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

Ahmed Laghribi, Nico Lorenz

Abstract

Our aim in this paper is to extend a work of Sivatski to characteristic 2. More precisely, for $F$ a field of characteristic $2$ and a central simple algebra $A$ of exponent 2 that splits over a triquadratic extension of $F$ of separability degree at least 4, we attach a cohomological invariant $\inv(A) \in H_2^3(F) / G$, where $H_2^3(F)$ is the third Kato-Milno cohomology group and $G$ is a subgroup of $H_2^3(F)$ divisible by the Brauer class of $A$. As an application, we will relate the decomposability of the algebra in degree 8 to the vanishing of $\inv(A)$. Moreover, we will use this invariant to prove some descent results for central simple algebras and quadratic forms over biquadratic extensions.

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

Abstract

Our aim in this paper is to extend a work of Sivatski to characteristic 2. More precisely, for a field of characteristic and a central simple algebra of exponent 2 that splits over a triquadratic extension of of separability degree at least 4, we attach a cohomological invariant , where is the third Kato-Milno cohomology group and is a subgroup of divisible by the Brauer class of . As an application, we will relate the decomposability of the algebra in degree 8 to the vanishing of . Moreover, we will use this invariant to prove some descent results for central simple algebras and quadratic forms over biquadratic extensions.
Paper Structure (9 sections, 16 theorems, 106 equations)

This paper contains 9 sections, 16 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Quadratic forms
  4. Central simple algebras
  5. Differential forms
  6. Construction of the invariant
  7. Triviality of the Invariant
  8. An application to Chow Groups and indecomposable algebras of degree $8$
  9. Descent Results

Key Result

Lemma 3.1

Let $K / F$, $s: K \to F$ and $\mathrm{N}_{K/F}$ be as in the preceding paragraph. We have the following:

Theorems & Definitions (35)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Remark 3.5
  • Remark 3.6
  • ...and 25 more