A cohomological invariant for algebras of degree 8 and exponent 2 in characteristic 2

1. Problem Statement and Motivation

The paper addresses two central problems in the theory of central simple algebras and quadratic forms over fields of characteristic 2:

1. Decomposability: Determining whether a central simple algebra $A$ of degree 8 and exponent 2 is decomposable (i.e., isomorphic to a tensor product of three quaternion algebras). While indecomposable examples exist (constructed by Rowen and Karpenko), a general criterion for decomposability in characteristic 2 was lacking, particularly extending the work of Sivatski [Siv22] which was restricted to characteristic $\neq 2$.
2. Descent: Determining if an algebra or quadratic form defined over an extension field $K$ can be "descended" to the base field $F$ (i.e., isomorphic to the scalar extension of an $F$-algebra/form).

The authors aim to extend Sivatski's cohomological invariant to characteristic 2, handling both separable and inseparable quadratic extensions, and apply this invariant to solve descent problems and construct specific indecomposable algebras with non-trivial Chow groups.

2. Methodology and Mathematical Framework

The authors utilize a combination of Kato-Milne cohomology, quadratic form theory, and central simple algebra theory specific to characteristic 2.

• Kato-Milne Cohomology ($H^n_2(F)$): The paper relies on the isomorphism between the graded Witt ring of quadratic forms and Kato-Milne cohomology groups. Specifically, $H^3_2(F)$ corresponds to the third power of the fundamental ideal of quadratic forms modulo higher powers.
• Differential Forms: The construction uses the space of 1-differential forms $\Omega^1_F$ and the operator $d\log(\alpha) = d\alpha/\alpha$. The invariant is constructed using the wedge product of $d\log$ terms and Brauer classes.
• Field Extensions: The study focuses on extensions of the form $L = K(\wp^{-1}(a), \wp^{-1}(b))$, where $K/F$ is an arbitrary quadratic extension (separable or inseparable). Here, $\wp(x) = x^2+x$ (Artin-Schreier) or $\wp(x)=x^2$ (inseparable).
• Transfer Maps (Scharlau Transfer): A crucial tool is the transfer map $s^*$ induced by a linear map $s: K \to F$ (trace for separable, specific linear map for inseparable). This allows moving invariants and forms from the extension back to the base field.
• Albert Forms: For biquaternion algebras, the authors utilize the associated Albert form (a 6-dimensional quadratic form) to encode the algebra's index and decomposability properties.

3. Key Contributions: The Invariant $\text{inv}(A)$

The core contribution is the construction of a cohomological invariant $\text{inv}(A)$ for a central simple algebra $A$ of exponent 2 that splits over a triquadratic extension $L/F$.

• Construction:
  Let $K/F$ be a quadratic extension and $L = K(\wp^{-1}(a), \wp^{-1}(b))$. If $A$ splits over $L$, then $A_K$ is Brauer equivalent to a biquaternion algebra $[a, x) \otimes [b, y)$ for some $x, y \in K^*$.
  The authors define an Albert form $\phi = [1, a+b] \perp x[1, a] \perp y[1, b]$. They apply the transfer map $s^*$ to $\phi$ to obtain a form over $F$, then map this to the cohomology group via the Arason invariant $e_3$.
  $$\text{inv}(A) := e_3(s^*(\phi)) \in H^3_2(F) / G$$
  where $G$ is a subgroup generated by terms involving the norm of the extension and the class of $A$.

• Well-definedness: The authors prove that $\text{inv}(A)$ is independent of the choice of $x$ and $y$ in the decomposition of $A_K$.

4. Main Results

A. Characterization of Decomposability (Section 4)

The paper establishes a precise criterion for the vanishing of the invariant:

• Theorem: $\text{inv}(A) = 0$ if and only if $A$ is Brauer equivalent to a tensor product of the form $Q \otimes [a+b, v) \otimes Q'$, where $Q, Q'$ are quaternion algebras and $Q'$ splits over the intermediate field $K$.
• Implication: If $\text{inv}(A) = 0$, the algebra $A$ is "close" to being decomposable over $F$. Specifically, the division algebra Brauer equivalent to $A$ over the intermediate field $E = K(\wp^{-1}(a+b))$ is defined over $F$.
• Counter-example: The authors construct an example (Example 4.6) of an algebra where $\text{inv}(A) \neq 0$ but a related descent invariant $\delta_{K/F}(A) = 0$. This demonstrates that $\text{inv}(A)$ detects decomposability properties that $\delta_{K/F}$ does not, proving the reverse implication of Corollary 4.4 is false in general.

B. Construction of Indecomposable Algebras with Non-trivial Chow Groups (Section 5)

Extending work by Sivatski and Karpenko, the authors construct a specific algebra $B$ over a field $K$ of cohomological dimension 3:

• Properties: $B$ has degree 8, exponent 2, and index 8 (hence indecomposable).
• Chow Group: The torsion part of the second Chow group of its Severi-Brauer variety, $\text{Tors}CH_2(SB(B))$, is isomorphic to $\mathbb{Z}/2\mathbb{Z}$.
• Explicit Description: The non-zero element in the Chow group is explicitly described as a symbol in $H^3_2(K)$, linking the geometric invariant directly to the cohomological invariant constructed in the paper.

C. Descent Results (Section 6)

The invariant is applied to solve descent problems for algebras and quadratic forms over mixed biquadratic extensions (extensions involving both separable and inseparable quadratic subfields) when combined with odd degree extensions.

• Theorem 6.1 (Algebra Descent): Let $M/F$ be an odd degree extension. If a central simple algebra $D$ becomes Brauer equivalent to a quaternion algebra over $M(\wp^{-1}(b), \sqrt{d})$, then $D$ is Brauer equivalent to a quaternion algebra over $F(\wp^{-1}(b), \sqrt{d})$.
• Theorem 6.6 (Quadratic Form Descent): Under similar conditions (odd degree extension $L/F$ and mixed biquadratic $E/F$), if the anisotropic part of a quadratic form $\phi$ over $LE$ is defined over $L$, then the anisotropic part of $\phi$ over $E$ is defined over $F$, provided specific dimension and Arf invariant constraints are met.
• Significance: These results generalize previous descent theorems to characteristic 2 and mixed extensions, utilizing the new invariant and transfer maps to bypass the failure of the trace map in inseparable cases.

5. Significance and Impact

1. Extension to Characteristic 2: The paper successfully bridges a gap in the theory of central simple algebras, adapting tools that were previously limited to characteristic $\neq 2$ (Sivatski's work) to the more complex setting of characteristic 2, where Artin-Schreier theory and inseparable extensions play a dominant role.
2. New Invariants: The introduction of $\text{inv}(A)$ provides a powerful new tool for distinguishing decomposable from indecomposable algebras of degree 8. It refines the understanding of the relationship between the Brauer group, quadratic forms, and field extensions.
3. Geometric Applications: By linking the invariant to the torsion of the second Chow group of Severi-Brauer varieties, the paper strengthens the connection between algebraic K-theory, cohomology, and algebraic geometry in characteristic 2.
4. Descent Theory: The results on descent over mixed biquadratic extensions are novel. They provide criteria for when algebras and forms defined over complex extensions actually originate from the base field, a problem of fundamental importance in the classification of algebraic structures.

In summary, this paper provides a robust framework for analyzing degree 8 algebras in characteristic 2, offering new invariants, counter-examples to naive generalizations, and constructive proofs for the existence of indecomposable algebras with specific geometric properties.