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On the relation between pseudocharacters and Chenevier's determinants

Amit Ophir

Abstract

Consider a commutative unital ring $A$ and a unital $A$-algebra $R$. Let $d$ be a positive integer. Chenevier proved that when $(2d)!$ is invertible in $A$, the map associating to a determinant its trace is a bijection between $A$-valued $d$-dimensional determinants of $R$ and $A$-valued $d$-dimensional pseudocharacters of $R$. In this paper, we show that assuming $d!$ is invertible in $A$ is sufficient. This assumption is already made in the definition of a $d$-dimensional pseudocharacter. Our proof involves establishing a product formula for pseudocharacters, which might be of independent interest.

On the relation between pseudocharacters and Chenevier's determinants

Abstract

Consider a commutative unital ring and a unital -algebra . Let be a positive integer. Chenevier proved that when is invertible in , the map associating to a determinant its trace is a bijection between -valued -dimensional determinants of and -valued -dimensional pseudocharacters of . In this paper, we show that assuming is invertible in is sufficient. This assumption is already made in the definition of a -dimensional pseudocharacter. Our proof involves establishing a product formula for pseudocharacters, which might be of independent interest.
Paper Structure (5 sections, 10 theorems, 35 equations)

This paper contains 5 sections, 10 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. A product formula
  3. Combinatorial preparation
  4. A product formula
  5. A product formula for pseudocharacters and the proof of \ref{['thm_main']}

Key Result

Theorem 1.1

Assume that $d!$ is invertible in $A$. The mapping $D\mapsto Tr_D$ is a bijection between $A$-valued $d$-dimensional determinants on $R$ and $A$-valued $d$-dimensional pseudocharacters on $R$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • ...and 14 more