Descent and Decomposability of Algebras of Exponent 2 in characteristic 2

Using a cohomological invariant of symplectic involutions, Garibaldi–Parimala–Tignol define in [14 §11] a cohomological invariant $\Delta$ of central simple algebras of degree 8 and exponent 2 over arbitrary fields of characteristic different from 2. This invariant vanishes when the algebra decomposes into a tensor product of three quaternion algebras. Examples where it does not vanish were given by Barry in [7]; Merkurjev–Neshitov–Zainoulline showed that it is nontrivial generically [22 §4a]. The purpose of the present paper is to describe a similar construction in characteristic 2, based on the discriminant Pfister form of symplectic involutions from [11] or [27]. Being rooted in quadratic forms, the invariant $\Delta$ in Definition 1.3 below takes values in a quotient of the group $I_q^3$ from the fundamental filtration of the Witt group of the base field. Kato’s isomorphisms from [18] allow an equivalent formulation in terms of Kato–Milne cohomology.

While the invariant $\Delta$ yields a necessary criterion for decomposability of central simple algebras of degree 8 and exponent 2, variants of the construction using discriminant Pfister forms yield necessary and sufficient criteria for the descent of biquaternion algebras over quadratic extensions, and for the descent of quaternion algebras over biquadratic extensions that are not

Key words and phrases. Incomposable division algebras, symplectic involutions, quadratic forms over discretely valued fields.

The authors acknowledge support from Wallonie–Bruxelles International and the French government in the framework of “Partenariats Hubert Curien” (Project 2-algèbres de degré 8 et d’exposant 2). Research leading to this work was initiated while the second author was visiting the Sorbonne Paris Nord university, whose hospitality is gratefully acknowledged. purely inseparable. These variants are an additional theme of the paper; they play a central role in the construction of examples showing that the invariant $\Delta$ is not trivial.

The definition of the invariant $\Delta$ is given in Section 1, see Definition 1.3. It relies on the proof that the discriminant 3-fold Pfister forms of all the symplectic involutions on a central simple algebra of degree 8 have the same image in a suitable quotient of the group $I_q^3$ of the center.

Section 2 discusses a necessary and sufficient condition for a 2-torsion Brauer class over a quadratic extension to descend to a 2-torsion Brauer class over the base field, i.e., to be obtained by scalar extension from a 2-torsion Brauer class over the base field. This condition is expressed by the vanishing of the corestriction when the extension is separable, and of a canonical map induced by the squaring map when it is not, see Theorem 2.1. The result is well-known when the characteristic is different from 2. For Brauer classes of quaternion algebras, a similar condition is given for the descent from a biquadratic extension that is not purely inseparable, see Theorem 2.3.

Even when the Brauer class of a biquaternion algebra over a quadratic extension descends, the algebra itself may not descend: it is Brauer-equivalent to the scalar extension of a central simple algebra defined over the base field, but this algebra may have larger degree. Given a quadratic extension $K$ of a base field $F$ of characteristic 2, section 3 introduces an invariant $\delta_{K/F}$ of biquaternion algebras over $K$, whose vanishing characterizes biquaternion algebras that descend, i.e., which are isomorphic to the scalar extension of a biquaternion algebra over $F$, see Definition 3.6 and Proposition 3.7. The definition of $\delta_{K/F}$ follows the same pattern as for the $\Delta$-invariant: it relies on the result that the given biquaternion algebra $B$ over $K$ is contained in a central simple $F$-algebra $A$ of degree 8, and that all the discriminant Pfister forms of symplectic involutions on $A$ that restrict to a symplectic involution on $B$ have the same image in a suitable quotient of $I_q^3(F)$, independently of the choice of the algebra $A$, see Definition 3.6. This invariant turns out to be the same as the one defined in cohomological terms by Barry–Chapman–Laghribi [8] when $K$ is separable over $F$, and by Laghribi–Lorenz [20 §6] when $K$ is inseparable over $F$ (and by Barry [7] when the characteristic is not 2), although the earlier constructions did not rely on the choice of an algebra $A$ containing $B$.

A similar strategy is used in Section 4 to define an invariant $\delta_{L/F}$ of quaternion algebras over a biquadratic extension $L$ of a base field $F$ of characteristic 2, assuming $L$ is not purely inseparable over $F$: when the Brauer class of the quaternion algebra $Q$ descends from $L$ to $F$, the algebra $Q$ can be embedded in a central simple $F$-algebra $A$ of degree 8, and all the discriminant Pfister forms of involutions on $A$ that extend the canonical conjugation involution on $Q$ have the same image in a suitable quotient of $I_q^3(F)$, independently of the choice of $A$. This common image defines an invariant $\delta_{L/F}(Q)$, which vanishes if and only if $Q$ descends to a quaternion algebra over $F$, see Definition 4.4 and Theorem 4.5. With a completely different method, this invariant has also been defined by Laghribi–Lorenz [20] (and by Sivatski [25] when the characteristic is not 2), although these authors consider it as an invariant of the central simple $F$-algebras that contain $Q$. The purely inseparable case is considered in the appendix. The situation there is quite different: a quaternion algebra over a purely inseparable biquadratic field extension descends if and only if its Brauer class descends, see Theorem A.1.

As this rough sketch makes clear, the various invariants defined in this paper are closely related: if a central simple $F$-algebra $A$ of degree 8 and exponent 2 contains a biquadratic field extension $L$ of $F$ that is not purely inseparable, and $K \subset L$ is a quadratic extension of $F$, then the centralizer of $L$ in $A$ is a quaternion $L$-algebra $Q$ and the centralizer of $K$ in $A$ is a biquaternion $K$-algebra $B$. The following inclusions are clear from the definitions:

Canonical homomorphisms induced by the inclusions carry the invariants as follows:

Therefore, examples where $\Delta(A) \ne0$ yield examples of biquaternion algebras $B$ such that $\delta_{K/F}(B) \ne0$ and of quaternion algebras $Q$ with $\delta_{L/F}(Q) \ne0$. We assume throughout that the characteristic is 2, but the reader may verify that all the results in Sections 1–4 are valid in arbitrary characteristic, with only minor alterations in the proofs involving separable field extensions.

Section 5 exhibits examples where the invariants do not vanish, using residue maps for quadratic forms as the main tool. A first construction takes as a starting ingredient any central simple $F$-algebra $A$ of degree 8 and exponent 2 that does not decompose into a tensor product of quaternion algebras. The $\delta_{K/F}$-invariant of the centralizer of any separable quadratic extension $K$ of $F$ in $A$ is shown to be nontrivial, even though it is not known whether $\Delta(A)$ vanishes or not. A central simple algebra of degree 8 and exponent 2 with nontrivial $\Delta$-invariant over the rational function field $F(t)$ in one indeterminate is built from $A$ and the quadratic extension $K$, see Theorem 5.1.

A second construction is more explicit: it centers on the quaternion algebra $[a_1a_2,a_1+a_2)_F$ over the rational function field $F=k(a_1,a_2)$ in two indeterminates over an arbitrary field $k$ of characteristic 2, and on the quadratic extension $K=F(\sqrt{a_1a_2})$ and the biquadratic extension $L=K(\wp^{-1}(a_2))$. With these ingredients, and a sequence of rational function field extensions $F \subset F(t_1) \subset F(t_1,t_2) \subset F(t_1,t_2,t_3)$, the following are successively constructed:

• a quaternion $L(t_1)$-algebra $Q$ such that $\delta_{L(t_1)/F(t_1)}(Q) \ne0$ (Theorem 5.2);

• a biquaternion $K(t_1,t_2)$-algebra $B$ such that $\delta_{K(t_1,t_2)/F(t_1,t_2)}(B) \ne0$ (Theorem 5.3);

• a central simple $F(t_1,t_2,t_3)$-algebra $A$ of degree 8 and exponent 2 such that $\Delta(A) \ne0$ (Theorem 5.6).

In addition, a biquaternion $K(t_1)$-algebra $B'$ containing $Q$ such that $\delta_{K(t_1)/F(t_1)}(B')=0$ is given in Remark 5.4, and a central simple $F(t_1,t_2)$-algebra $A'$ of degree 8 and exponent 2 containing $B$ such that $\Delta(A')=0$ is given in Remark 5.7.

Since $\Delta(A) \ne0$, the algebra $A$ in Theorem 5.6 does not decompose into a tensor product of quaternion algebras, but it is Brauer-equivalent to a tensor product of four quaternion algebras, which is particularly easy to describe explicitly.

Besides Rowen’s early examples in [23], few methods to construct indecomposable division algebras of degree 8 and exponent 2 in characteristic 2 are known. A plausibly exhaustive list is given by the contributions of Karpenko [17], McKinnie [21] and Aravire–Jacob–O’Ryan [5]. McKinnie’s examples have the noteworthy property of not containing any inseparable quadratic extension of the center. In [5], Theorem 4.1(iii) asserts that any central division algebra of exponent 2 with a maximal subfield of the form $F(\wp^{-1}(b_1),\wp^{-1}(b_2),\sqrt{x})$ over a field $F$ of characteristic 2 decomposes as a tensor product of three quaternion algebras. The algebra from Theorem 5.6 is a counterexample to this assertion. In addition, as explained in the appendix, [5 Theorem 4.1(ii)] fails for purely inseparable quartic field extensions of exponent 2: Remark A.3 provides an example of an indecomposable algebra of degree 8 and exponent 2 which contains a maximal subfield $F(\wp^{-1}(a),\sqrt[4]{x})$.

Notation. Throughout the paper, $F$ is a field of characteristic 2. For quadratic and bilinear forms over $F$, we use the same notation as [13]: the Witt group of quadratic forms and the Witt ring of bilinear forms over $F$ are denoted by $I_q(F)$ and $W(F)$ respectively, and for every integer $n \ge1$ the $n$-th power of the fundamental ideal $I(F)$ of $W(F)$ is denoted by $I^n(F)$. We write $I_q^{n+1}(F)$ for $I^n(F) \cdot I_q(F)$ and let

For simplicity, we do not distinguish notationally between a form from its Witt-equivalence class. For $\varphi\in I_q^n(F)$, we write $\overline{\varphi}$ for the image of $\varphi$ in $\overline{I}_q^n(F)$.

Recall from [13 Th. 14.3] that the Clifford invariant defines a map $e_2$ from $I_q^2(F)$ to $\operatorname{Br}_2(F)$, the 2-torsion subgroup of the Brauer group. A theorem of Sah [24] (see also [13 Th. 16.3]) asserts that $e_2$ induces an isomorphism

For any central simple $F$-algebra $A$ such that $2[A] = 0$ in $\operatorname{Br}(F)$, we write $q_A$ for a quadratic form in $I_q^2(F)$ such that $e_2(q_A) = [A]$. Its image $\overline{q}_A \in\overline{I}_q^2(F)$ is thus uniquely determined by $A$. In particular, when $A$ is a biquaternion algebra, the form $q_A$ can be chosen to be 6-dimensional. The 6-dimensional quadratic forms $q_A$ are known as Albert forms of the biquaternion algebra $A$, see [19 §16.A].

For any central simple $F$-algebra $A$ and any field extension $L$ of $F$, we write $A_L$ for the $L$-algebra $A \otimes_F L$ obtained by scalar extension from $F$ to $L$. We also write $[A]_L$ for $[A_L] \in\operatorname{Br}(L)$. Similarly, for any quadratic space $(V,\varphi)$ over $F$, we write $(V_L,\varphi_L)$ for the quadratic space over $L$ obtained by extending scalars.

For central simple algebras with involution, we use the notation of [19]. In particular, for every involution $\sigma$ on a central simple $F$-algebra $A$, we let