DECOMPOSABLE AND INDECOMPOSABLE ALGEBRAS OF DEGREE 8 AND EXPONENT 2

DEMBA BARRY WITH AN APPENDIX BY ALEXANDER S. MERKURJEV

Abstract. We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let B be a biquaternion algebra over F (\sqrt{a}) with trivial corestriction. A degree 3 cohomological invariant is defined and we show that it determines whether B has a descent to F. This invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2-cohomological dimension 3 and over a field 𝕄(t) where the u-invariant of 𝕄 is 8 and t is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by A. S. Merkurjev in Appendix.

1. Introduction

Let A be a central simple algebra over a field F. We say that A is decomposable if A ≃ A_{1} ⊗_{F} A_{2} for two central simple F -algebras A_{1} and A_{2} both non isomorphic to F; otherwise A is called indecomposable. Let K = F (\sqrt{a}) be a quadratic separable extension of F. We say that A admits a decomposition adapted to K if K is in a quaternion subalgebra of A, that is, A ≃ (a, a′) ⊗_{F} A′ for some a′ ∈ F and some subalgebra A′ ⊂ A. If A is isomorphic to a tensor product of quaternion algebras, we will say that A is totally decomposable. Let B be a central simple algebra over K. The algebra B has a descent to F if there exists an F -algebra B′ such that B ≃ B′ ⊗_{F} K. It is clear that the algebra A admits a decomposition adapted to K if and only if the centralizer C_{A}K of K in A has a descent to F.

The first example of a non-trivial indecomposable central simple algebra of exponent 2 was given by Amitsur-Rowen-Tignol. More precisely, an explicit central division algebra of degree 8 and exponent 2 which has no quaternion subalgebra is constructed in [2]. Other examples of such algebras were given by Karpenko (see [11] and [12]) by computing torsion in Chow groups. In fact, 8 is the smallest possible degree for such an algebra by a well-known theorem of Albert which asserts that every algebra of exponent 2 and degree 4 is decomposable.

The decomposability question depends on the cohomological dimension of the ground field. Indeed, for obvious reasons there is no indecomposable algebra of exponent 2 over a field of cohomological dimension 0 or 1 (since the Brauer group is trivial in these cases). It follows from a result of Merkurjev (Theorem 3.1) that over a field of cohomological dimension 2 any central simple algebra of exponent 2 is isomorphic to a tensor product of quaternion algebras. On the other hand, the known examples of indecomposable algebras of exponent 2 are constructed over fields of cohomological dimension greater than or equal to 5.

The main goal of this article is to extend the existence of indecomposable algebras of exponent 2 to some fields of cohomological dimension smaller than or equal to 4. We also give an example over a field of rational functions in one variable over a field of u-invariant 8. The problem will be addressed through the study of the decomposability adapted to a quadratic extension of the ground field. More precisely, let K = F (\sqrt{a}) ⊂ A be a quadratic extension field. We first prove (Section 3) that if cd_{2}(F) ≤ 2 then K lies in a quaternion subalgebra of A. If deg A = 8, we define a degree 3 cohomological invariant, depending only on the centralizer C_{A}K of K in A, which determines whether A admits a decomposition adapted to K, that is, whether C_{A}K has a descent to F (see Section 4). This invariant is used to give examples of indecomposable algebras of exponent 2 (Theorem 1.2 and Theorem 1.3). Although our invariant depends only on C_{K}A, it is nothing but a refinement of the invariant Δ(A) defined by Garibaldi-Parimala-Tignol [8, §11] (see Remark 4.3). Through Remark 4.10 and the proof of Theorem 1.2, our invariant provides an example of indecomposable algebra A of degree 8 and exponent 2 such that Δ(A) is nonzero, this is an answer to Garibaldi-Parimala-Tignol’s question [8, Question 11.2].

For the proofs of Theorem 1.2 and Theorem 1.3 we need some results — injections (5.1) and (5.2) — on Chow groups of cycles of codimension 2. These results follow by Theorem A.9 and Theorem A.7 provided by A. S. Merkurjev in Appendix.

Statement of main results. Standard examples of fields of cohomological dimension 3 are k(t_{1}, t_{2}, t_{3}) or k((t_{1}))((t_{2}))((t_{3})) where k is an algebraically closed field and t_{1}, t_{2}, t_{3} are independent indeterminates over k. Recall also that the u-invariant of such fields is 8 by Tsen-Lang (see for instance [9, Theorem 1]). The following theorem shows that there is no indecomposable algebra of degree 8 and exponent 2 over these standard fields. We recall that the u-invariant of a field F is defined as

u(F) = max{dim φ | φ anisotropic form over F }.

If no such maximum exists, u(F) is defined to be ∞.

Theorem 1.1. There exists no indecomposable algebra of degree 8 and exponent 2 over a field of u-invariant smaller than or equal to 8.

On the other hand, examples of indecomposable algebras do exist over a function field of one variable over a suitable field of u-invariant 8:

Theorem 1.2. Let A be a central simple algebra of degree 8 and exponent 2 over a field F and let K = F (\sqrt{a}) be a quadratic field extension of F contained in A. If K is not in a quaternion subalgebra of A then there exists an extension 𝕄 of F with u(𝕄) = 8 such that the division algebra Brauer equivalent to

A_{𝕄} ⊗_{𝕄} (a, t)_{𝕄(t)}

is an indecomposable algebra of degree 8 and exponent 2 over 𝕄(t), where t is an indeterminate.

To produce explicit examples, one may take for A any indecomposable algebra, as those constructed in [2] or [12]. Alternately, we give in [3] an example of a decomposable algebra satisfying the hypothesis of Theorem 1.2.

The following theorem shows that there exists an indecomposable algebra of degree 8 and exponent 2 over some field of 2-cohomological dimension 3. Let us recall that CH^{2}(SB(A))_{tors} is the torsion in the Chow group of cycles of codimension 2 over the Severi-Brauer variety SB(A) of A modulo rational equivalence. If A is of prime exponent p and index p^{n} (except the case p = 2 = n) Karpenko showed in [12, Proposition 5.3] that if CH^{2}(SB(A))_{tors} ≠ 0 then A is indecomposable. Examples of such indecomposable algebras are given in [12, Corollary 5.4].

Theorem 1.3. Let A be a central simple algebra of degree 8 and exponent 2 such that CH^{2}(SB(A))_{tors} ≠ 0. Then there exists an extension 𝕄 of F with cd_{2}(𝕄) = 3 such that A_{𝕄} is indecomposable.

2. Notations and preliminaries

Throughout this paper the characteristic of the base field F is assumed to be different from 2 and all algebras are associative and finite-dimensional over F. The main tools in this paper are central simple algebras, quadratic forms and Galois cohomology. Pierce’s book [21] is a reference for the general theory of central simple algebras, references for quadratic form theory over fields are [5], [14] and [22].

Let F_{s} be a separable closure of F. For any integer n ≥ 0, H^{n}(F, μ_{2}) denotes the Galois cohomology group

H^{n}(F, μ_{2}) := H^{n}(Gal(F_{s}/F), μ_{2})

where μ_{2} = {±1}. The group H^{1}(F, μ_{2}) is identified with F^{×}/F^{×2} by Kummer theory. For any a ∈ F^{×}, we write (a) for the element of H^{1}(F, μ_{2}) corresponding to aF^{×2}. The group H^{2}(F, μ_{2}) is identified with the 2-torsion Br_{2}(F) in the Brauer group Br(F) of F, and we write [A] for the element of H^{2}(F, μ_{2}) corresponding to a central simple algebra A of exponent 2. For more details on Galois cohomology the reader can consult Serre’s book [24].

For any quadratic form q, we denote by C(q) and d(q) the Clifford algebra and the discriminant of q. If q has dimension 2m + 2, it is easy to see that C(q) is a tensor product of m + 1 quaternion algebras. Conversely, any tensor product of quaternion algebras is a Clifford algebra. Let I^{n}F be the n-th power of the fundamental ideal IF of the Witt ring W F. Abusing notations, we write q ∈ I^{n}F if the Witt class of q lies in I^{n}F. We shall use frequently the following property: if q ∈ I^{2}F (i.e, dim q is even and its discriminant is trivial) the Clifford algebra of q has the form C(q) ≃ M_{2}(E(q)) for some central simple algebra E(q) which is totally decomposable.

Now, let E/F be an extension of F and let q be a quadratic form defined over F. We say that E/F is excellent for q if the anisotropic part (q_{E})_{an} of q_{E} is defined over F. If E/F is excellent for every quadratic form defined over F, the extension E/F is called excellent.

Generally, the biquadratic extensions are not excellent (see for instance [6, §5]) but we have the following result:

Lemma 2.1. Assume that I^{3}F = 0 and let L = F (\sqrt{a_{1}}, \sqrt{a_{2}}) be a biquadratic extension of F. Then, L/F is excellent for the quadratic forms q ∈ I^{2}F. More precisely, for all q ∈ I^{2}F there exists q_{0} ∈ I^{2}F such that (q_{L})_{an} ≃ (q_{0})_{L}.

Proof. Let s : K = F (\sqrt{a_{1}}) → F be the F -linear map such that s(1) = 0 and s(\sqrt{a_{1}}) = 1. The corresponding Scharlau’s transfer will be denoted by s_{∗}. Notice that I^{3}K = 0 because of I^{3}F = 0 and the exactness of the sequence (see [5, Theorem 40.3])

\xymatrix{⟨1, −a_{1}⟩I^{2}F \ar[r] & I^{3}F \ar[r] & I^{3}K \ar[r]^{s_{∗}} & I^{3}F.}

Let q ∈ I^{2}F be an anisotropic form such that q_{L} is isotropic. We first show that there exists a quadratic form q_{0} defined over F such that (q_{0})_{L} ≃ (q_{L})_{an}. If q_{L} is hyperbolic, there is nothing to show. We suppose q_{L} is not hyperbolic. It is wellknown (see e.g. [14, Theorem 3.1, p.197]) that (q_{K})_{an} is defined over F. If (q_{K})_{an} remains anisotropic over L, we take q_{0} = (q_{K})_{an}. Otherwise, one has

(q_{K})_{an} = ⟨1, −a_{2}⟩ ⊗ ⟨γ_{1}, …, γ_{r}⟩ ⊥ q′

for some γ_{1}, …, γ_{r} ∈ K and some subform q′ of (q_{K})_{an} defined over K with q′_{L} anisotropic (see for instance [10, Proposition 3.2.1]). On the other hand, the form (q_{K})_{an} being in I^{2}K and I^{3}K = 0, one has (q_{K})_{an} ⊗ ⟨1, −γ_{1}⟩ = 0, that is, (q_{K})_{an} ≃ γ_{1}(q_{K})_{an}. Thus, we may suppose γ_{1} = 1.

We claim that the dimension of the form ⟨1, γ_{2} …, γ_{r}⟩ is 1. Indeed, assume that r ≥ 2 and write (q_{K})_{an} = ⟨1, −a_{2}⟩ ⊗ ⟨1, γ_{2}⟩ ⊥ φ for some φ over K. Since φ_{L} is nonzero, it is clear that φ is nonzero. Let α ∈ K^{×} be represented by φ. The form ⟨1, −a_{2}⟩ ⊗ ⟨1, γ_{2}⟩ ⊗ ⟨1, α⟩ is hyperbolic since I^{3}K = 0, hence ⟨1, −a_{2}⟩ ⊗ ⟨1, γ_{2}⟩ represents −α. It follows that (q_{K})_{an} is isotropic, a contradiction. Hence r = 1 and so (q_{K})_{an} ≃ ⟨1, −a_{2}⟩ ⊥ q′.

Now, we are going to show that q′_{L} ≃ (q_{0})_{L} for some q_{0} defined over F. The Scharlau transfers s_{∗}((q_{K})_{an}) and s_{∗}(⟨1, −a_{2}⟩) are both hyperbolic because (q_{K})_{an} and ⟨1, −a_{2}⟩ are defined over F. This implies that s_{∗}(q′) is hyperbolic, so q′ is Witt equivalent to (q_{0})_{K} for some q_{0} defined over F. We deduce from the excellence of K/L that q′ ≃ (q_{0})_{K}. Whence (q_{L})_{an} ≃ (q_{0})_{L}.

It remains to prove that q_{0} may be chosen to be in I^{2}F. Since (q_{0})_{L} ∈ I^{2}L, the discriminant d(q_{0}) ∈ {F^{×2}, a_{1}.F^{×2}, a_{2}.F^{×2}, a_{1}a_{2}.F^{×2}}. If d(q_{0}) = 1, we are done. Otherwise, assume for example d(q_{0}) = a_{1} and write q_{0} ≃ ⟨c_{1}, …, c_{m}⟩. The quadratic form q_{1} ≃ ⟨a_{1}c_{1}, c_{2}, …, c_{m}⟩ is such that (q_{1})_{L} ≃ (q_{0})_{L} and d(q_{1}) = 1; that is q_{1} ∈ I^{2}F. It suffices to replace q_{0} by q_{1}. This concludes the proof. □

3. Adapted decomposition under cd_{2}(F) ≤ 2

In this section we assume that the 2-cohomological dimension cd_{2}(F) ≤ 2. If the characteristic of F is different from 2, Merkurjev proved that division algebras of exponent 2 over F are totally decomposable. We use this observation to show that any quadratic field extension of F in a central simple algebra over F of exponent 2 lies in a quaternion subalgebra. We also prove a related result for a biquadratic extension of F. First, let us recall the following result:

Theorem 3.1 (Merkurjev). Assume that cd_{2}(F) ≤ 2. Then

(1) Any central division algebra over F whose class is in Br_{2}(F) is totally decomposable.

(2) A quadratic form q ∈ I^{2}F is anisotropic if and only if E(q) is a division algebra.

Proof. See [9, Theorem 3]. □

Let A be a 2-power dimensional central simple algebra over F and let K ⊂ A be a quadratic extension of F. If A is not a division algebra, K is in any split quaternion algebra. We have the following general result in the division case:

Proposition 3.2. Assume that cd_{2}(F) ≤ 2. Let A be a central division algebra of exponent 2 over F. Every square-central element of A lies in a quaternion subalgebra.

Proof. Assume that deg A = 2^{m}. By Theorem 3.1, the algebra A decomposes into a tensor product of m quaternion algebras. We may associate with M_{2}(A) a (2m+2)- dimensional quadratic form q ∈ I^{2}F in such a way that C(q) ≃ M_{2}(A). This form q is anisotropic by Theorem 3.1 (2). The form q being in I^{2}F, the center of the Clifford algebra C_{0}(q) is F × F and C_{0}(q) ≃ C_{+}(q) × C_{−}(q) with A ≃ C_{+}(q) ≃ C_{−}(q). Let x ∈ A − F be such that x^{2} = a ∈ F^{×}. The element z = x^{−1}\sqrt{a}, which is different from 1 and −1, is such that z^{2} = 1; hence A_{F (\sqrt{a})} is not a division algebra. Then Theorem 3.1 and [14, Theorem 3.1, p.197] indicate that the form q has a subform α⟨1, −a⟩ for some α ∈ F^{×}. Write q ≃ α⟨1, −a⟩ ⊥ q_{1} and let V_{q} be the underlying space of q. Let e_{1}, …, e_{2m+2} be an orthogonal basis of V_{q} which corresponds to this diagonalization. One has:

\{ \begin{matrix}e_{1}e_{2}, e_{1}e_{3} ∈ C_{0}(q) \\ (e_{1}e_{2})^{2} = α^{2}a, (e_{1}e_{3})^{2} = −α^{2}q(e_{3}) \\ (e_{1}e_{2})(e_{1}e_{3}) = −(e_{1}e_{3})(e_{1}e_{2}). \end{matrix}

Let z = e_{1} … e_{2m+2}. Since q ∈ I^{2}F, we have z^{2} ∈ F^{×2}. Let z^{2} = λ^{2} with λ ∈ F^{×} and set z_{+} = \frac{1}{2}(1+λ^{−1}z) and z_{−} = \frac{1}{2}(1−λ^{−1}z). So, z_{+}, z_{−} are the primitive central idempotents in C_{0}(q). The elements (e_{1}e_{2})z_{+} and (e_{1}e_{3})z_{+} generate a quaternion subalgebra isomorphic to (a, a′) in C_{+}(q) ≃ A where a′ = (e_{1}e_{3})^{2}. Hence, A contains some element y such that y^{2} = a and y lies in a quaternion subalgebra of A. By the Skolem-Noether Theorem x and y are conjugated. Therefore, x is in a quaternion subalgebra. □

Recall that if cd_{2}(F) ≤ 2 then I^{3}F = 0 ([15, Theorem A5’]). This fact allows to use Lemma 2.1 in the proof of the following results.

Theorem 3.3. Assume that cd_{2}(F) ≤ 2. Let A be a central division algebra of degree 2^{n} and exponent 2 over F. Let L = F (\sqrt{a_{1}}, \sqrt{a_{2}}) be a biquadratic extension of F contained in A. There exist quaternion algebras Q_{1}, Q_{2} and a subalgebra A′ ⊂ A over F such that F (\sqrt{a_{i}}) ⊂ Q_{i} (i = 1, 2) and A ≃ Q_{1} ⊗ Q_{2} ⊗ A′.

Proof. Let q ∈ I^{2}F be an anisotropic quadratic form such that C(q) ≃ M_{2}(A) (Theorem 3.1). Notice that the index of A_{L} is 2^{n−2}. Since cd_{2}(L) = 2, it follows by [25, Lemma 8] that

dim(q_{L})_{an} = 2 log_{2} ind(A_{L}) + 2 = 2n − 2.

By Lemma 2.1, there is q′ ∈ I^{2}F such that (q_{L})_{an} ≃ q′_{L}. Put ψ = q − q′. The form ψ_{L} being hyperbolic it follows by [14, Theorem 4.3, p.444] that there exist quadratic forms φ_{1}, φ_{2} such that

ψ = ⟨1, −a_{1}⟩ ⊗ φ_{1} ⊥ ⟨1, −a_{2}⟩ ⊗ φ_{2} in W F.

Since ψ ∈ I^{2}F, we must have φ_{1}, φ_{2} ∈ IF. Taking the Witt-Clifford invariant of each side of the identity above, we obtain

A ⊗ A′ ∼ (a_{1}, d(φ_{1})) ⊗ (a_{2}, d(φ_{2}))

where A′ is such that M_{2}(A′) = C(q′). This yields that

A ≃ (a_{1}, d(φ_{1})) ⊗ (a_{2}, d(φ_{2})) ⊗ A′.
□

If deg A = 8, the above theorem holds without the division condition on A as shows the following:

Proposition 3.4. Assume that cd_{2}(F) ≤ 2. Let A be a central simple algebra of degree 8 and exponent 2 over F. Let L = F (\sqrt{a_{1}}, \sqrt{a_{2}}) be a biquadratic extension of F contained in A. There exist quaternion algebras Q_{1}, Q_{2} and Q over F such that F (\sqrt{a_{i}}) ⊂ Q_{i} (i = 1, 2) and A ≃ Q_{1} ⊗ Q_{2} ⊗ Q.

Proof. We are going to argue on the index ind(A) of A: if ind(A) ≤ 2, the statement is clear since L lies in any split algebra of degree 4. If ind(A) = 8, the result follows from Theorem 3.3.

Suppose ind(A) = 4. Let q ∈ I^{2}F be an anisotropic quadratic form such that C(q) ∼ M_{2}(A). Such a form exists by Merkurjev’s Theorem. Since A_{L} is split or of index 2, the form q_{L} is either hyperbolic or equivalent to a multiple of the anisotropic norm form, n_{Q}, of some quaternion algebra Q defined over L. Notice that every multiple of n_{Q} is isometric to n_{Q}. We may consider n_{Q} as being defined over F because of Lemma 2.1; so Q is defined over F. Moreover, (q − n_{Q})_{L} is hyperbolic. Put ψ = q if ind(A_{L}) = 1 and ψ = q −n_{Q} if ind(A_{L}) = 2; the form ψ_{L} is hyperbolic. As in the proof of Theorem 3.3 there exist quadratic forms φ_{1}, φ_{2} ∈ IF such that

A ⊗ Q ∼ (a_{1}, d(φ_{1})) ⊗ (a_{2}, d(φ_{2})) if ind(A_{L}) = 2,

and

A ∼ (a_{1}, d(φ_{1})) ⊗ (a_{2}, d(φ_{2})) if ind(A_{L}) = 1.

This yields that

A ≃ (a_{1}, d(φ_{1})) ⊗ (a_{2}, d(φ_{2})) ⊗ M_{2}(F) or A ≃ (a_{1}, d(φ_{1})) ⊗ (a_{2}, d(φ_{2})) ⊗ Q.

□

Remark 3.5. For central simple algebras of degree 8 and exponent 2 over F with cd_{2}(F) = 2, Proposition 4.9 cannot be generalized to the triquadratic extensions of F. For example, let F = ℂ(x, y) where ℂ is the field of complex numbers and x, y are independent indeterminates over ℂ. Notice that cd_{2}(F) = 2. Since all 5-dimensional quadratic forms over F are isotropic by Tsen-Lang (see for instance [14, p.376]), Theorem 3.1 yields that index equals 2 for every division algebra of exponent 2 over F. Let a_{1} = x^{2}(1 + y)^{2} − 4xy, a_{2} = 1 − x, a_{3} = y and M = F (\sqrt{a_{1}}, \sqrt{a_{2}}, \sqrt{a_{3}}). It is shown in [6, (5.8)] that there exists an algebra A of exponent 2 and degree 8 over F containing M such that A admits no decomposition of the form (a_{1}, a′_{1}) ⊗ (a_{2}, a′_{2}) ⊗ (a_{3}, a′_{3}) with a′_{1}, a′_{2}, a′_{3} ∈ F.

4. Degree 3 invariant

In this section, we assume no condition on the 2-cohomological dimension of F. Let K = F (\sqrt{a}) be a quadratic field extension of F. Throughout this section B is a biquaternion algebra over K such that cor_{K/F} (B) is trivial. In fact, this condition on cor_{K/F} (B) means that B has a descent to F up to Brauer equivalence because of the exactness of the sequence (see for instance [1, (4.6)])

\xymatrix{Br_{2}(F) \ar[r]^{r_{K/F}} & Br_{2}(K) \ar[r]^{^{cor^{K/F}}} & Br_{2}(F).}

We are going to attach to B a cohomological invariant which determines whether B has a descent to F. This descent problem is another way of studying the question of adapted decomposition. Indeed, let A be a central simple algebra over F of degree greater than or equal to 8 whose restriction is B. As explained in the introduction, B has a descent to F if and only if A admits a decomposition adapted to K.

Let φ be an Albert form of B, that is, φ is a 6-dimensional form in I^{2}K such that C_{0}(φ) ≃ C_{+}(φ) × C_{−}(φ) with C_{+}(φ) ≃ C_{−}(φ) ≃ B. Recall that an Albert form is unique up to a scalar. We have the following lemma where e_{3} denotes the Arason invariant I^{3}F → H^{3}(F, μ_{2}):

Lemma 4.1. We keep the above notations. One has the following statements:

(1) Scharlau’s transfer s_{∗}(φ) lies in I^{3}F.

(2) For any λ ∈ K^{×}, we have e_{3}(s_{∗}(φ)) = e_{3}(s_{∗}(⟨λ⟩ ⊗ φ)) + cor_{K/F} ((λ) · [B]).

(3) For λ ∈ K^{×}, the following are equivalent:

(i) s_{∗}(⟨λ⟩ ⊗ φ) = 0.

(ii) e_{3}(s_{∗}(φ)) = cor_{K/F} ((λ) · [B]).

Proof. (1) Since the diagram

\xymatrix{I^{2}K \ar[r]^{s_{∗}} \ar[d]_{e_{2}} & I^{2}F \ar[d]^{e_{2}} \\ Br_{2}(K) \ar[r]^{_{cor_{K/F}}} & Br_{2}(F)}

commutes (see [1, (4.18)]) and cor_{K/F} (B) = 0, we have e_{2}(s_{∗}(φ)) = 0. Hence,

s_{∗}(φ) ∈ Ker(e_{2} : I^{2}F → Br_{2}(F)) = I^{3}F.

(2) From the identity

s_{∗}(φ) = s_{∗}(⟨λ⟩ ⊗ φ + ⟨1, −λ⟩ ⊗ φ) = s_{∗}(⟨λ⟩ ⊗ φ) + s_{∗}(⟨1, −λ⟩ ⊗ φ)

we have

e_{3}(s_{∗}(φ)) = e_{3}(s_{∗}(⟨λ⟩ ⊗ φ)) + e_{3}(s_{∗}(⟨1, −λ⟩ ⊗ φ)).  (4.1)

On the other hand, since the diagram

\xymatrix{I^{3}K \ar[r]^{s_{∗}} \ar[d]_{e_{3}} & I^{3}F \ar[d]^{e_{3}} \\ H^{3}(K, μ_{2}) \ar[r]^{_{cor_{K/F}}} & H^{3}(F, μ_{2})}

commutes ([1, (5.7)]), one has

e_{3}(s_{∗}(⟨1, −λ⟩ ⊗ φ)) = cor_{K/F} (e_{3}(⟨1, −λ⟩ ⊗ φ))
= cor_{K/F} ((λ) · e_{2}(φ))
= cor_{K/F} ((λ) · [C_{+}(φ)])
= cor_{K/F} ((λ) · [B]).

Therefore the relation (4.1) becomes

e_{3}(s_{∗}(φ)) = e_{3}(s_{∗}(⟨λ⟩ ⊗ φ)) + cor_{K/F} ((λ) · [B])

as desired.

(3) This point follows immediately from (2) because s_{∗}(⟨λ⟩ ⊗ φ) = 0 if and only if e_{3}(s_{∗}(⟨λ⟩ ⊗ φ) = 0 by the Arason-Pfister Hauptsatz (see for instance [5, Chap. 4]) □

It follows from Lemma 4.1 (2) that e_{3}(s_{∗}(φ)) modulo cor_{K/F} ((K^{×}) · [B]) does not depend on the choice of the Albert form φ. Therefore we may define an invariant:

Definition 4.2. Let B be a biquaternion algebra over K such that cor_{K/F} (B) = 0. The invariant

δ_{K/F} (B) ∈ \frac{H^{3}(F, μ_{2})}{cor_{K/F} ((K^{×}) · [B])}

is the class of e_{3}(s_{∗}(φ)).

Remark 4.3. If A is a central simple F -algebra of degree 8 containing K, and if B = C_{A}K, then δ_{K/F} (B) is the image in H^{3}(F, μ_{2})/ cor_{K/F} ((K^{×})·[B]) of the discriminant Δ(A, σ) ∈ H^{3}(F, μ_{2}) of any symplectic involution σ on A which leaves K elementwise fixed. This follows by comparing the definition of δ_{K/F} (B) with Garibaldi-Parimala-Tignol [8, Proposition 8.1]. Therefore, the image of δ_{K/F} (B) in H^{3}(F, μ_{2})/(F^{×})·[A] is the invariant Δ(A) defined in [8, §11] (Note that [B] = res_{K/F} [A], so by the projection formula cor_{K/F} ((K^{×}) · [B]) = (N_{K/F} (K^{×})) · [A] ⊂ (F^{×}) · [A]).

The main property of the invariant is the following result:

Proposition 4.4. The algebra B has a descent to F if and only if δ_{K/F} (B) = 0.

Proof. It follows from the definition that δ_{K/F} (B) = 0 if and only if there exists λ ∈ K^{×} such that the equivalent conditions (i) and (ii) of Lemma 4.1 hold. We are going to prove that B has a descent to F if and only if (i) holds. We know that B has a descent to F if and only if there is φ_{0} ∈ I^{2}F with dim φ_{0} = 6 such that

B ≃ C_{+}(φ) ≃ C_{+}(φ_{0}) ⊗ K ≃ C_{+}((φ_{0})_{K}).

In other words, B has a descent to F if and only if there exists λ ∈ K^{×} such that φ ≃ ⟨λ⟩ ⊗ (φ_{0})_{K} because of the uniqueness of the Albert form up to similarity. Therefore to get the statement, it suffices to show for given λ ∈ K^{×}, there exists φ_{0} ∈ I^{2}F with dim φ_{0} = 6 and φ ≃ ⟨λ⟩ ⊗ (φ_{0})_{K} if and only if s_{∗}(⟨λ⟩ ⊗ φ) = 0.

Suppose B has a descent to F, that is, φ ≃ ⟨λ⟩ ⊗ (φ_{0})_{K}. We have automatically s_{∗}(⟨λ⟩ ⊗ φ) = 0. Conversely, assume that s_{∗}(⟨λ⟩ ⊗ φ) = 0. Then, as in the proof of Lemma 2.1 there exists a quadratic form φ_{0} ∈ I^{2}F with ⟨λ⟩ ⊗ φ ≃ (φ_{0})_{K}. That concludes the proof. □

Now, let us denote by X the Weil transfer of SB(B) over F. Such a transfer exists (see for instance [4, (2.8)] or [23, Chapter 4]) and is projective ([13, Corollary 2.4]) since SB(B) is a projective variety. Moreover, we have

X_{K} ≃ SB(B) × SB(B)

(see [4, (2.8)]). We denote by F (X) the function field of X. Notice that F (X) splits B; that also means φ is hyperbolic over F (X).

For any integer d ≥ 1, let ℚ/ℤ(d − 1) = \underrightarrow{lim} μ_{n}^{⊗(d−1)}, where μ_{n} is the group of n-th roots of unity in F_{s}. We let (see [7, Appendix A]) H^{d}(F, ℚ/ℤ(d − 1)) be the Galois cohomology group with coefficients in ℚ/ℤ(d − 1). By definition, one has the canonical map H^{3}(F, μ_{2n}^{⊗2}) → H^{3}(F, ℚ/ℤ(2)). On the other hand, using the surjectivity of the map H^{3}(F, μ_{2n}^{⊗2}) → H^{3}(F, μ_{n}^{⊗2}) (see [18]), the infinite long exact sequence in cohomology induced by the natural exact sequence of Galois modules

1 → μ_{2} → μ_{2n}^{⊗2} → μ_{n}^{⊗2} → 1

shows that the canonical map H^{3}(F, μ_{2}) → H^{3}(F, ℚ/ℤ(2)) is injective. Since φ is hyperbolic over F (X), it is clear that

e_{3}(s_{∗}(φ)) ∈ Ker (H^{3}(F, ℚ/ℤ(2)) → H^{3}(F (X), ℚ/ℤ(2))).

On the other hand, since F (X) splits B, we have

cor_{K/F} ((K^{×}) · [B]) ⊂ Ker (H^{3}(F, ℚ/ℤ(2)) → H^{3}(F (X), ℚ/ℤ(2))).

We have just proved the following consequence:

Corollary 4.5. The invariant δ_{K/F} (B) is in the quotient group

\frac{Ker (H^{3}(F, ℚ/ℤ(2)) → H^{3}(F (X), ℚ/ℤ(2)))}{cor_{K/F} ((K^{×}) · [B])}.  (4.2)

Remark 4.6. The quotient (4.2) is canonically identified with the torsion CH^{2}(X)_{tors} (see [20] or [19]). Therefore, we may view δ_{K/F} (B) as belonging to this group.

We now study the behavior of δ_{K/F} (B) under an odd degree extension. Let 𝔽 be an odd degree extension of F. We denote by K𝔽 the quadratic extension 𝔽(\sqrt{a}). The following result shows that if B has no descent to F, then the same holds for B_{𝔽}.

Proposition 4.7. The scalar extension map

\frac{Ker (H^{3}(F, ℚ/ℤ(2)) → H^{3}(F (X), ℚ/ℤ(2)))}{cor_{K/F} ((K^{×}) · [B])} ⟶

\frac{Ker (H^{3}(𝔽, ℚ/ℤ(2)) → H^{3}(𝔽(X_{𝔽}), ℚ/ℤ(2)))}{cor_{K𝔽/𝔽}(((K𝔽)^{×}) · [B_{K𝔽}])}

is an injection.

Proof. Let M ⊃ K ⊃ F be a separable biquadratic extension of K in B which splits B; so [M : F] = 8. Since M splits B, we have

X_{M} ≃ SB(B)_{M} × SB(B)_{M} ≃ P_{M}^{3} × P_{M}^{3}

where P_{M}^{3} denotes the projective 3-space considered as variety. The function field M(P_{M}^{3}) of P_{M}^{3} being a purely transcendental extension of M, the function field M(X_{M}) is a purely transcendental extension of M. So, we have

H^{3}(M, ℚ/ℤ(2)) ↪ H^{3}(M(X_{M}), ℚ/ℤ(2)).

Let ξ ∈ Ker (H^{3}(F, ℚ/ℤ(2)) → H^{3}(F (X), ℚ/ℤ(2))) and denote by ξ_{𝔽} its image in

H^{3}(𝔽, ℚ/ℤ(2)) under the scalar extension. Suppose ξ_{𝔽} = cor_{K𝔽/𝔽}((λ) · [B_{K𝔽}]) for some λ ∈ (K𝔽)^{×}. Let A be a central simple F -algebra such that r_{K/F} (A) = B. Note that such an algebra A exists because the sequence

\xymatrix{Br_{2}(F) \ar[r]^{r_{K/F}} & Br_{2}(K) \ar[r]^{^{cor^{K/F}}} & Br_{2}(F)}

is exact. One has

[𝔽 : F]ξ = cor_{𝔽/F} (ξ_{𝔽}) = cor_{𝔽/F} ( cor_{K𝔽/𝔽}((λ) · [B_{K𝔽}]))
= cor_{𝔽/F} (N_{K𝔽/𝔽}(λ) · [A_{𝔽}])
= N_{𝔽/F} (N_{K𝔽/𝔽}(λ)) · [A]
= N_{K/F} (N_{K𝔽/K}(λ)) · [A]
= cor_{K/F} (N_{K𝔽/K}(λ) · [A_{K}]) ∈ cor_{K/F} ((K^{×}) · [B]).

This implies that the order of ξ is odd in

\frac{Ker (H^{3}(F, ℚ/ℤ(2)) → H^{3}(F (X), ℚ/ℤ(2)))}{cor_{K/F} ((K^{×}) · [B])}

since [𝔽 : F] is odd. On the other hand, consider the following commutative diagram where the vertical maps are given by scalar extension and the horizontal maps are restriction and corestriction maps

\xymatrix{H^{3}(F, ℚ/ℤ(2)) \ar[r] \ar[d] & H^{3}(M, ℚ/ℤ(2)) \ar[r] \ar@{^{(}->}[d] & H^{3}(F, ℚ/ℤ(2)) \ar[d] \\ H^{3}(F (X), ℚ/ℤ(2)) \ar[r] & H^{3}(M(X_{M}), ℚ/ℤ(2)) \ar[r] & H^{3}(F (X), ℚ/ℤ(2))}

The image of ξ by the top row is [M : F]ξ = 8ξ. Moreover, a diagram chase shows that 8ξ is trivial in H^{3}(F, ℚ/ℤ(2)) because ξ is in the kernel of the left vertical map, and the central vertical map is injective. So, the order of ξ is then prime to [𝔽 : F]. It follows that ξ ∈ cor_{K/F} ((K^{×}) · [B]). The proof is complete. □

Now, assume that A a central simple algebra of degree 8 and exponent 2 over F. Let K = F (\sqrt{a}) be a quadratic field extension of F contained in A and let 𝔽 be an odd degree extension of F. Denote by B = C_{A}K the centralizer of K in A.

Remark 4.8. (1) The algebra A admits a decomposition adapted to K if and only if B has a descent to F. In other words, A admits a decomposition adapted to K if and only if δ_{K/F} (B) = 0. So, δ_{K/F} (B) ≠ 0 if A is indecomposable.

(2) One deduces from Proposition 4.7 that if K is not in a quaternion subalgebra of A, then the same holds over 𝔽.

(3) Let Ker(res) be the kernel of the restriction map

H^{3}(F, ℚ/ℤ(2)) ⟶ H^{3}(F (SB(A)), ℚ/ℤ(2)).

The quotient group

Ker(res)/[A] · H^{1}(F, ℚ/ℤ(2))

is identified with CH^{2}(SB(A))_{tors} in [19]. Arguing as in the proof of Proposition 4.7, we may see that the scalar extension map

CH^{2}(SB(A))_{tors} ⟶ CH^{2}(SB(A)_{𝔽})_{tors}

is injective. This injection may be also deduced from [12, Corollary 1.2 and Proposition 1.3].

Let A′ be the division algebra Brauer equivalent to

A′ ∼ A ⊗_{F} (a, t)_{F (t)}

where t is an indeterminate over F. Note that

A′_{K} ∼ A_{K} ⊗ K(t) ∼ B_{K(t)}.

The following proposition shows that if δ_{K/F} (B) is nonzero then the invariant Δ(A′), defined in [8, §11], is nonzero.

Proposition 4.9. The scalar extension map

\frac{H^{3}(F, μ_{2})}{cor_{K/F} ((K^{×}) · [B])} ⟶ \frac{H^{3}(F (t), μ_{2})}{(F (t)^{×}) · [A′]}  (4.3)

is an injection.

Proof. The map is clearly well-defined because

cor_{K/F} (K^{×} · [B]) ⊂ cor_{K(t)/F (t)}(K(t)^{×} · [B_{K(t)}]) ⊂ (F (t)^{×}) · [A′].

Let ξ ∈ H^{3}(F, μ_{2}). Suppose ξ = f[A′] = f([A] + [(a, t)]) for some f ∈ F (t)^{×}. Consider the residue map

∂_{t} : H^{3}(F (t), μ_{2}) ⟶ H^{2}(F, μ_{2})

where F (t) is equipped with its t-adic valuation v_{t} (see for instance [7, §7]). We have

∂_{t}(ξ) = ∂_{t}(f[A] + f[(a, t)]) = 0.

Since f is taken modulo F (t)^{×2}, we may assume that v_{t}(f) is either 0 or 1. If v_{t}(f) = 1, that is, f = tf_{0} for some t-adic unit f_{0}, then

∂_{t}(ξ) = ∂_{t}(f[A] + f[(a, t)]) = [A] + ∂_{t}((tf_{0}, a, t))
= [A] + ∂_{t}((−f_{0}, a, t))
= [A] + [(−f_{0}(0), a)] = 0.

Therefore A is Brauer equivalent to (−f_{0}(0), a). It follows that the two quotients of the map (4.3) are trivial. In this case, there is nothing to show.

Suppose v_{t}(f) = 0. One has

∂_{t}(f[A] + f[(a, t)]) = [(f(0), a)] = 0.

On the other hand, ξ = f(0)[A] by the specialization map ker ∂_{t} → H^{3}(F, μ_{2}) associated with t at 0. Since (f(0), a) = 0, we deduce that ξ ∈ (N_{K/F} (K^{×})) · [A] = cor_{K/F} ((K^{×}) · [B]). The proof is complete. □

Remark 4.10. According to Remark 4.3, the image of δ_{K/F} (B) by the scalar extension map (4.3) is Δ(A′). The proof of Theorem 1.2 provides an example of indecomposable algebra A′ of degree 8 and exponent 2 such that Δ(A′) is nonzero.

5. Proofs of the main statements

5.1. Proof of Theorem 1.1. We start out by the following lemma (a part of [9, Theorem 2]):

Lemma 5.1. Every central simple F -algebra of exponent 2 is Brauer equivalent to a tensor product of at most \frac{1}{2}u(F) − 1 quaternion algebras.

Proof. Let A be a central simple F -algebra of exponent 2. Let φ ∈ I^{2}F be an anisotropic quadratic form such that C(φ) ∼ A (by Merkurjev’s Theorem [16]). The form φ being anisotropic, dim φ ≤ u(F). Let φ′ be a subform of φ of codimension 1. The algebra C(φ) is Brauer equivalent to C_{0}(φ′) which is a tensor product of \frac{1}{2}(dim(φ′) − 1) = \frac{1}{2}(dim φ) − 1 quaternion algebras. □

Now, let A be a degree 8 and exponent 2 algebra over a field of u-invariant smaller than or equal to 8. By Lemma 5.1, A is Brauer equivalent to a tensor product of three quaternion algebras. Therefore, this equivalence is an isomorphism by dimension count; this proves that A is decomposable. This concludes the proof of Theorem 1.1.

5.2. A consequence. Let U and V be smooth complete geometrically irreducible varieties over F. In the appendix, Merkurjev gives conditions under which the scalar extension map CH(V) ⟶ CH(V_{F (U)}) is injective. For instance, let A be central simple F -algebra of degree 8 and exponent 2 and let K be a quadratic separable extension of F contained in A. Denote by X the Weil transfer of SB(C_{A}K) over F. Let q be a quadratic form over F with dim q ≥ 9. It follows from Theorem A.9 that the scalar extension map

CH^{2}(X)_{tors} ⟶ CH^{2}(X_{F (q)})_{tors}  (5.1)

is injective, where F (q) is the function field of the projective quadric defined by q = 0. This property does not hold anymore if one replaces X by SB(A). Indeed, Theorem 1.1 implies the following:

Corollary 5.2. Let A be a central simple algebra of degree 8 and exponent 2 over F. Assume that CH^{2}(SB(A))_{tors} ≠ 0. Then there exist an extension F′ of F and a 9-dimensional quadratic form q defined over F_{ℓ} such that the scalar extension map

CH^{2}(SB(A)_{F′})_{tors} ⟶ CH^{2}(SB(A)_{F′(q)})_{tors}

is not injective.

For the proof we need the following particular class of fields of u-invariant at most 8: starting with any field F over which there is an anisotropic quadratic form of dimension 8, one defines a tower of fields

F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{∞} = ⋃_{i} F_{i} =: M_{8}^{u}(F)

inductively as follows: if F_{i−1} is already given, the field F_{i} is the composite of all function fields F_{i−1}(φ), where φ ranges over (the isometry classes of) all 9dimensional forms over F_{i−1}. Clearly, any 9-dimensional form over 𝕄 is isotropic. Therefore u(M_{8}^{u}(F)) ≤ 8. Such a construction is due to Merkujev (see [17]).

Proof of Corollary 5.2. Let 𝕄 = M_{8}^{u}(F) be an extension as above; recall that u(𝕄) ≤ 8. The algebra A_{𝕄} being decomposable by Theorem 1.1, we have CH^{2}(SB(A)_{𝕄})_{tors} = 0. Since 𝕄 = ⋃_{i} F_{i}, there exists ℓ such that CH^{2}(SB(A)_{F_{ℓ}})_{tors} ≠ 0 and CH^{2}(SB(A)_{F_{ℓ+1}})_{tors} = 0. By definition, F_{ℓ+1} is the composite of all function fields F_{ℓ}(q), where q ranges over all 9-dimensional forms over F_{ℓ}. Hence, there exists an extension F′ of F_{ℓ} and a 9-dimensional form q over F_{ℓ} such that CH^{2}(SB(A)_{F′})_{tors} ≠ 0 and CH^{2}(SB(A)_{F′(q)})_{tors} = 0 as was to be shown.

□

5.3. Proof of Theorem 1.2. Put 𝕄 = M_{8}^{u}(F) and B = C_{A}K. As in the previous section we denote by A′ be the division algebra Brauer equivalent to A ⊗_{F} (a, t)_{F (t)}.

First proof. Recall that δ_{K/F} (C_{A}K) is in CH^{2}(X)_{tors} by Remark 4.6, where X is the Weil transfer of SB(C_{A}K). Since δ_{K/F} (C_{A}K) ≠ 0 (see Remark 4.8), it follows by injection (5.1) that the extension 𝕄(\sqrt{a}) is not in a quaternion subalgebra of A_{𝕄}. By [26, Proposition 2.10] (or [3]) the algebra A′_{𝕄} is an indecomposable algebra of degree 8 and exponent 2. This concludes the proof. □

Second proof. Consider the following injections

\frac{H^{3}(F, μ_{2})}{cor_{K/F} ((K^{×}) · [B])} ↪ \frac{H^{3}(𝕄, μ_{2})}{cor_{𝕄K/𝕄}(((𝕄K)^{×}) · [B_{𝕄K}])} ↪ \frac{H^{3}(𝕄(t), μ_{2})}{(𝕄(t)^{×}) · [A′_{𝕄}]}

where the first is due to Merkurjev (injection (5.1)) and the second by Proposition 4.9. Since δ_{K/F} (B) ≠ 0 and its image by the composite of these above injections is Δ(A′_{𝕄}), we have Δ(A′_{𝕄}) ≠ 0. This also means A′_{𝕄} is indecomposable by Garibaldi-Parimala-Tignol [8, §11]. □

5.4. Proof of Theorem 1.3. Here, we also need a particular class of fields of cohomological dimension at most 3: starting with any field F over which there is an anisotropic 3-fold Pfister form, one defines a tower of fields

F = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{∞} = ⋃_{i} F_{i} =: M_{3}^{cd}(F)

inductively as follows: the field F_{2i+1} is the maximal odd degree extension of F_{2i}; the field F_{2i+2} is the composite of all the function fields F_{2i+1}(π), where π ranges over all 4-fold Pfister forms over F_{2i+1}. The arguments used by Merkurjev in [17], show that cd_{2}(M_{3}^{cd}(F)) ≤ 3. Merkurjev used such a technique for constructing fields of cohomological dimension 2 over which there exist anisotropic quadratic forms of dimension 2d for an arbitrary integer d, i.e, counterexamples to Kaplansky’s conjecture in the theory of quadratic forms.

Now, let A be a central simple algebra of degree 8 and exponent 2 such that CH^{2}(SB(A))_{tors} ≠ 0. Put 𝕄 = M_{3}^{cd}(F) and let 𝔽 be an odd degree extension of F. The scalar extension map

CH^{2}(SB(A))_{tors} ⟶ CH^{2}(SB(A)_{𝔽})_{tors}

is injective by Remark 4.8. On the other hand, let π be a 4-fold Pfister form over 𝔽. It follows from Theorem A.7 that the scalar extension map

CH^{2}(SB(A)_{𝔽})_{tors} ⟶ CH^{2}(SB(A)_{𝔽(π)})_{tors}  (5.2)

is injective. We deduce from these two latter injections that CH^{2}(SB(A)_{𝕄})_{tors} ≠ 0; and so A_{𝕄} is indecomposable. The algebra A_{𝕄} being indecomposable, we must have cd_{2}(𝕄) > 2 by Theorem 3.1. Hence, cd_{2}(𝕄) = 3. This completes the proof.

Acknowledgements. This work is part of my PhD thesis at Université catholique de Louvain and Université Paris 13. I would like to thank my thesis supervisors, Anne Quéguiner-Mathieu and Jean-Pierre Tignol, for directing me towards this problem. I would also like to thank Karim Johannes Becher for suggesting Theorem 1.1 and the idea of the proof of Lemma 2.1. I am particularly grateful to Alexander S. Merkurjev for providing the appendix of the paper.

References

[1] J. K. Arason, Cohomologische invarianten quadratischer Formen, J. Algebra, 36, 448–491, (1975).

[2] S. A. Amitsur, L. H. Rowen, J.-P. Tignol, Division algebras of degree 4 and 8 with involution, Isreal J. Math., 33, 133–148, (1979),

[3] D. Barry, Square-central elements in tensor products of quaternion algebras, In preparation.

[4] A. Borel, J.-P.Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv., 39, 111–164, (1964-65).

[5] R. Elman, N. Karpenko, S. A. Merkurjev, The algebraic and geometric theory of quadratic forms, Colloquium Publ., vol. 56, AMS, Providence, RI, (2008).

[6] R. Elman, T. Y. Lam, J.-P. Tignol, A. Wadsworth, Witt rings and Brauer groups under multiquadratic extensions, I, Amer. J. Math., 105, 1119–1170, (1983).

[7] S. Garibaldi, A. Merkurjev, J.-P. Serre, Cohomological invariants in Galois cohomology, vol. 28 of University Lecture Series. American Mathematical Society, Providence, RI, (2003).

[8] S. Garibaldi, R. Parimala, J.-P. Tignol, Discriminant of symplectic involutions, Pure App. Math. Quart., 5, 349 - 374, (2009).

[9] B. Kahn, Quelques remarques sur le u-invariant, Sém. Théor. Nombres Bordeaux, 2, 155–161, (1990). Erratum in: Sém. Théor. Nombres Bordeaux 3, 247,(1991).

[10] B. Kahn, Formes quadratiques sur un corps, Société Mathématique de France, Paris, (2008).

[11] N. A. Karpenko, Torsion in CH^{2} of Severi-Brauer varieties and indecomposability of generic algebras, Manuscripta Math., 88 (1), 109–117, (1995).

[12] N. A. Karpenko, Codimension 2 cycles on Severi-Brauer varieties, K-Theory, 13, 305–330, (1998).

[13] N. A. Karpenko, Weil transfer of algebraic cycles, Indag. Mathem., 11 (1), 73–86, (2000).

[14] T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, AMS, Providence, (2005).

[15] T. Y. Lam, D. B. Leep, Tignol, J.-P., Biquaternion algebras and quartic extensions, Publ. Math. I.H.E.S, 77, 63--102, (1993).

[16] A. S. Merkurjev, On the norm residue symbol of degree 2, Sov. Math. Dokl., 24, 546–551, (1981).

[17] A. S. Merkurjev, Simple algebras and quadratic forms, Izv. Akad. Nauk. SSSR Ser. Mat 55, no. 1, 218–224, (1991). English translation: Math. USSR-Ivz. 38, no.1, 215–221, (1992).

[18] A. S. Merkurjev, A. A. Suslin, K-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk. SSSR Ser. Mat 46, no. 5, 1011-1046, (1982). English translation: Math. USSR-Ivz. 21 , no.2, 307–340, (1983).

[19] E. Peyre, Products of Severi-Brauer varieties and Galois cohomology, Proc. Sympos. Pure Math., 58, 369–401, AMS, Providence, (1995).

[20] E. Peyre, Corps de fonction de variétés homogènes et cohomologie galoisienne, C. R. Acad. Sci. Paris, t. 321, Série I , 891–896, (1995).

[21] R. S. Pierce, Associative Algebras, Graduate Texts in Mathematics, vol. 88. Springer-Verlag, New York-Heidelberg-Berlin, (1982).

[22] W. Scharlau, Quadratic and Hermitian forms, Grundlehren, vol. 270. Springer, Berlin, (1985).

[23] C. Scheiderer, Real and étale cohomology, Lecture notes in mathematics 1588, Springer, (1994).

[24] J.-P. Serre, Cohomologie Galoisienne, 5e éd., révisée et complétée, Lecture notes in mathematics 5, Springer, (1997).

[25] A. S. Sivatski, Applications of Clifford algebras to involutions and quadratic forms, Com. in Algebra, 33, 937–951, (2005).

[26] J.-P. Tignol, Algèbres indécomposables d’exposant premier, Adv. in Math, 65, 205–228, (1987).

Appendix A.

On the Chow Group of Cycles of Codimension 2

by Alexander S. Merkurjev

Let X be an algebraic variety over F. We write A^{i}(X, K_{n}) for the homology group of the complex

∐_{x∈X^{(i−1)}} K_{n−i+1}(F (x)) ⟶^{∂} ∐_{x∈X^{(i)}} K_{n−i}(F (x)) ⟶^{∂} ∐_{x∈X^{(i+1)}} K_{n−i−1}(F (x)),

where K_{j} are the Milnor K-groups and X^{(i)} is the set of points in X of codimension i (see [7, §5]). In particular, A^{i}(X, K_{i}) = CH^{i}(X) is the Chow group of classes of codimension i algebraic cycles on X.

Let X and Y be smooth complete geometrically irreducible varieties over F.

Proposition A.1. Suppose that for every field extension K/F we have:

(1) The natural map CH^{1}(X) ⟶ CH^{1}(X_{K}) is an isomorphism of torsion free groups,

(2) The product map CH^{1}(X_{K}) ⊗ K^{×} ⟶ A^{1}(X_{K}, K_{2}) is an isomorphism.

Then the natural sequence

0 ⟶ (CH^{1}(X) ⊗ CH^{1}(Y)) ⊕ CH^{2}(Y) ⟶ CH^{2}(X × Y) ⟶ CH^{2}(X_{F (Y)})

is exact.

Proof. Consider the spectral sequence

E_{1}^{p,q} = ∐_{y∈Y^{(p)}} A^{q}(X_{F (y)}, K_{2−p}) ⟹ A^{p+q}(X × Y, K_{2})

for the projection X × Y ⟶ Y (see [7, Cor. 8.2]). The nonzero terms of the first page are the following:

\xymatrix{CH^{2}(X_{F (Y)}) \\ A^{1}(X_{F (Y)}, K_{2}) \ar[r] & ∐_{y∈Y^{(1)}} CH^{1}(X_{F (y)}) \\ A^{0}(X_{F (Y)}, K_{2}) \ar[r] & ∐_{y∈Y^{(1)}} A^{0}(X_{F (y)}, K_{1}) \ar[r] & ∐_{y∈Y^{(2)}} CH^{0}(X_{F (y)}).}

Then E_{1}^{2,0} = ∐_{y∈Y^{(2)}} ℤ is the group of cycles on X of codimension 2 and E_{1}^{1,0} = ∐_{y∈Y^{(1)}} F (y)^{×} as X is complete. It follows that E_{2}^{2,0} = CH^{2}(Y).

By assumption, the differential E_{1}^{0,1} ⟶ E_{1}^{1,1} is identified with the map

CH^{1}(X) ⊗ (F (Y)^{×} ⟶ ∐_{y∈Y^{(1)}} ℤ).

Since Y is complete and CH^{1}(X) is torsion free, we have E_{2}^{0,1} = CH^{1}(X) ⊗ F^{×} and E_{∞}^{1,1} = E_{2}^{1,1} = CH^{1}(X) ⊗ CH^{1}(Y).

The edge map

A^{1}(X × Y, K_{2}) ⟶ E_{2}^{0,1} = CH^{1}(X) ⊗ F^{×}

is split by the product map

CH^{1}(X) ⊗ F^{×} = A^{1}(X, K_{1}) ⊗ A^{0}(Y, K_{1}) ⟶ A^{1}(X × Y, K_{2}),

hence the edge map is surjective. Therefore, the differential E_{2}^{0,1} ⟶ E_{2}^{2,0} is trivial and hence E_{∞}^{2,0} = E_{2}^{2,0} = CH^{2}(Y). Thus, the natural homomorphism

CH^{2}(Y) ⟶ Ker(CH^{2}(X × Y) ⟶ CH^{2}(X_{F (Y)}))

is injective and its cokernel is isomorphic to CH^{1}(X) ⊗ CH^{1}(Y). The statement follows. □

Example A.2. Let X be a projective homogeneous variety of a semisimple algebraic group over F. There exist an étale F -algebra E and an Azumaya E-algebra A such that for i = 0 and 1, we have an exact sequence

0 ⟶ A^{1}(X, K_{i+1}) ⟶ K_{i}(E) ⟶^{ρ} H^{i+2}(F, ℚ/ℤ(i + 1)),

where ρ(x) = N_{E/F} ((x) ∪ [A]) (see [4] and [5]). If the algebras E and A are split, then ρ is trivial and for every field extension K/F,

CH^{1}(X) ≃ K_{0}(E) ≃ K_{0}(E ⊗ K) ≃ CH^{1}(X_{K}),
A^{1}(X_{K}, K_{2}) ≃ K_{1}(E ⊗ K) ≃ K_{0}(E) ⊗ K^{×} ≃ CH^{1}(X_{K}) ⊗ K^{×}.

Therefore, the condition (1) and (2) in Proposition A.1 hold. For example, if X is a smooth projective quadric of dimension at least 3, then E = F and A is split.

Now consider the natural complex

CH^{2}(X) ⊕ (CH^{1}(X) ⊗ CH^{1}(Y)) ⟶ CH^{2}(X × Y) ⟶ CH^{2}(Y_{F (X)}).  (A.1)

Proposition A.3. Suppose that (1) The Grothendieck group K_{0}(Y) is torsion-free,

(2) The product map K_{0}(X) ⊗ K_{0}(Y) ⟶ K_{0}(X × Y) is an isomorphism. Then the sequence (A.1) is exact.

Proof. It follows from the assumptions that the map K_{0}(Y) ⟶ K_{0}(Y_{F (X)}) is injective and the kernel of the natural homomorphism K_{0}(X × Y) ⟶ K_{0}(Y_{F (X)}) coincides with

I_{0}(X) ⊗ K_{0}(Y),

where I_{0}(X) is the kernel of the rank homomorphism K_{0}(X) ⟶ ℤ.

The kernel of the second homomorphism in the sequence (A.1) is generated by the classes of closed integral subschemes Z ⊂ X × Y that are not dominant over X. By Riemann-Roch (see [2]), we have [Z] = −c_{2}([O_{Z}]) in CH^{2}(X × Y), where c_{i} : K_{0}(X × Y) ⟶ CH^{i}(X × Y) is the i-th Chern class map. As

[O_{Z}] ∈ Ker(K_{0}(X × Y) ⟶ K_{0}(Y_{F (X)})) = I_{0}(X) ⊗ K_{0}(Y),

it suffices to to show that c_{2}(I_{0}(X) ⊗ K_{0}(Y)) is contained in the image M of the first map in the sequence (A.1).

The formula c_{2}(x + y) = c_{2}(x) + c_{1}(x)c_{1}(y) + c_{2}(y) shows that it suffices to prove that for all a, a′ ∈ I_{0}(X) and b, b′ ∈ K_{0}(Y), the elements c_{1}(ab) · c_{1}(a′b′) and c_{2}(ab) are contained in M. This follows from the formulas (see [1, Remark 3.2.3 and Example 14.5.2]): c_{1}(ab) = mc_{1}(a) + nc_{1}(b) and

c_{2}(ab) = \frac{m^{2} − m}{2}c_{1}(a)^{2} + mc_{2}(a) + (nm − 1)c_{1}(a)c_{1}(b) + \frac{n^{2} − n}{2}c_{1}(b)^{2} + nc_{2}(b),

where n = rank(a) and m = rank(b). □

Example A.4. If Y is a projective homogeneous variety, then the condition (1) holds by [6]. If X is a projective homogeneous variety of a semisimple algebraic group G over F and the Tits algebras of G are split, then it follows from [6] that the condition (2) also holds for any Y. For example, if the even Clifford algebra of a nondegenerate quadratic form is split, then the corresponding projective quadric X satisfies (2) for any Y.

For any field extension K/F, let K^{s} denote the subfield of elements that are algebraic and separable over F.

Proposition A.5. Suppose that for every field extension K/F we have: (1) The natural map CH^{1}(X) ⟶ CH^{1}(X_{K}) is an isomorphism,

(2) The natural map CH^{1}(Y_{K^{s}}) → CH^{1}(Y_{K}) is an isomorphism. Then the sequence (A.1) is exact.

Proof. Consider the spectral sequence

E_{1}^{p,q}(F) = ∐_{x∈X^{(p)}} A^{q}(Y_{F (x)}, K_{2−p}) ⟹ A^{p+q}(X × Y, K_{2})  (A.2)

for the projection X × Y ⟶ X. The nonzero terms of the first page are the following:

\xymatrix{CH^{2}(Y_{F (X)}) \\ A^{1}(Y_{F (X)}, K_{2}) \ar[r] & ∐_{x∈X^{(1)}} CH^{1}(Y_{F (x)}) \\ A^{0}(Y_{F (X)}, K_{2}) \ar[r] & ∐_{x∈X^{(1)}} A^{0}(Y_{F (x)}, K_{1}) \ar[r] & ∐_{x∈X^{(2)}} CH^{0}(Y_{F (x)}).}

As in the proof of Proposition A.1, we have E_{2}^{2,0}(F) = CH^{2}(X). For a field extension K/F, write C(K) for the factor group

Ker(CH^{2}(X_{K} × Y_{K}) ⟶ CH^{2}(Y_{K(X)}))/ Im(CH^{2}(X_{K}) ⟶ CH^{2}(X_{K} × Y_{K})).

The spectral sequence (A.2) for the varieties X_{K} and Y_{K} over K yields an isomorphism C(K) ≃ E_{2}^{1,1}(K). We have a natural composition

CH^{1}(X_{K}) ⊗ CH^{1}(Y_{K}) ⟶ E_{1}^{1,1}(K) ⟶ E_{2}^{1,1}(K) ≃ C(K).

We claim that the group C(F) is generated by images of the compositions

CH^{1}(X_{K}) ⊗ CH^{1}(Y_{K}) ⟶ C(K) ⟶^{N_{K/F}} C(F)

over all finite separable field extensions K/F (here N_{K/F} is the norm map for the extension K/F).

The group C(F) is generated by images of the maps

φ_{x} : CH^{1}(Y_{F (x)}) ⟶ E_{2}^{1,1}(F) ≃ C(F)

over all points x ∈ X^{(1)}. Pick such a point x and let K := F (x)^{s} be the subfield of elements that are separable over F. Then K/F is a finite separable field extension. Let x′ ∈ X_{K}^{(1)} be a point over x such that K(x′) ≃ F (x). Then φ_{x} coincides with the composition

CH^{1}(Y_{K(x′)}) ⟶ C(K) ⟶^{N_{K/F}} C(F).

By assumption, the map CH^{1}(Y_{K}) ⟶ CH^{1}(Y_{K(x′)}) is an isomorphism, hence the image of φ_{x} coincides with the image of

[x′] ⊗ CH^{1}(Y_{K}) ⟶ C(K) ⟶^{N_{K/F}} C(F),

whence the claim.

As CH^{1}(X) ⟶ CH^{1}(X_{K}) is an isomorphism for every field extension K/F, the projection formula shows that the map CH^{1}(X) ⊗ CH^{1}(Y) ⟶ C(F) is surjective. The statement follows. □

Example A.6. Let Y be a projective homogeneous variety with the F -algebras E and A as in Example A.2. If A is split, then CH^{1}(Y_{K}) = K_{0}(E ⊗ K) for every field extension K/F. As K^{s} is separably closed in K, the natural map K_{0}(E ⊗ K^{s}) ⟶ K_{0}(E ⊗ K) is an isomorphism, therefore, the condition (2) holds.

Write C̃H^{2}(X × Y) for the cokernel of the product map CH^{1}(X) ⊗ CH^{1}(Y) ⟶ CH^{2}(X × Y). We have the following commutative diagram:

\xymatrix{ &  & CH^{2}(X) \ar[d] \ar[dr] \\ 0 \ar[r] & CH^{2}(Y) \ar[r] \ar[dr] & C̃H^{2}(X × Y) \ar[r] \ar[d] & CH^{2}(X_{F (Y)}) \\  &  & CH^{2}(Y_{F (X)})}

Proposition A.1 gives conditions for the exactness of the row in the diagram and Propositions A.3 and A.5 - for the exactness of the column in the diagram.

A diagram chase yields together with Propositions A.1, A.3 and A.5 yields the following statements.

Theorem A.7. Let X and Y be smooth complete geometrically irreducible varieties such that for every field extension K/F:

(1) The natural map CH^{1}(X) ⟶ CH^{1}(X_{K}) is an isomorphism of torsion free groups,

(2) The natural map CH^{2}(X) ⟶ CH^{2}(X_{K}) is injective, (3) The product map CH^{1}(X_{K}) ⊗ K^{×} ⟶ A^{1}(X_{K}, K_{2}) is an isomorphism,

(4) The Grothendieck group K_{0}(Y) is torsion-free,

(5) The product map K_{0}(X) ⊗ K_{0}(Y) ⟶ K_{0}(X × Y) is an isomorphism. Then the natural map CH^{2}(Y) ⟶ CH^{2}(Y_{F (X)}) is injective.

Remark A.8. The conditions (1) − (3) hold for a smooth projective quadric X of dimension at least 7 by [3, Theorem 6.1] and Example A.2. By Example A.4, the conditions (4) and (5) hold if the even Clifford algebra of X is split and Y is a projective homogeneous variety.

Theorem A.9. Let X and Y be smooth complete geometrically irreducible varieties such that for every field extension K/F:

(1) The natural map CH^{1}(X) ⟶ CH^{1}(X_{K}) is an isomorphism of torsion free groups,

(2) The natural map CH^{2}(X) ⟶ CH^{2}(X_{K}) is injective,

(3) The product map CH^{1}(X_{K}) ⊗ K^{×} ⟶ A^{1}(X_{K}, K_{2}) is an isomorphism,

(4) The natural homomorphism CH^{1}(Y_{K^{s}}) → CH^{1}(Y_{K}) is an isomorphism. Then the natural map CH^{2}(Y) ⟶ CH^{2}(Y_{F (X)}) is injective.

Remark A.10. The conditions (1) − (3) hold for a smooth projective quadric X of dimension at least 7 and a projective homogeneous variety Y with the split Azumaya algebra by Remark A.8 and Example A.6.

References

[1] W. Fulton, Intersection theory, Springer-Verlag, Berlin, 1984.

[2] A. Grothendieck, La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137–154.

[3] N. A. Karpenko, Algebro-geometric invariants of quadratic forms, Algebra i Analiz 2 (1990), no. 1, 141–162.

[4] A. S. Merkurjev, The group H^{1}(X, K_{2}) for projective homogeneous varieties, Algebra i Analiz 7 (1995), no. 3, 136–164.

[5] A. S. Merkurjev and J.-P. Tignol, The multipliers of similitudes and the Brauer group of homogeneous varieties, J. Reine Angew. Math. 461 (1995), 13–47.

[6] I. A. Panin, On the algebraic K-theory of twisted flag varieties, K-Theory 8 (1994), no. 6, 541–585.

[7] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393 (electronic).

ICTEAM Institute, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

   E-mail address: demba.barry@uclouvain.be

Department of Mathematics, University of California, Los Angeles, CA, USA

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