{"title":"论伪字符与切内维埃行列式之间的关系","authors":"Amit Ophir","doi":"10.1112/blms.13162","DOIUrl":null,"url":null,"abstract":"<p>Consider a commutative unital ring <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> and a unital <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>-algebra <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>. Let <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math> be a positive integer. Chenevier proved that when <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <mn>2</mn>\n <mi>d</mi>\n <mo>)</mo>\n <mo>!</mo>\n </mrow>\n <annotation>$(2d)!$</annotation>\n </semantics></math> is invertible in <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>, the map associating to a determinant its trace is a bijection between <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>-valued <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-dimensional determinants of <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math>-valued <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-dimensional pseudocharacters of <span></span><math>\n <semantics>\n <mi>R</mi>\n <annotation>$R$</annotation>\n </semantics></math>. In this paper, we show that assuming <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>!</mo>\n </mrow>\n <annotation>$d!$</annotation>\n </semantics></math> is invertible in <span></span><math>\n <semantics>\n <mi>A</mi>\n <annotation>$A$</annotation>\n </semantics></math> is sufficient. This assumption is already made in the definition of a <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-dimensional pseudocharacter. Our proof involves establishing a product formula for pseudocharacters, which might be of independent interest.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 12","pages":"3721-3730"},"PeriodicalIF":0.8000,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13162","citationCount":"0","resultStr":"{\"title\":\"On the relation between pseudocharacters and Chenevier's determinants\",\"authors\":\"Amit Ophir\",\"doi\":\"10.1112/blms.13162\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Consider a commutative unital ring <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> and a unital <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math>-algebra <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math>. Let <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$d$</annotation>\\n </semantics></math> be a positive integer. Chenevier proved that when <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <mn>2</mn>\\n <mi>d</mi>\\n <mo>)</mo>\\n <mo>!</mo>\\n </mrow>\\n <annotation>$(2d)!$</annotation>\\n </semantics></math> is invertible in <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math>, the map associating to a determinant its trace is a bijection between <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math>-valued <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$d$</annotation>\\n </semantics></math>-dimensional determinants of <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math>-valued <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$d$</annotation>\\n </semantics></math>-dimensional pseudocharacters of <span></span><math>\\n <semantics>\\n <mi>R</mi>\\n <annotation>$R$</annotation>\\n </semantics></math>. In this paper, we show that assuming <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>!</mo>\\n </mrow>\\n <annotation>$d!$</annotation>\\n </semantics></math> is invertible in <span></span><math>\\n <semantics>\\n <mi>A</mi>\\n <annotation>$A$</annotation>\\n </semantics></math> is sufficient. This assumption is already made in the definition of a <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$d$</annotation>\\n </semantics></math>-dimensional pseudocharacter. Our proof involves establishing a product formula for pseudocharacters, which might be of independent interest.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"56 12\",\"pages\":\"3721-3730\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-09-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13162\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.13162\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13162","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
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。
On the relation between pseudocharacters and Chenevier's determinants
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.
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