Decomposable and indecomposable algebras of degree $8$ and exponent 2

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 $\mathbb M(t)$ where the $u$-invariant of $\mathbb M$ is $8$ and $t$ is an indeterminate. The construction of these indecomposable algebras uses Chow group computations provided by Merkurjev in “Appendix”.

Appendix 1: On the Chow group of cycles of codimension 2

A. S. Merkurjev Department of Mathematics, University of California, Los Angeles, CA, USA email: merkurev@math.ucla.edu

Let $X$ be an algebraic variety over $F$. We write $A^i(X,K_n)$ for the homology group of the complex

$$\begin{aligned} \coprod _{x\in X^{(i-1)}}K_{n-i+1}\bigl (F(x)\bigr )\mathop {\longrightarrow }\limits ^{\partial } \coprod _{x\in X^{(i)}}K_{n-i}\bigl (F(x)\bigr )\mathop {\longrightarrow }\limits ^{\partial } \coprod _{x\in X^{(i+1)}}K_{n-i-1}\bigl (F(x)\bigr ), \end{aligned}$$

where $K_j$ are the Milnor $K$-groups and $X^{(i)}$ is the set of points in $X$ of codimension $i$ (see [29, §5]). In particular, $A^i(X,K_i)={\mathrm{CH }}^i(X)$ 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 6.1

Suppose that for every field extension $K/F$ we have:

• (1) The natural map ${\mathrm{CH }}^1(X)\longrightarrow {\mathrm{CH }}^1(X_K)$ is an isomorphism of torsion free groups,

• (2) The product map ${\mathrm{CH }}^1(X_K)\otimes K^\times \longrightarrow A^1(X_K,K_2)$ is an isomorphism.

Then the natural sequence

$$\begin{aligned} 0\longrightarrow \bigl ({\mathrm{CH }}^1(X)\otimes {\mathrm{CH }}^1(Y)\bigr )\oplus {\mathrm{CH }}^2(Y)\longrightarrow {\mathrm{CH }}^2(X\times Y)\longrightarrow {\mathrm{CH }}^2(X_{F(Y)}), \end{aligned}$$

is exact.

Proof

Consider the spectral sequence

$$\begin{aligned} E_1^{p,q}=\coprod _{y\in Y^{(p)}} A^q(X_{F(y)},K_{2-p})\Longrightarrow A^{p+q}(X\times Y,K_2), \end{aligned}$$

for the projection $X\times Y\longrightarrow Y$ (see [29, Cor. 8.2]). The nonzero terms of the first page are the following:

Then $E_1^{2,0}=\coprod _{y\in Y^{(2)}} {\mathbb {Z}}$ is the group of cycles on $X$ of codimension 2 and $E_1^{1,0}=\coprod _{y\in Y^{(1)}} F(y)^\times$ as $X$ is complete. It follows that $E_2^{2,0}={\mathrm{CH }}^2(Y)$.

By assumption, the differential $E_1^{0,1}\longrightarrow E_1^{1,1}$ is identified with the map

$$\begin{aligned} {\mathrm{CH }}^1(X)\otimes \left( F(Y)^\times \longrightarrow \coprod _{y\in Y^{(1)}} {\mathbb {Z}}\right) . \end{aligned}$$

Since $Y$ is complete and ${\mathrm{CH }}^1(X)$ is torsion free, we have $E_2^{0,1}={\mathrm{CH }}^1(X)\otimes F^\times$ and $E_\infty ^{1,1}=E_2^{1,1}={\mathrm{CH }}^1(X)\otimes {\mathrm{CH }}^1(Y)$.

The edge map

$$\begin{aligned} A^1(X\times Y,K_2)\longrightarrow E_2^{0,1}={\mathrm{CH }}^1(X)\otimes F^\times , \end{aligned}$$

is split by the product map

$$\begin{aligned} {\mathrm{CH }}^1(X)\otimes F^\times =A^1(X,K_1)\otimes A^0(Y,K_1)\longrightarrow A^1(X\times Y,K_2), \end{aligned}$$

hence the edge map is surjective. Therefore, the differential $E_2^{0,1}\longrightarrow E_2^{2,0}$ is trivial and hence $E_\infty ^{2,0}= E_2^{2,0}={\mathrm{CH }}^2(Y)$. Thus, the natural homomorphism

$$\begin{aligned} {\mathrm{CH }}^2(Y)\longrightarrow {\mathrm{Ker }}\bigl ({\mathrm{CH }}^2(X\times Y)\longrightarrow {\mathrm{CH }}^2(X_{F(Y)})\bigr ) \end{aligned}$$

is injective and its cokernel is isomorphic to ${\mathrm{CH }}^1(X)\otimes {\mathrm{CH }}^1(Y)$. The statement follows. $\square$

Example 6.2

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

$$\begin{aligned} 0\longrightarrow A^1(X,K_{i+1})\longrightarrow K_i(E) \overset{\rho }{\longrightarrow } H^{i+2}\bigl (F,{\mathbb {Q}}/{\mathbb {Z}}(i+1)\bigr ), \end{aligned}$$

where $\rho (x)=N_{E/F}\bigl ((x)\cup [A]\bigr )$ (see [23] and [24]). If the algebras $E$ and $A$ are split, then $\rho$ is trivial and for every field extension $K/F$,

$$\begin{aligned}&{\mathrm{CH }}^1(X)\simeq K_0(E)\simeq K_0(E\otimes K)\simeq {\mathrm{CH }}^1(X_K),\\&A^1(X_K,K_{2})\simeq K_1(E\otimes K)\simeq K_0(E)\otimes K^\times \simeq {\mathrm{CH }}^1(X_K)\otimes K^\times . \end{aligned}$$

Therefore, the condition $(1)$ and $(2)$ in Proposition 6.1 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

$$\begin{aligned} {\mathrm{CH }}^2(X)\oplus \bigl ({\mathrm{CH }}^1(X)\otimes {\mathrm{CH }}^1(Y)\bigr )\longrightarrow {\mathrm{CH }}^2(X\times Y)\longrightarrow {\mathrm{CH }}^2(Y_{F(X)}). \end{aligned}$$

(6.1)

Proposition 6.3

Suppose that

• (1) The Grothendieck group $K_0(Y)$ is torsion-free,

• (2) The product map $K_0(X)\otimes K_0(Y)\longrightarrow K_0(X\times Y)$ is an isomorphism.

Then the sequence (6.1) is exact.

Proof

It follows from the assumptions that the map $K_0(Y)\longrightarrow K_0(Y_{F(X)})$ is injective and the kernel of the natural homomorphism $K_0(X\times Y)\longrightarrow K_0(Y_{F(X)})$ coincides with

$$\begin{aligned} I_0(X)\otimes K_0(Y), \end{aligned}$$

where $I_0(X)$ is the kernel of the rank homomorphism $K_0(X)\longrightarrow {\mathbb {Z}}$.

The kernel of the second homomorphism in the sequence (6.1) is generated by the classes of closed integral subschemes $Z\subset X\times Y$ that are not dominant over $X$. By Riemann–Roch (see [11]), we have $[Z]=-c_2\bigl ([O_Z]\bigr )$ in ${\mathrm{CH }}^2(X\times Y)$, where $c_i:K_0(X\times Y)\longrightarrow {\mathrm{CH }}^i(X\times Y)$ is the $i$th Chern class map. As

$$\begin{aligned}{}[O_Z]\in {\mathrm{Ker }}(K_0(X\times Y)\longrightarrow K_0(Y_{F(X)})\bigr )=I_0(X)\otimes K_0(Y), \end{aligned}$$

it suffices to to show that $c_2\bigl (I_0(X)\otimes K_0(Y)\bigr )$ is contained in the image $M$ of the first map in the sequence (6.1).

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 $a,a'\in I_0(X)$ and $b,b'\in K_0(Y)$, the elements $c_1(ab)\cdot c_1 (a'b')$ and $c_2(ab)$ are contained in $M$. This follows from the formulas (see [8, Remark 3.2.3 and Example 14.5.2]): $c_1(ab)=mc_1(a)+nc_1(b)$ and

$$\begin{aligned} c_2(ab)=\frac{m^2-m}{2}c_1(a)^2+mc_2(a)+(nm-1)c_1(a)c_1(b)+\frac{n^2-n}{2}c_1(b)^2+nc_2(b), \end{aligned}$$

where $n=\mathrm{rank }(a)$ and $m=\mathrm{rank }(b)$. $\square$

Example 6.4

If $Y$ is a projective homogeneous variety, then the condition $(1)$ holds by [25]. 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 [25] 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 6.5

Suppose that for every field extension $K/F$ we have:

• (1) The natural map ${\mathrm{CH }}^1(X)\longrightarrow {\mathrm{CH }}^1(X_K)$ is an isomorphism,

• (2) The natural map ${\mathrm{CH }}^1(Y_{K^s})\rightarrow {\mathrm{CH }}^1(Y_K)$ is an isomorphism.

Then the sequence (6.1) is exact.

Proof

Consider the spectral sequence

$$\begin{aligned} E_1^{p,q}(F)=\coprod _{x\in X^{(p)}} A^q(Y_{F(x)},K_{2-p})\Longrightarrow A^{p+q}(X\times Y,K_2) \end{aligned}$$

(6.2)

for the projection $X\times Y\longrightarrow X$. The nonzero terms of the first page are the following:

As in the proof of Proposition 6.1, we have $E_2^{2,0}(F)={\mathrm{CH }}^2(X)$. For a field extension $K/F$, write $C(K)$ for the factor group

$$\begin{aligned} {\mathrm{Ker }}\bigl ({\mathrm{CH }}^2(X_K\times Y_K)\longrightarrow {\mathrm{CH }}^2(Y_{K(X)})\bigr )/\mathfrak {I}\bigl ({\mathrm{CH }}^2(X_K)\longrightarrow {\mathrm{CH }}^2(X_K\times Y_K)\bigr ). \end{aligned}$$

The spectral sequence (6.2) for the varieties $X_K$ and $Y_K$ over $K$ yields an isomorphism $C(K)\simeq E_2^{1,1}(K)$. We have a natural composition

$$\begin{aligned} {\mathrm{CH }}^1(X_K)\otimes {\mathrm{CH }}^1(Y_K)\longrightarrow E_1^{1,1}(K)\longrightarrow E_2^{1,1}(K)\simeq C(K). \end{aligned}$$

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

$$\begin{aligned} {\mathrm{CH }}^1(X_K)\otimes {\mathrm{CH }}^1(Y_K)\longrightarrow C(K)\overset{N_{K/F}}{\longrightarrow } C(F) \end{aligned}$$

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

$$\begin{aligned} \varphi _x:{\mathrm{CH }}^1(Y_{F(x)})\longrightarrow E_2^{1,1}(F)\simeq C(F) \end{aligned}$$

over all points $x\in X^{(1)}$. 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 $x'\in X_K^{(1)}$ be a point over $x$ such that $K(x')\simeq F(x)$. Then $\varphi _x$ coincides with the composition

$$\begin{aligned} {\mathrm{CH }}^1(Y_{K(x')})\longrightarrow C(K)\overset{N_{K/F}}{\longrightarrow } C(F). \end{aligned}$$

By assumption, the map ${\mathrm{CH }}^1(Y_{K})\longrightarrow {\mathrm{CH }}^1(Y_{K(x')})$ is an isomorphism, hence the image of $\varphi _x$ coincides with the image of

$$\begin{aligned}{}[x']\otimes {\mathrm{CH }}^1(Y_K)\longrightarrow C(K)\overset{N_{K/F}}{\longrightarrow } C(F), \end{aligned}$$

whence the claim.

As ${\mathrm{CH }}^1(X)\longrightarrow {\mathrm{CH }}^1(X_K)$ is an isomorphism for every field extension $K/F$, the projection formula shows that the map ${\mathrm{CH }}^1(X)\otimes {\mathrm{CH }}^1(Y)\longrightarrow C(F)$ is surjective. The statement follows. $\square$

Example 6.6

Let $Y$ be a projective homogeneous variety with the $F$-algebras $E$ and $A$ as in Example 6.2. If $A$ is split, then ${\mathrm{CH }}^1(Y_K)=K_0(E\otimes K)$ for every field extension $K/F$. As $K^s$ is separably closed in $K$, the natural map $K_0(E\otimes K^s)\longrightarrow K_0(E\otimes K)$ is an isomorphism, therefore, the condition $(2)$ holds.

Write $\widetilde{{\mathrm{CH }}}^2(X\times Y)$ for the cokernel of the product map ${\mathrm{CH }}^1(X)\otimes {\mathrm{CH }}^1(Y)\longrightarrow {\mathrm{CH }}^2(X\times Y)$. We have the following commutative diagram:

Proposition 6.1 gives conditions for the exactness of the row in the diagram and Propositions 6.3 and 6.5—for the exactness of the column in the diagram.

A diagram chase yields together with Propositions 6.1, 6.3 and 6.5 yields the following statements.

Theorem 6.7

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

• (1) The natural map ${\mathrm{CH }}^1(X)\longrightarrow {\mathrm{CH }}^1(X_K)$ is an isomorphism of torsion free groups,

• (2) The natural map ${\mathrm{CH }}^2(X)\longrightarrow {\mathrm{CH }}^2(X_K)$ is injective,

• (3) The product map ${\mathrm{CH }}^1(X_K)\otimes K^\times \longrightarrow A^1(X_K,K_2)$ is an isomorphism,

• (4) The Grothendieck group $K_0(Y)$ is torsion-free,

• (5) The product map $K_0(X)\otimes K_0(Y)\longrightarrow K_0(X\times Y)$ is an isomorphism.

Then the natural map ${\mathrm{CH }}^2(Y)\longrightarrow {\mathrm{CH }}^2(Y_{F(X)})$ is injective.

Remark 6.8

The conditions (1)–(3) hold for a smooth projective quadric $X$ of dimension at least $7$ by [14, Theorem 6.1] and Example 6.2. By Example 6.4, the conditions $(4)$ and $(5)$ hold if the even Clifford algebra of $X$ is split and $Y$ is a projective homogeneous variety.

Theorem 6.9

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

• (1) The natural map ${\mathrm{CH }}^1(X)\longrightarrow {\mathrm{CH }}^1(X_K)$ is an isomorphism of torsion free groups,

• (2) The natural map ${\mathrm{CH }}^2(X)\longrightarrow {\mathrm{CH }}^2(X_K)$ is injective,

• (3) The product map ${\mathrm{CH }}^1(X_K)\otimes K^\times \longrightarrow A^1(X_K,K_2)$ is an isomorphism,

• (4) The natural homomorphism ${\mathrm{CH }}^1(Y_{K^s})\rightarrow {\mathrm{CH }}^1(Y_K)$ is an isomorphism.

Then the natural map ${\mathrm{CH }}^2(Y)\longrightarrow {\mathrm{CH }}^2(Y_{F(X)})$ is injective.

Remark 6.10

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 6.8 and Example 6.6.