has

E_{i}^{i,−i}(X) = τ^{i/i+1}(X).

In particular, for X a SL_{1}(A)-torsor, with A of prime degree p, one has E_{p+1}^{p+1,−p−1}(X) = 0, i.e the differential δ : A^{1}(X, K_{2}) → CH^{p+1}(X) is surjective.

2.3. On the group A^{1}(X, K_{2}). The proof in the next section will use the work of A. Merkurjev on the Rost invariant of simply connected algebraic groups (see [6, Part II]). Let X be a SL_{1}(A)-torsor over F. The group A^{1}(X_{F (X)}, K_{2}) is infinite cyclic with generator q and isomorphic to A^{1}(SL_{n}, K_{2}) under restriction (where n = deg(A)). Furthermore, the restriction map r : A^{1}(X, K_{2}) → A^{1}(X_{F (X)}, K_{2}) is injective with finite cokernel of same order as the element R_{SL_{1}(A)}(X), where

R_{SL_{1}(A)} : H^{1}(F, SL_{1}(A)) → H^{3}(F, ℚ/ℤ(2))

is the Rost invariant of SL_{1}(A) (see [6, Theorem 9.10]). Moreover, the homomorphism R_{SL_{1}(A)} is of order exp(A) by [6, Theorem 11.5].

If char(F) = l is prime then the modulo l component H^{3}(F, ℤ/lℤ(2)) of the Galois cohomology group H^{3}(F, ℚ/ℤ(2)) is the group H_{l}^{3}(F) defined by K. Kato in [10] by means of logarithmic differential forms.

3. Proof of the result

In this section, we prove the result of this note.

Proof of Theorem 1.1. We use notations and materials introduced in the previous section. One can assume that X does not have any rational point over F (or equivalently X does not have any closed point of prime to p degree, by the result [1, Theorem 3.3] of J. Black), if else there is nothing to prove. Note that in this situation, the central simple algebra A is necessarily a division algebra. We recall that conclusion (i) has already been proved (see Remark 2.1). According to Proposition 1.2, it suffices to show that CH^{p+1}(X_{F (ζ)}) = 0 for each generic point ζ ∈ Y to get conclusion (ii). Since X_{F (ζ)} does not have any closed point of prime to p degree, it is enough to prove that CH^{p+1}(X) = 0.

Assume on the contrary that CH^{p+1}(X) ≠ 0. Then δ : A^{1}(X, K_{2}) → CH^{p+1}(X) is nonzero (since δ is surjective by Remark 2.2), i.e E_{p+1}^{1,−2}(X) is strictly included in E_{p}^{1,−2}(X) = A^{1}(X, K_{2}). We claim that this implies that, by denoting as q_{X} the generator of A^{1}(X, K_{2}), one has r(q_{X}) = q. Indeed, otherwise one has r(q_{X}) = p · q by §2.3. Consecutively, by denoting as c the corestriction morphism A^{1}(SL_{p}, K_{2}) → A^{1}(X, K_{2}), for any i ≥ 2, one has c(E_{i}^{1,−2}(SL_{p})) = c(A^{1}(SL_{p}, K_{2})) = A^{1}(X, K_{2}) (where the first identity is due to CH^{i}(SL_{p}) = 0 for any i ≥ 2). In particular, one has E_{p}^{1,−2}(X) = c(E_{p+1}^{1,−2}(SL_{p})) ⊂ E_{p+1}^{1,−2}(X), which is a contradiction.

Therefore, we have shown that under the assumption CH^{p+1}(X) ≠ 0, the generator q of A^{1}(X_{F (X)}, K_{2}) is rational. Then it follows that the generator g of CH^{p+1}(X_{F (X)}) is also rational.

However, since A_{F (X)} is still a division algebra (see [17, Corollary 6.5]), by [9, Theorem 7.2 and Theorem 8.2] the cycle g^{p−1} in CH_{0}(SL_{1}(A_{F (X)})) is nonzero and the latter group is cyclic of order p generated by the class of the identity of SL_{1}(A_{F (X)}). Thus, the degree of the rational cycle g^{p−1} is prime to p.

It follows that X has a closed point of prime to p degree, which is a contradiction. The Theorem is proved. □

Remark 3.1. Conclusion (i) of Theorem 1.1 holds for central simple algebras of p-primary degree (with the same proof). Over a field F of characteristic ≠ p, one can extend conclusion (ii) of Theorem 1.1 to central simple algebras A of p-primary degree and of index p because the kernel of the Rost invariant R_{SL_{1}(A)} is trivial by the result [12, Theorem 12.2] of A. Merkurjev and A. Suslin.

Remark 3.2. The end of the above proof shows in particular that for a division algebra A of prime degree p over a field F of arbitrary charateristic, the kernel of the Rost invariant R_{SL_{1}(A)} is trivial as well. Indeed, let ξ ∈ H^{1}(F, SL_{1}(A)) and let X be the associated SL_{1}(A)-torsor. Assume that R_{SL_{1}(A)}(ξ) is trivial. It follows then by §2.3 that the generator of A^{1}(X_{F (X)}, K_{2}) is rational. As we have seen in the above proof, this implies that X has a rational point over F, i.e the cocycle ξ is trivial.

Note also that for a division algebra A of prime degree p over a field F, the Rost invariant R_{SL_{1}(A)} coincides, up to sign, with the normalized invariant given by the cup product [A] ∪ (c) ∈ H^{3}(F, ℤ/pℤ(2)) for any class c Nrd(A^{×}), where [A] is the class of the algebra A in the Brauer group Br(F), see [6, §11].

4. Exceptional projective homogeneous varieties

In this section, we describe how Theorem 1.1 implies a similar version of it for projective homogeneous varieties under a group of type F_{4} or E_{8}. Namely, we give an alternative proof of Theorem 4.1 below. The following proof requires the characteristic of the base field to be different from p, with p = 3 when G is of type F_{4} and p = 5 when G is of type E_{8}, although the original result [4, Theorem 1.1] is valid for arbitrary characterisitic.

Let X be a nonsplit SL_{1}(A)-torsor over a field F, with A a division algebra of prime degree p. There exists a smooth compactification X̃ of X such that the Chow motive M(X̃, ℤ/pℤ) decomposes as a direct sum R_{p} ⊕ N, where R_{p} is the indecomposable Rost motive associated with the symbol [A] ∪ (c) ∈ H^{3}(F, ℤ/pℤ(2)), with c ∈ F^{×}\Nrd(A^{×}) giving X, see [9, Theorem 1.1]. Note that the projective variety X̃ is a norm variety of s.

Theorem 4.1. Let G be a linear algebraic group of type F_{4} or E_{8} over a field F of characteristic different from p, with p = 3 when G is of type F_{4} and p = 5 when G is of type E_{8}, and let X′ be a projective homogeneous G-variety. For any equidimensional variety Y, the change of field homomorphism

Ch(Y) → Ch(Y_{F (X′)}),

where Ch is the Chow group modulo p, is surjective in codimension < p + 1.

It is also surjective in codimension p+1 for a given Y provided that 1 ∉ deg Ch_{0}(X′_{F (ζ)}) for each generic point ζ ∈ Y.

Proof. Since the F -variety X′ is A-trivial in the sense of [8, Definition 2.3], one can assume that G has no splitting field of degree coprime to p. Indeed, otherwise 1 ∈ deg Ch_{0}(X′) by corestricition and this implies that Ch(Y) → Ch(Y_{F (X′)}) is an isomorphism in any codimension by A-triviality, see [8, Lemma 2.9].

Let us now write G = _{ξ}G_{0} for a nontrivial cocycle ξ ∈ H^{1}(F, G_{0}), with G_{0} a split group of the same type as G. Then the motive R_{p}(G) living on the Chow motive (with coefficients in ℤ/pℤ) of X′ given in [15, Theorem 5.17] is the Rost motive of the symbol R_{G_{0},p}(ξ) = [A] ∪ (c) ∈ H^{3}(F, ℤ/pℤ(2)), where R_{G_{0},p} is the the modulo p component of the Rost invariant R_{G_{0}}, A is a division algebra of degree p and c ∈ F^{×}\Nrd(A^{×}) – see [13, §4] and [5, §14] (here the assumption char(F) ≠ p is needed).

Let us denote as X the nonsplit SL_{1}(A)-torsor over F associated with c and as X̃ its smooth compactification. We claim that X′ has a closed point of prime to p degree over F (X̃) and vice versa.

Indeed, since X̃ is a norm variety for [A] ∪ (c), the motive R_{p}(G) decomposes as a sum of Tate motives over F (X̃). Therefore, the group G_{F (X̃)} is split by an extension of degree coprime to p and it follows that X′ has a closed point of prime to p degree over F (X̃) (this is more generally true for any extension L/F over which X̃ has a closed point of prime to p degree). Moreover, the motive R_{p}(G) decomposes as a sum of Tate motives over F (X′) because G is split by F (X′). Consequently, X̃ has a closed point of prime to p degree over F (X′).

It follows then (note that X̃ is A-trivial by [8, Example 5.7]) that the right and the bottom homomorphisms in the commutative square

\xymatrix{Ch(Y) \ar[r] \ar[d] & Ch(Y_{F (X′)}) \ar[d] \\ Ch(Y_{F (X̃)}) \ar[r] & Ch(Y_{F (X̃×X′)})}

are isomorphisms. Since F (X̃) = F (X), Theorem 4.1 is now a direct consequence of Theorem 1.1. □

The following was pointed out to me by Philippe Gille.

Remark 4.2. Let G_{0} a split group of type E_{8} over a 5-special field F (i.e F has no proper extension of degree coprime to 5) of characteristic ≠ 5. The above proof gives rise to a new argument for the triviality of the kernel of the Rost invariant modulo 5

H^{1}(F, G_{0}) → H^{3}(F, ℤ/5ℤ(2)).

This result is originally due to Vladimir Chernousov (under the assumption char(F) ≠ 2, 3, 5, see [2, Theorem]).

Indeed, since F is 5-special, for any nontrivial cocycle ξ ∈ H^{1}(F, G_{0}), the group _{ξ}G_{0} has no splitting field of degree coprime to 5. Then, as we have seen in the proof, there is a division algebra A of degree 5 such that R_{G_{0},5}(ξ) is equal to a symbol [A] ∪ (c) associated with a nonsplit SL_{1}(A)-torsor X. The injectivity of R_{G_{0},5} follows now from Remark 3.2.

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UPMC Sorbonne Universités, Institut de Mathématiques de Jussieu, Paris, FRANCE

  Web page: www.math.jussieu.fr/˜ fino
  E-mail address: raphael.fino at imj-prg.fr