On the relation between pseudocharacters and Chenevier's determinants

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-09-25 DOI:10.1112/blms.13162

Amit Ophir

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引用次数: 0

Abstract

Consider a commutative unital ring $A$ and a unital $A$ -algebra $R$ . Let $d$ be a positive integer. Chenevier proved that when $(2 d)!$ 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.

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论伪字符与切内维埃行列式之间的关系

考虑一个交换一元环a $ a $和一个一元a $ a $ -代数R$ R$。设d$ d$为正整数。Chenevier证明了当（2d）！美元(2 d) !$在A$中是可逆的，与行列式相关联的映射，它的迹线是a $ a $值的d$ d$维的R$ R$行列式和a $ a $值的d$ d$维伪字符之间的双射R$ R$。在本文中，我们证明了假设d ！$ d !$在A中可逆，A$是充分的。这个假设已经在d$ d$维伪字符的定义中做出了。我们的证明包括建立伪字符的乘积公式，这可能是独立的兴趣。

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