On the relation between pseudocharacters and Chenevier's determinants
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914693742854144 |
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| author | Ophir, Amit |
| author_facet | Ophir, Amit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18034 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the relation between pseudocharacters and Chenevier's determinants Ophir, Amit Number Theory Representation Theory 11F80 (Primary) 16R30 (Secondary) 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. |
| title | On the relation between pseudocharacters and Chenevier's determinants |
| topic | Number Theory Representation Theory 11F80 (Primary) 16R30 (Secondary) |
| url | https://arxiv.org/abs/2402.18034 |