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Signature maps from positive cones on algebras with involution

Vincent Astier, Thomas Unger

TL;DR

The paper develops a robust framework to define and compute signatures of hermitian forms over central simple algebras with involution by grounding them in positive cones on $(A,\sigma)$. It provides a direct Morita-based construction to obtain signatures from a given positive cone, fixes sign ambiguities with a fixed reference form $oldsymbol{ extmu}$, and rectifies a flaw in the previous A-U-pos work through a new, complete proof and diagonalization approach. It also delivers a comprehensive description of the space of positive cones $X_{(A,\sigma)}$, its topology, and the behavior of signatures under field embeddings and extensions, including maximum signatures and the extension of positive cones. Collectively, the results supply a coherent real-algebraic toolkit for ordering and signature theory in algebras with involution, with precise control over nil-orderings, prime m-ideals, and Morita-equivalence classes.

Abstract

We introduced positive cones in an earlier paper as a notion of ordering on central simple algebras with involution that corresponds to signatures of hermitian forms. In the current paper we describe signatures of hermitian forms directly out of positive cones, and also use this approach to rectify a problem that affected some results in the previously mentioned paper.

Signature maps from positive cones on algebras with involution

TL;DR

The paper develops a robust framework to define and compute signatures of hermitian forms over central simple algebras with involution by grounding them in positive cones on . It provides a direct Morita-based construction to obtain signatures from a given positive cone, fixes sign ambiguities with a fixed reference form , and rectifies a flaw in the previous A-U-pos work through a new, complete proof and diagonalization approach. It also delivers a comprehensive description of the space of positive cones , its topology, and the behavior of signatures under field embeddings and extensions, including maximum signatures and the extension of positive cones. Collectively, the results supply a coherent real-algebraic toolkit for ordering and signature theory in algebras with involution, with precise control over nil-orderings, prime m-ideals, and Morita-equivalence classes.

Abstract

We introduced positive cones in an earlier paper as a notion of ordering on central simple algebras with involution that corresponds to signatures of hermitian forms. In the current paper we describe signatures of hermitian forms directly out of positive cones, and also use this approach to rectify a problem that affected some results in the previously mentioned paper.

Paper Structure

This paper contains 10 sections, 32 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signatures of hermitian forms over quadratic field extensions and quaternions
  4. Signatures of hermitian forms over $F$-algebras with involution
  5. Signatures under ordered field embeddings
  6. Positive cones
  7. Reduction to diagonal forms
  8. Signature maps from positive cones
  9. Description of positive cones and the topology of $X_{(A,\sigma)}$
  10. Maximum signatures and extension of positive cones

Key Result

Lemma 2.6

Let $F_P \subseteq L$ be a field extension with $L$ real closed. We denote by $\mathfrak{m}$ the hermitian Morita equivalence between $(A \otimes_F F_P,\sigma \otimes \mathrm{id})$ and $(D_P, \vartheta_P)$ as well as the induced isomorphism of Witt groups. Then $\mathfrak{m}$ extends to a hermitian

Theorems & Definitions (76)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3: See also \ref{['sign7']} below
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 66 more