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A Knebusch trace formula for Azumaya algebras with involution

Vincent Astier, Thomas Unger

Abstract

We establish a trace formula for signatures of hermitian forms over Azumaya algebras with involution, extending Knebusch's work on symmetric bilinear forms over finite étale extensions of commutative base rings. As an application when the base ring is semilocal, we obtain an exact sequence for total signatures, related to Pfister's local-global principle and the notion of stability index.

A Knebusch trace formula for Azumaya algebras with involution

Abstract

We establish a trace formula for signatures of hermitian forms over Azumaya algebras with involution, extending Knebusch's work on symmetric bilinear forms over finite étale extensions of commutative base rings. As an application when the base ring is semilocal, we obtain an exact sequence for total signatures, related to Pfister's local-global principle and the notion of stability index.
Paper Structure (14 sections, 25 theorems, 92 equations)

This paper contains 14 sections, 25 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hermitian forms
  4. Finite étale algebras
  5. Azumaya algebras with involution
  6. The trace map
  7. Real algebra
  8. Signatures of hermitian forms
  9. The Knebusch trace formula
  10. Finite étale extensions and real closed fields
  11. The trace formula
  12. A reference form of $2$-power signature
  13. An exact sequence for total signatures in the semilocal case
  14. Hermitian Morita equivalence

Key Result

Proposition 2.2

With the same notation as in Definition direct-product, assume that there are commutative rings $R_i$ such that $A_i$ is a finitely generated $R_i$-module, for $i=1, \ldots, t$. Then

Theorems & Definitions (55)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3: Saltman99
  • Definition 2.4: first23
  • Proposition 2.5: first23
  • Lemma 2.6
  • Proposition 2.7: Change of base ring
  • Remark 2.8
  • Definition 2.9: See first23
  • ...and 45 more