SUSLIN’S CONJECTURE ON THE REDUCED WHITEHEAD GROUP OF A SIMPLE ALGEBRA

ALEXANDER MERKURJEV

Abstract. In 1991, A. Suslin conjectured that if the index of a central simple algebra A is not square-free, then the reduced Whitehead group of A is nontrivial generically. We prove this conjecture in the present paper.

1. Introduction

Let A be a central simple algebra over a field F. The reduced norm homomorphism A^{×} → F^{×} yields a homomorphism

Nrd : K_{1}(A) → F^{×} = K_{1}(F).

The kernel SK_{1}(A) of Nrd is the reduced Whitehead group of A. Wang proved in [25] that if ind(A) is a square-free integer, then SK_{1}(A) = 0. He also proved that the reduced Whitehead group is always trivial if F is a number field. Platonov found examples of A with nontrivial SK_{1}(A) (see [17]).

In 1991, Suslin conjectured in [23] that if ind(A) is not square-free, then the reduced Whitehead group SK_{1}(A) of A is generically nontrivial, i.e., there is a field extension L/F such that SK_{1}(A ⊗_{F} L) ≠ 0.

Suslin’s Conjecture was proved in the case when ind(A) is divisible by 4 (see [13] and [15]).

In this paper we prove Suslin’s Conjecture (Theorem 8.1):

Theorem. Let A be a central simple F -algebra. If ind(A) is not square-free, then there is a field extension L/F such that SK_{1}(A ⊗_{F} L) ≠ 0.

Note that the group SK_{1}(A) coincides with the group of R-equivalence classes in the special linear group SL_{1}(A). In particular, generic non-triviality of the reduced Whitehead group of A implies that SL_{1}(A) is not a retract rational variety (Corollary 8.2).

2. Cycle modules and spectral sequences

Let Z be a variety over a field F and let M_{∗} be a cycle module over Z (see [20, §2]). This is a collection of group M_{n}(z) for n ∈ ℤ and a point z : L → Z over F having certain compatibility properties. We write K_{∗} for the cycle module over Spec F given by (Quillen’s) K-groups (see [20, Remark 2.5]).

Date: November, 2014.

Key words and phrases. simple algebras; reduced Whitehead group; algebraic K-theory; algebraic cycles. 2010 Mathematical Subject Classifications: 19B99, 14C25 .

The work has been supported by the NSF grant DMS #1160206 and the Guggenheim Fellowship. The author thanks the Max Planck Institute (Bonn) for hospitality.

For every integer r ≥ 0, denote by Z^{(r)} the set of points of Z of codimension r. We write A^{r}(Z, M_{n}) for the homology group of the complex [20, §5]

∐_{z∈Z^{(r−1)}} M_{n−r+1}F (z) ⟶^{∂} ∐_{z∈Z^{(r)}} M_{n−r}F (z) ⟶^{∂} ∐_{z∈Z^{(r+1)}} M_{n−r−1}F (z).

For example, A^{r}(Z, K_{r}) is the Chow group CH^{r}(Z) of classes of codimension r algebraic cycles on Z. If Z is smooth, A^{∗}(Z, K_{∗}) is a bi-graded commutative ring.

If f : Y → Z is a flat morphism of equidimensional varieties and M a cycle module over Y, for every n ∈ ℤ, there is a spectral sequence [20, Corollary 8.2]

E_{1}^{r,s}(f, n) = ∐_{z∈Z^{(r)}} A^{s}(f^{−1}(z), M_{n−r}) ⇒ A^{r+s}(Y, M_{n}),

where f^{−1}(z) is the fiber of f over z ∈ Z.

Very often we will be considering the projections f : W × Z → Z with Z and W smooth varieties. In this case f^{−1}(z) = W_{F (z)}. The associated spectral sequences have the following functorial properties. A morphism h : W → W′ of smooth varieties yields a pull-back morphism of spectral sequences

h^{∗} : E_{∗}^{∗,∗}(f′, n) → E_{∗}^{∗,∗}(f, n)

for every n (here f′ : W′×Z → Z is the projection). If h is a closed embedding of codimension c, we have a push-forward morphism of spectral sequences

h_{∗} : E_{∗}^{∗,∗}(f, n) → E_{∗}^{∗,∗+c}(f′, n + c).

More generally, every correspondence λ between W and W′ of degree d (see [3, §63]) yields a morphism

λ^{∗} : h^{∗} : E_{∗}^{∗,∗}(f′, n) → E_{∗}^{∗,∗−d}(f, n − d).

This is because the four basic maps of complexes of W ×Z and W′ ×Z respect the filtration when projected to Z (see [20, §3]).

3. Chern classes

Let X be a smooth variety. There are Chern classes (see [7]):

c_{i,n} : K_{n}(X) → A^{i−n}(X, K_{i})

for i ≥ n ≥ 0. We will only need the classes

c_{i} := c_{i+1,1} : K_{1}(X) → A^{i}(X, K_{i+1}).

There is the following product formula (see [21]):

Proposition 3.1. If x ∈ K_{0}(X) is the class of a line bundle L and y ∈ K_{1}(X), we have

c_{i}(xy) = ∑_{j=0}^{i}(−1)^{j}\binom{i}{j} h^{j} ∪ c_{i−j}(y),

where h is the first (classical) Chern class of L in A^{1}(X, K_{1}) = CH^{1}(X).

Let E → X be a vector bundle of rank n and SL(E) the group scheme over Z of determinant 1 automorphisms of E. We will be using the following result due to Suslin [22, Th. 4.2].

Proposition 3.2. If X is a smooth variety, the ring A^{∗}(SL(E), K_{∗}) is almost exterior algebra over A^{∗}(X, K_{∗}) with generators c_{1}(β), c_{2}(β), …, c_{n−1}(β), where β ∈ K_{1}(SL(E)) is the generic element. In particular,

CH(SL(E)) ≃ CH(X).

4. Severi-Brauer varieties

Let A be a central simple algebra of degree n over F, X = SB(A) the Severi-Brauer variety of rank n right ideals of A. If A is split, i.e., A = End(V) for a vector space of dimension n, the variety X is isomorphic to the projective space ℙ(V).

The variety X has a point over a field extension L/F if and only if A is split over L, i.e., A_{L} := A ⊗_{F} L ≃ M_{n}(L).

Write h for the class of a hyperplane section in CH^{1}(ℙ^{n−1}). The Chow group CH^{i}(ℙ^{n−1}) for i = 0, 1, …, n − 1 is infinite cyclic generated by h^{i}.

In the general case, the kernel of the degree homomorphism

deg : CH^{i}(X) → CH^{i}(X_{sep}) = ℤh^{i}

coincides with the torsion part of CH^{i}(X). The group CH_{0}(X) is torsion free (see [16] or [1, Corollary 7.3]). Therefore, the classes in CH_{0}(X) of every two points of the same degree are equal.

If A = M_{m}(B) for a central simple algebra B over F and S = SB(B), then S is a closed subvariety of X = SB(A). Moreover, the Chow motive M(X) of X is isomorphic to the direct sum M(S) ⊕ M(S){k} ⊕ ⋯ ⊕ M(S){(m − 1)k}, where k = n/m.

Let I → X be the tautological rank n vector bundle. The fiber of this bundle over a right ideal in A, a point of X, is the ideal itself. In the split case A = End(V), where V is a vector space of dimension n, a line l ⊂ V as a point of X = ℙ(V) corresponds to the right ideal Hom(V, l) = V^{∨} ⊗ l. Therefore, I = V^{∨} ⊗ L_{t}, where L_{t} is the tautological line bundle over ℙ(V). The canonical bundle J over X, the dual of I, is equal then to V ⊗ L_{c}, where L_{c} is the canonical line bundle, dual of L_{t}. We have in the split case

X × X = X × ℙ(V) = ℙ_{X}(V) = ℙ_{X}(V ⊗ L_{c}) = ℙ_{X}(J).

Note that the projective linear group PGL(V) acts on ℙ(V) and the vector bundles I and J. In the general case, twisting by the PGL(V)-torsor corresponding to the algebra A, we get an isomorphism

X × X ≃ ℙ_{X}(J),

i.e., X × X is a projective vector bundle of J over X (with respect to the first of the two projections q_{1}, q_{2} : X × X → X).

The tautological line bundle L_{t} over X × X = ℙ_{X}(J) is the sub-bundle q_{1}^{∗}(L_{t}) ⊗ q_{2}^{∗}(L_{c}) of the bundle q_{1}^{∗}(J) = V ⊗ q_{2}^{∗}(L_{c}) in the split case. Therefore,

(4.1)  L_{c} = q_{1}^{∗}(L_{c}) ⊗ q_{2}^{∗}(L_{t}),

where L_{c} is the canonical bundle over over X × X.

Lemma 4.2. Let x ∈ X be a closed point. Then the push-forward homomorphism ℤ = CH(X_{F (x)}) → CH(X × X) for the closed embedding

i : X_{F (x)} = X × Spec F (x) ↪ X × X

depends only on the degree of x.

Proof. The canonical line bundle L over the projective space is the pull-back of the canonical bundle L on X × X. Hence the class h_{1} = c_{1}(L) is equal to i^{∗}(h), where h = c_{1}(L). By the projection formula,

i_{∗}(h_{1}^{i}) = i_{∗}(i^{∗}(h^{i})) = i_{∗}(1) · h^{i} = [X_{F (x)}] · h^{i}.

The class of X_{F (x)} in CH(X × X) is the image of [X] × [x] under the exterior product map

CH(X) ⊗ CH_{0}(X) → CH(X × X).

Finally, the class of x in CH_{0}(X) depends only on the degree of x. □

Choose a splitting field extension L/F of the smallest degree ind(A). We have X_{L} ≃ ℙ_{L}^{n−1}. Let l_{i} ∈ CH_{i}(X_{L}) be the class of a projective linear subspace of dimension i and e_{i} = e_{i}(A) the image of l_{i} under the norm homomorphism

N_{L/F} : CH_{i}(X_{L}) → CH_{i}(X).

Then e_{i} is independent of the choice of L. Indeed, choose a closed point x ∈ X such that F (x) ≃ L. Then e_{i} is the image of l_{i} under the composition

CH_{i}(X_{L}) = CH_{i}(X_{F (x)}) → CH_{i}(X × X) → CH_{i}(X),

where the last map is induced by the first projection. By Lemma 4.2, the composition does not depend on the choice of x.

The proof of Lemma 4.2 shows that for every closed point x ∈ X, we have

(4.3)  N_{F (x)/F} (l_{i}) = \frac{deg(x)}{ind(A)} e_{i}(A).

Lemma 4.4. If K/F is a finite extension, then

N_{K/F} (e_{i}(A_{K})) = \frac{[K : F] ind(A_{K})}{ind(A)} e_{i}(A).

Proof. Let L/K be a splitting field of A_{K} of degree ind(A_{K}). Choose an Lpoint Spec(L) → X. Let {x} be the image of this morphism. We have by (4.3),

(4.5) N_{K/F} (e_{i}(A_{K})) = N_{L/F} (e_{i}(A_{L})) = [L : F (x)] · N_{F (x)/F} (e_{i}(A_{L}))

= \frac{[L : F]}{ind(A)} e_{i}(A) = \frac{[K : F] ind(A_{K})}{ind(A)} e_{i}(A). □

Proposition 4.6. Let p be a prime integer and A a central simple F -algebra of p-primary degree, X = SB(A) the Severi-Brauer variety of A. Then CH_{i}(X) = ℤe_{i} for i = 0, 1, …, p − 2. In particular, these groups have no torsion.

Proof. If D is a division algebra Brauer equivalent to A, the Severi-Brauer variety Y = SB(D) is a closed subscheme of X. The push-forward map CH_{i}(Y) → CH_{i}(X) is an isomorphism for i ≤ dim(Y) taking e_{i}(D) to e_{i}(A). Thus, in the proof of the proposition it suffices to assume that A is a division algebra.

We prove the proposition by induction on ind(A). The case ind(A) = p was considered in [12, Corollary 8.7.2]. A standard restriction-corestriction argument reduces the proof to the case when F is a p-special field, i.e., the degree of every finite field extension of F is a power of p.

Let A be a central division algebra of p-primary degree n and L ⊂ A a maximal subfield (of degree n over F). The torus T = R_{L/F} (𝔾_{m})/𝔾_{m} acts naturally on X making X a toric variety. Write U for the open T -invariant orbit and Z for X \ U. Thus, U is a T -torsor over Spec(F).

Conversely, let U be a T -torsor over Spec(F) and let A be a central simple algebra degree n over F with class in the relative Brauer group Br(L/F) = H^{1}(F, T) corresponding to the class of U. Then U is the open orbit of the T -action on SB(A).

In the split case, X = ℙ^{n−1} and T is the torus of invertible diagonal matrices modulo the scalar matrices. Then U consists of all points in ℙ^{n−1} with all coordinates ≠ 0. The T -orbits are the subsets in ℙ^{n−1} with zeros on the fixed set of coordinates.

Let Σ be the set of all n primitive idempotents of L⊗_{F} F_{sep} = F_{sep}×⋯×F_{sep}. Every σ ∈ Σ yields a co-character χ_{σ} : 𝔾_{m}_{,F_{sep}} → T_{sep} which belongs to an edge (1-dimensional cone) in the fan of the toric variety X_{sep}. Moreover, the correspondence σ ↦ χ_{σ} yields a bijection between the set of nonempty subsets in Σ and the set of cones in the fan (or the set of T -orbits in X_{sep}). The absolute Galois group Γ = Gal(F_{sep}/F) of F acts transitively on the set Σ.

Lemma 4.7. We have CH_{i}(U) = 0 for i = 0, 1, …, p − 2.

Proof. If ind(A) = p, every cycle c in CH_{i}(U) comes by restriction from CH_{i}(X) = pℤ and therefore, by the norm, comes from CH_{i}(X_{L}). Hence c comes by the norm from CH_{i}(U_{L}). But U_{L} ≃ T_{L}, hence CH_{i}(U_{L}) = 0.

In the general case, since F is a p-special field, there is a subfield K ⊂ L of degree p over F. Consider the subtorus S := R_{K/F} (𝔾_{m})/𝔾_{m} of T, the S-torsor

f : U → X := U/S

and Rost’s spectral sequence for f converging to CH_{i}(U). On the zero diagonal, we have the groups ∐_{x∈X_{(j)}} CH_{k}(f^{−1}(x)) with j + k = i. Note that f^{−1}(x) is an S-torsor over Spec F (x). Since k ≤ i ≤ p − 2, by the first part of the proof, CH_{k}(f^{−1}(x)) = 0. □

The T -orbits in Z_{sep} correspond to proper subsets of the set of Σ. No such subset is fixed by Γ, hence no orbit in Z_{sep} is fixed by Γ.

We have a sequence of closed T -invariant subsets

(4.8)  Z = Z_{0} ⊃ Z_{1} ⊃ ⋯ ⊃ Z_{m} ⊃ Z_{m+1} = ∅

such that every variety (Z_{j} \ Z_{j+1})_{sep} is the disjoint union of T -orbits of the same dimension which are permuted by Γ. It follows that each Z_{j} \ Z_{j+1} is a disjoint union of varieties defined over finite separable field extensions K/F corresponding to the stabilizers Γ′ ⊂ Γ of T -orbits. The group Γ′ does not act transitively on the set Σ, hence L ⊗_{F} K is not a field and therefore, A_{K} is not a division algebra, i.e., ind(A_{K}) < ind(A).

If W is a scheme over a finite separable field extension K/F, the norm map CH(W ⊗_{F} K) → CH(W) is surjective, since K is a direct factor of K ⊗_{F} K.

Fix an integer i = 0, 1, …, p − 2. We say that a variety W over F satisfies the condition (∗) if CH_{i}(W) is generated by the images of the norm maps CH_{i}(W ⊗_{F} K) → CH_{i}(W) over finite field extensions K/F with ind(A_{K}) < ind(A). We have proved that all the differences Z_{j} \ Z_{j+1} satisfy (∗).

Let W′ be a closed subvariety of W. The exactness of the localization sequence

CH_{i}(W′) → CH_{i}(W) → CH_{i}(W \ W′) → 0

shows that if W′ and W \ W′ satisfy (∗), then so does W. It follows from (4.8) that Z satisfies (∗). By Lemma 4.7, U satisfies (∗), hence so does X.

By the induction hypothesis, CH_{i}(X_{K}) for K as above, is generated by e_{i}(A_{K}). By Lemma 4.4, CH_{i}(X) is generated by e_{i}(A). □

Corollary 4.9. The degree map CH_{i}(X) → CH_{i}(X_{sep}) = ℤl_{i} is injective, it takes e_{i} to ind(A)l_{i}. Thus, CH_{i}(X) is identified with the subgroup ind(A)ℤl_{i} in ℤl_{i}.

By the Projective Bundle Theorem, for every j ≥ 0, we have

CH^{d−j}(X × X) ≃ CH^{d−j}(X) ⊕ CH^{d−j−1}(X)h ⊕ ⋯ ⊕ CH^{0}(X)h^{d−j},

where h ∈ CH^{1}(X × X) is the first Chern class of the canonical line bundle L_{c} over X×X. The element λ_{j} := h^{d−j} can be viewed as a degree j correspondence from X to itself and hence λ_{j} yields the homomorphism (see Section 2):

λ_{j}^{∗} : CH_{0}(X) → CH_{j}(X).

Lemma 4.10. The maps λ_{j}^{∗} are isomorphisms for j = 0, 1, …, p − 2, taking e_{0}(A) to e_{j}(A).

Proof. By (4.1), in the split case, h = h_{2} − h_{1}, where h_{i} are the pull-backs to X × X of the classes of the hyperplanes in X, hence λ_{j} = (h_{2} − h_{1})^{d−j}. Therefore, λ_{j}^{∗} takes the generator l_{0} of the infinite cyclic group CH_{0}(X) to the generator l_{j} of CH_{j}(X).

By Proposition 4.6, in the general case, the degree map CH_{j}(X) → CH(X_{sep}) = ℤl_{j} identifies the group CH_{j}(X) with ind(A)l_{j} by Corollary 4.9. The result follows. □

5. Two cycle modules

Let A be a central simple algebra over F. The first cycle module K_{∗}^{QA} is defined by

K_{n}^{QA}(L) = K_{n}(A_{L})

for a field extension L/F. The reduced norm map Nrd : K_{n}(A_{L}) → K_{n}(L) is defined for n = 0, 1, 2 (see [12, §6]).

Let G = SL_{1}(A) be the algebraic group of reduced norm 1 elements in A. There is a canonical isomorphism (see [6, Proposition 7.3])

A^{1}(G, K_{2}) ≃ ℤ.

The group A^{1}(G, K_{2}) does not change under field extensions.

In particular, we have a homomorphism

Nrd^{QA} : A^{1}(G, K_{2}^{QA}) → A^{1}(G, K_{2}) = ℤ.

Let X be the Severi-Brauer variety of A of dimension d. We will be using another cycle module K_{∗}^{A} over F defined by

K_{n}^{A}(L) = A^{d}(X_{L}, K_{d+n}).

The push-forward homomorphism for the morphism X_{L} → Spec(L) yields a map A^{d}(X_{L}, K_{d+n}) → K_{n}(L) and therefore, a morphism of cycle modules K_{∗}^{A} → K_{∗}. In particular, we have a homomorphism

Nrd^{A} : A^{1}(G, K_{2}^{A}) → A^{1}(G, K_{2}) = ℤ.

There is a natural homomorphism A^{d}(X_{L}, K_{d+n}) → K_{n}(A_{L}) which is an isomorphism for n = 0 and 1 (see [14]). Thus, we have a morphism of cycle modules K_{∗}^{A} → K_{∗}^{QA} that is isomorphism in degree 0 and 1. It follows that the images of the maps Nrd^{QA} and Nrd^{A} coincide.

If A is split, K_{∗}^{QA} = K_{∗}^{A} = K_{∗}.

6. A reduction

Recall that G = SL_{1}(A) for a central simple algebra A of degree n over F. For every commutative F -algebra R there is a natural composition

G(R) ↪ A_{R}^{×} → K_{1}(A_{R}),

where A_{R} = A ⊗_{F} R.

Consider the generic point ξ ∈ G(F [G]) and its image ξ_{F (G)} in G(F (G)). Let α be the image of ξ under the map

G(F [G]) → K_{1}(A_{F [G]}),

and let α_{F (G)} be the image of ξ_{F (G)} under the map

G(F (G)) → K_{1}(A_{F (G)}).

We will prove that α_{F (G)} is nontrivial in K_{1}(A_{F (G)}) when A is a central simple algebra with ind(A) not square-free.

Filtering the category of coherent A ⊗_{F} O_{G}-modules by codimension of support as in [18, §7.5], we get the Brown-Gersten-Quillen spectral sequence (see [18, §7])

E_{1}^{r,s} = ∐_{g∈G^{(r)}} K_{−r−s}(A_{F (g)}) ⇒ K_{−r−s}(A_{F [G]}),

where the limit is the K-group of the category of coherent A ⊗_{F} O_{G}-modules equipped with the topological filtration (by codimension of support). In particular,

E_{2}^{r,s} = A^{r}(G, K_{−s}^{QA})

and the first term of the topological filtration on K_{1}(A_{F [G]}) is equal to

K_{1}(A_{F [G]})^{(1)} = Ker(K_{1}(A_{F [G]}) → K_{1}(A_{F (G)})).

The spectral sequence gives then a homomorphism

ε : K_{1}(A_{F [G]})^{(1)} → A^{1}(G, K_{2}^{QA}).

If α_{F (G)} is trivial in K_{1}(A_{F (G)}), then α ∈ K_{1}(A_{F [G]})^{(1)}. Therefore, we have an element ε(α) ∈ A^{1}(G, K_{2}^{QA}).

We compute ε(α) in the split case. We have G = SL_{n} and

α ∈ K_{1}(A_{F [G]}) = K_{1}(F [G]) = K_{1}(G).

By [22, Th. 2.7], the first Chern class c_{1}(α) of α generates the group A^{1}(G, K_{2}^{QA}) = A^{1}(G, K_{2}) = ℤ.

Lemma 6.1. In the split case, ε(α) = c_{1}(α).

Proof. Let H := GL_{n} and β ∈ K_{1}(H) be the element given by the generic matrix. By [22, Th. 3.10], γ_{i+1}(β) ∈ K_{1}(H)^{(i)} for all i ≥ 0, where γ is the gamma operation, and the image of −γ_{2}(β) under the canonical homomorphism

K_{1}(H)^{(1)} → A^{1}(H, K_{2})

is equal to c_{1}(β). On the other hand, the sum of γ_{i}(β) for all i ≥ 1 coincides with Λ^{n}(β) = det(β) by [22, p. 65]. Hence −γ_{2}(β) ≡ β − det(β) modulo K_{1}(H)^{(2)}.

Pulling back with respect to the embedding of G into H we have −γ_{2}(α) ≡ α modulo K_{1}(G)^{(2)} since det(α) is trivial and therefore, the image of α under the homomorphism K_{1}(G)^{(1)} → A^{1}(G, K_{2}) is equal to c_{1}(α). □

Let L/F be a splitting field of A. We have a commutative diagram

[figure: K_{1}(A_{F [G]})^{(1)}; ε; A^{1}(G, K_{2}^{QA}); K_{1}(A_{L[G]})^{(1)}; ε; A^{1}(G_{L}, K_{2}^{QA}); ℤ.]

The right vertical homomorphism factors as follows:

A^{1}(G, K_{2}^{QA}) ⟶^{Nrd^{QA}} A^{1}(G, K_{2}) →^{∼} A^{1}(G_{L}, K_{2}) = A^{1}(G_{L}, K_{2}^{QA}).

Assume that α_{F (G)} is trivial in K_{1}(A_{F (G)}), hence α ∈ K_{1}(A_{F [G]})^{(1)}. By Lemma 6.1, ε(α)_{L} in A^{1}(G_{L}, K_{2}^{QA}) = ℤ is a generator. It follows that the image of ε(α) under the map Nrd^{QA} : A^{1}(G, K_{2}^{QA}) → A^{1}(G, K_{2}) = ℤ is equal to ±1, hence Nrd^{QA} is surjective.

We have proved:

Proposition 6.2. Suppose that the map Nrd^{QA} : A^{1}(G, K_{2}^{QA}) → A^{1}(G, K_{2}) = ℤ is not surjective. Then Suslin’s Conjecture holds for A.

Let A be a central simple F -algebra such that ind(A) is not square-free, i.e., ind(A) is divisible by p^{2} for a prime integer p. We want to prove that SK_{1}(A) is nontrivial generically. Replacing F by a field extension over which A has index exactly p^{2} and replacing A by a Brauer equivalent division algebra, we may assume that A is a division algebra of degree p^{2}. Moreover, an application of the index reduction formula shows that we may assume that A is decomposable, i.e., A is a tensor product of two algebras of degree p (see [19, Theorem 1.20]).

We will prove that if A is a decomposable division algebra of degree p^{2}, then the map Nrd^{QA} is not surjective. Recall that the maps Nrd^{QA} and Nrd^{A} have the same images. Therefore, it suffices to prove that the map Nrd^{A} : A^{1}(G, K_{2}^{A}) → A^{1}(G, K_{2}) = ℤ is not surjective.

7. A spectral sequence

Let A be a central simple F -algebra of degree p^{2} and X the Severi-Brauer variety of A with dim(X) = d = p^{2} − 1. We would like to find a reasonable description the group A^{1}(G, K_{2}^{A}) via algebraic cycles on G × X.

Consider the spectral sequence associated with the projection q : G×X → G (see Section 2):

(7.1) E_{1}^{r,s} = E_{1}^{r,s}(q, d + 2) = ∐_{g∈G^{(r)}} A^{s}(X_{F (g)}, K_{d+2−r}) ⇒ A^{r+s}(G × X, K_{d+2}).

We have E_{1}^{r,s} = 0 if s > d and

E_{2}^{r,d} = A^{r}(G, K_{2}^{A}).

There are no nontrivial differentials arriving at E_{∗}^{r,d}.

Proposition 7.2. We have E_{2}^{i,d+2−i} = 0 for i = 2, 3, …, p. In particular,

A^{1}(G, K_{2}^{A}) = E_{2}^{1,d} = E_{3}^{1,d} = ⋯ = E_{p}^{1,d}.

Proof. Let j = i − 2 and λ_{j} be the correspondence on X × X of degree j considered in Section 4. By Lemma 4.10, the maps

λ_{j}^{∗} : CH_{0}(X_{L}) → CH_{j}(X_{L}).

are isomorphisms for j = 0, 1, …, p − 2 and every field extension L/F.

Consider the spectral sequence

(7.3) Ê_{1}^{r,s} := E_{1}^{r,s}(q, d + i) = ∐_{g∈G^{(r)}} A^{s}(X_{F (g)}, K_{d+i−r}) ⇒ A^{r+s}(G × X, K_{d+i}).

The edge homomorphism

CH^{d+i}(G × X) = A^{d+i}(G × X, K_{d+i}) → Ê_{2}^{i,d}

is surjective. By Proposition 3.2,

CH^{d+i}(G × X) = CH^{d+i}(X) = 0

since d + i > dim(X). It follows that Ê_{2}^{i,d} = 0.

The correspondence λ_{j} yields a morphism between the spectral sequences (7.1) and (7.3). In particular, we have a homomorphism

λ̃_{j} : Ê_{2}^{i,d} → E_{2}^{i,d−i+2}.

Since λ_{j}^{∗} is an isomorphism for X_{L} for every field extension L/F, the map λ̃_{j} is surjective. As Ê_{2}^{i,d} = 0, we have E_{2}^{i,d+2−i} = 0. □

By Proposition 7.2, we have a differential

A^{1}(G, K_{2}^{A}) = E_{p}^{1,d} ⟶^{δ} E_{p}^{p+1,d+1−p}.

Proposition 7.4. If ind(A) = p, the image of Ker(δ) under the homomorphism

Nrd^{A} : A^{1}(G, K_{2}^{A}) → A^{1}(G, K_{2}) = ℤ

is equal to pℤ.

We will prove this proposition in Section 10.

Let A be a division algebra of degree p^{2} over F. Choose a field extension K/F such that ind(A_{K}) = p and set Ã = A_{K}, X̃ = X_{K}, G̃ = G_{K} and write Ẽ_{∗}^{r,s} for the terms of the spectral sequence associated with the projection G̃ × X̃ → G̃. We have the following commutative diagram

A^{1}(G, K_{2}^{A})  E_{p}^{1,d} ⟶^{δ} E_{p}^{p+1,d+1−p}
↓  ↓  ↓^{κ}
A^{1}(G̃, KÃ ; 2)  Ẽ_{p}^{1,d} \underrightarrow{δ̃} Ẽ_{p}^{p+1,d+1−p},

where δ and δ̃ are the differentials in the p-th pages of the spectral sequences.

Proposition 7.5. If A is decomposable degree p^{2} division algebra, then κ : E_{p}^{p+1,d+1−p} → Ẽ_{p}^{p+1,d+1−p} is the zero map.

We will prove this proposition in Section 11.

8. Main theorem

We deduce the following theorem from Propositions 7.4 and 7.5.

Theorem 8.1. Let A be a central simple F -algebra. If ind(A) is not squarefree, then there is a field extension L/F such that SK_{1}(A_{L}) ≠ 0.

Proof. We may assume that A is a decomposable division algebra of degree p^{2} for a prime integer p. Note that ind(Ã) = p.

By Propositions 7.4 (applied to the algebra Ã) and 7.5, the image of the composition

E_{p}^{1,d} → Ẽ_{p}^{1,d} = A^{1}(G̃, KÃ ; 2) ⟶^{NrdÃ} A^{1}(G̃, K_{2}) = ℤ

is contained in pℤ. On the other hand, this composition coincides with

E_{p}^{1,d} = A^{1}(G, K_{2}^{A}) ⟶^{Nrd^{A}} A^{1}(G, K_{2}) = ℤ.

Therefore, the norm homomorphism Nrd^{A} : A^{1}(G, K_{2}^{A}) → A^{1}(G, K_{2}) is not surjective and this finishes the proof by Proposition 6.2 since Im(Nrd^{QA}) = Im(Nrd^{A}). □

An irreducible variety Z over F is called a retract rational variety if there exist rational morphisms α : Z −−→ ℙ^{m} and β : ℙ^{m} −−→ Z for some m such that the composition β ∘ α is defined and equal to the identity of Z.

Corollary 8.2. Let A be a central simple algebra over F. Then the following are equivalent:

(1) The group SL_{1}(A) is a retract rational variety;

(2) SK_{1}(A_{L}) = 0 for every field extension L/F;

(3) The index ind(A) is square-free.

Proof. (1) ⇒ (2): If G := SL_{1}(A) is a retract rational variety, then G_{L} is so for every field extension L/F. By [2, Proposition 11], the group of R-equivalence classes G(L)/R is trivial. But G(L)/R is isomorphic to SK_{1}(A_{L}) by [24, §18.2].

(2) ⇒ (1): This is proved in [5, Proposition 2.4] and [10, Proposition 5.1].

(2) ⇔ (3): This is Theorem 8.1. □

9. Chow ring of G

Let G = SL_{1}(A) for a central simple algebra A of p-primary degree.

Lemma 9.1. The Chow groups CH^{i}(G) are trivial for i = 1, 2, …, p and p · CH^{p+1}(G) = 0.

Proof. Since CH(G_{sep}) = ℤ by [22, Theorem 2.7], the groups CH^{i}(G) are pprimary torsion if i > 0. As K_{0}(G) = ℤ (see [22, Theorem 4.1]), by [4, Example 15.3.6], we have (i − 1)! CH^{i}(G) = 0 for i > 0. The result follows. □

Consider the Brown-Gersten-Quillen spectral sequence

E_{2}^{r,s} = A^{r}(G, K_{−s}) ⇒ K_{−r−s}(G).

It follows from Lemma 9.1 that

A^{1}(G, K_{2}) = E_{2}^{1,−2} = E_{3}^{1,−2} = ⋯ = E_{p}^{1,−2}.

Moreover, by [8, §3],

CH^{p+1}(G) = E_{2}^{p+1,−p−1} = E_{3}^{p+1,−p−1} = ⋯ = E_{p}^{p+1,−p−1}.

We have then a differential

δ : A^{1}(G, K_{2}) = E_{p}^{1,−2} → E_{p}^{p+1,−p−1} = CH^{p+1}(G).

Write h ∈ CH^{p+1}(G) for the image under δ of the canonical generator of the group A^{1}(G, K_{2}) = ℤ.

Proposition 9.2. Suppose that ind(A) = p, i.e., A = M_{n}(B) and G = SL_{n}(B) for some n and a central division algebra B of degree p. Then

CH^{∗}(G) = ℤ ⊕ (ℤ/pℤ)h ⊕ (ℤ/pℤ)h^{2} ⊕ ⋯ ⊕ (ℤ/pℤ)h^{p−1}.

Proof. Induction on n. The case n = 1 is done in [8, Theorem 9.7].

Let H = SL_{n−1}(B). We view H as a subgroup of G with respect to the embedding x ↦ diag(1, x). Consider the closed subvariety V of the affine space B^{2n} consisting of tuples (b_{1}, …, b_{n}, c_{1}, …, c_{n}) such that ∑ b_{i}c_{i} = 1. Define the morphism

f : G → V, a = (a_{ij}) ↦ (a_{11}, …, a_{1n}, a′_{11}, …, a′_{n1}),

where (a′_{ij}) = a^{−1}. Clearly, f is an H-torsor over V. For any field extension L/F, in the exact sequence of Galois cohomology

G(L) ⟶^{f(L)} V (L) → H^{1}(L, H) ⟶^{r} H^{1}(L, G)

the map r is a bijection (both sets are identified with L^{×}/ Nrd(B_{L}^{×}) and r is the identity map by [11, Cor. 2.9.4]). Hence f is surjective on L-points.

Let W be the open subset of the affine space B^{n} consisting of all tuples (b_{1}, …, b_{n}) such that ∑ b_{i}B = B. We have CH^{i}(W) = 0 for i > 0. The obvious projection V → W is an affine bundle, hence by the homotopy invariance property,

(9.3)  CH^{i}(V) ≃ CH^{i}(W) = 0

for every i > 0.

For every m, consider the spectral sequence associated with the morphism f:

E_{1}^{r,s} = E_{1}^{r,s}(f, m) = ∐_{v∈V^{(r)}} A^{s}(f^{−1}(v), K_{m−r}) ⇒ A^{r+s}(G, K_{m}).

Since f is surjective on L-points, f^{−1}(v) ≃ H_{F (v)}.

We claim that E_{2}^{r,s} = 0 if r + s = m and r > 0. By induction, the group A^{s}(G_{v}, K_{s}) = CH^{s}(G_{v}) is trivial unless s = (p + 1)i for i = 0, 1, …, p − 1. In the latter case the map CH^{r}(V) → E_{2}^{r,s} of multiplication by h^{i} is surjective by the induction hypothesis. The claim follows from the triviality of CH^{r}(V) for r > 0.

By the claim, CH(G) ≃ CH(H_{F (V)}). The statement of the proposition follows by induction. □

Corollary 9.4. Let A be a central simple algebra of degree p^{2}. Then for every field extension L/F such that ind(A_{L}) ≤ p, the map

CH(G) → CH(G_{L})

is surjective.

Proof. The element h belongs to CH^{l+1}(G). As ind(A_{L}) ≤ p, by Proposition 9.2, the element h_{L} generates the ring CH(G_{L}), whence the result. □

10. Proof of Proposition 7.4

In this section, A is a central simple algebra of degree p^{2} and index p, so that A = M_{p}(B), where B is a division algebra of degree p. We write S for the Severi-Brauer variety SB(B) of dimension p − 1. Recall that the variety S can be viewed as a closed subvariety of X. Moreover, the Chow motive M(X) of X is isomorphic to M(S) ⊕ M(S){p} ⊕ ⋯ ⊕ M(S){(p − 1)p}.

Consider the spectral sequence associated with the projection t : G×S → G:

(10.1) Ê_{1}^{r,s} := E_{1}^{r,s}(t, p + 1) = ∐_{g∈G^{(r)}} A^{s}(S_{F (g)}, K_{p+1−r}) ⇒ A^{p+q}(G × S, K_{p+1}).

The embedding of S into X induces the push-forward morphisms between the spectral sequences (10.1) and (7.1). Moreover, (10.1) is a direct summand of (7.1). More precisely, the maps

Ê_{∗}^{r,s} → E_{∗}^{r,s+d+1−p}

are isomorphisms for s = 0, 1, …, p − 1.

By Proposition 7.2, we have E_{2}^{i,d+2−i} = 0 for i = 2, 3, …, p. It follows that Ê_{2}^{i,p+1−i} = 0 for i = 2, 3, …, p, i.e., all the terms but Ê_{2}^{p+1,0} on the diagonal r + s = p + 1 on page Ê_{2}^{∗,∗} are zero. Moreover,

(10.2)  E_{2}^{p+1,d+1−p} = Ê_{2}^{p+1,0} = CH^{p+1}(G).

It follows that

A^{1}(G, K_{2}^{A}) = Ê_{2}^{1,p−1} = Ê_{3}^{1,p−1} = ⋯ = Ê_{p}^{1,p−1}

and the only potentially nonzero differential starting in Ê_{≥2}^{1,p−1} appears on page p:

A^{1}(G, K_{2}^{A}) = Ê_{p}^{1,p−1} ⟶^{δ̂} Ê_{p}^{p+1,0}.

The spectral sequence (10.1) yields then an exact sequence

(10.3)  A^{l}(G × S, K_{p+1}) → Ê_{p}^{1,p−1} ⟶^{δ̂} Ê_{p}^{p+1,0}.

The differential δ : E_{p}^{1,d} → E_{p}^{p+1,d+1−p} in (10.1) is identified with the differential δ̂: Ê_{p}^{1,p−1} → Ê_{p}^{p+1,0} in (7.1). Thus, to prove the proposition, it suffices to show that the image of the composition

A^{p}(G × S, K_{p+1}) → Ê_{p}^{1,p−1} = A^{1}(G, K_{2}^{A}) ⟶^{Nrd^{A}} A^{1}(G, K_{2}) = ℤ

is equal to pℤ.

This composition is the push-forward homomorphism

t_{∗} : A^{p}(G × S, K_{p+1}) → A^{1}(G, K_{2}) = ℤ

with respect to the projection t : G × S → G.

Over S, the algebra A is isomorphic to End_{O_{S}}(J^{p}), where J is the canonical vector bundle over S of rank p. By Proposition 3.2,

A^{p}(G × S, K_{p+1}) = CH^{p−1}(S) · c_{1}(β) ⊕ CH^{p−2}(S) · c_{2}(β) ⊕ ⋯ ⊕ CH^{0}(S) · c_{p}(β),

where β ∈ K_{1}(G × S) is the generic element.

Since the group A^{1}(G, K_{2}) does not change under field extensions, it is sufficient to compute the image over a field extension L/F splitting A. Over such a field extension the group G_{L} is isomorphic to SL_{p^{2}}. Let β′ ∈ K_{1}(G_{L}) be the class of the generic matrix, so that β = [L_{c}] · t^{∗}(β′), where L_{c} is the canonical line bundle over X. By Proposition 3.1,

c_{i}(β) = ∑_{j=0}^{i}(−1)^{j}\binom{i}{j} h^{j} c_{i−j}(t^{∗}β′)

for every i = 1, 2, …, p, where h ∈ CH^{1}(S_{L}) is the first Chern class of L_{c}. Note that c_{1}(β′) is the canonical generator of A^{1}(G_{L}, K_{2}). By the projection formula, the image of t_{∗} is the sum of the subgroups

\binom{i}{j} · t_{∗} [CH^{p−i}(S) · h^{j}] · c_{i−j}(β′)

over all i = 1, 2, …, p and j = 0, 1, …, i. By dimension consideration, the subgroup is trivial if j ≠ i − 1. Consider the case j = i − 1. If p − i > 0, then the image of CH^{p−i}(S) in ℤ (when splitting S) is equal to pℤ. Finally, if i = p, the multiple \binom{i}{j} is equal to p. The proposition is proved.

11. Proof of Proposition 7.5

In this section we assume that A is a decomposable division algebra of degree p^{2}.

Since for every i and j with i + j = d + 2 the natural homomorphism E_{2}^{i,j} → E_{p}^{i,j} is surjective, it is sufficient to prove that the homomorphism

E_{2}^{p+1,d+1−p} → Ẽ_{2}^{p+1,d+1−p}

is trivial.

Let L/F be a field extension. Considering X over a separable closure of L we get the homomorphisms

A^{i}(X_{L}, K_{i+n}) → A^{i}(X_{L_{sep}}, K_{i+n}) = K_{n}(L)

for i = d+1−p and n = 0, 1. These homomorphisms induce the vertical maps in the following commutative diagram

E_{2}^{p+1,d+1−p} ⟶ Ẽ_{2}^{p+1,d+1−p}
φ↓  ↓φ̃
CH^{p+1}(G) ⟶ CH^{p+1}(G̃).

Since ind(Ã) = p, it follows from (10.2) (applied to Ã) that φ̃ is an isomorphism. Thus, it is sufficient to prove that φ = 0.

Recall that A is a decomposable algebra. By a theorem of Karpenko [9, Th. 1],

(11.1)  Im(CH^{d+1−p}(X_{F (g)}) ⟶^{deg} ℤ) = \{ \begin{matrix}pℤ, & if ind A_{F (g)} = p^{2}; \\ ℤ, & if ind A_{F (g)} ≤ p. \end{matrix}

Let Y = SB(p, A) be the generalized Severi-Brauer variety and set m := dim(Y) = p^{3} − p^{2}. Consider the cycle module M_{∗} over F defined by

M_{n}(L) = A^{m}(Y_{L}, K_{n+m}).

There is the norm morphism N : M_{∗} → K_{∗} well defined. The variety Y has a point over a field extension L/F if and only if ind(A_{L}) ≤ p. It follows that

Im(A^{m}(Y_{L}, K_{m}) ⟶^{N} ℤ) = \{ \begin{matrix}pℤ, & if ind A_{L} = p^{2}; \\ ℤ, & if ind A_{L} ≤ p. \end{matrix}

Therefore the image of φ coincide with the image of the map

ψ : A^{p+1}(G, M_{p+1}) → A^{p+1}(G, K_{p+1}) = CH^{p+1}(G)

induced by the norm map N. It is sufficient to prove that ψ = 0.

The spectral sequence for the projection G × Y → G,

E_{1}^{r,s} = ∐_{g∈G^{(r)}} A^{s}(Y_{F (g)}, K_{m+p+1−r}) ⇒ A^{r+s}(G × Y, K_{m+p+1})

yields a surjective homomorphism CH^{m+p+1}(G × Y) → A^{p+1}(G, M_{p+1}). The composition

CH^{m+p+1}(G × Y) → A^{p+1}(G, M_{p+1}) ⟶^{ψ} A^{p+1}(G, K_{p+1}) = CH^{p+1}(G)

is the push-forward homomorphism with respect to the projection G×Y → G. Thus, it is sufficient to show that the push-forward homomorphism

CH^{m+p+1}(G × Y) → CH^{p+1}(G)

is zero.

Since ind(A_{F (y)}) ≤ p for every y ∈ Y, it follows from Corollary 9.4 that the map CH(G) → CH(G_{F (y)}) is surjective. Then the proof of [3, Lemma 88.5] yields the following lemma.

Lemma 11.2. The product homomorphism

CH(G) ⊗ CH(Y) → CH(G × Y)

is surjective.

Lemma 11.3. For every closed point y ∈ Y, the norm homomorphism

N_{F (y)} : CH^{p+1}(G_{F (y)}) → CH^{p+1}(G)

is trivial.

Proof. The first map in the composition

CH^{p+1}(G) → CH^{p+1}(G_{F (y)}) ⟶^{N_{F (y)/F}} CH^{p+1}(G)

is surjective by Corollary 9.4 since ind(A_{F (y)}) ≤ p. The composition is multiplication by deg(y). Note that deg(y) is divisible by p since ind(A) = p^{2}. The result follows from Lemma 9.1. □

Proposition 11.4. If A is a division algebra, the push-forward homomorphism

CH^{m+p+1}(G × Y) → CH^{p+1}(G)

is trivial.

Proof. By Lemma 11.2, it is sufficient to show that for every closed point y ∈ Y the norm homomorphism CH^{p+1}(G_{F (y)}) → CH^{p+1}(G) is trivial. This is proved in Lemma 11.3. □

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

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