{"title":"等变上同调环的一个度公式","authors":"Mark Blumstein, J. Duflot","doi":"10.4310/hha.2023.v25.n1.a18","DOIUrl":null,"url":null,"abstract":"This paper generalizes a result of Lynn on the\"degree\"of an equivariant cohomology ring $H^*_G(X)$. The degree of a graded module is a certain coefficient of its Poincar\\'{e} series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree: $$deg(H^*_G(X)) = \\sum_{[A,c] \\in \\mathcal{Q'}_{max}(G,X)}\\frac{1}{|W_G(A,c)|} \\deg(H^*_{C_G(A,c)}(c)).$$ We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algebraic and geometric character of the degree invariant.","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2022-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A degree formula for equivariant cohomology rings\",\"authors\":\"Mark Blumstein, J. Duflot\",\"doi\":\"10.4310/hha.2023.v25.n1.a18\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper generalizes a result of Lynn on the\\\"degree\\\"of an equivariant cohomology ring $H^*_G(X)$. The degree of a graded module is a certain coefficient of its Poincar\\\\'{e} series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree: $$deg(H^*_G(X)) = \\\\sum_{[A,c] \\\\in \\\\mathcal{Q'}_{max}(G,X)}\\\\frac{1}{|W_G(A,c)|} \\\\deg(H^*_{C_G(A,c)}(c)).$$ We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algebraic and geometric character of the degree invariant.\",\"PeriodicalId\":55050,\"journal\":{\"name\":\"Homology Homotopy and Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2022-02-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Homology Homotopy and Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/hha.2023.v25.n1.a18\",\"RegionNum\":4,\"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":"Homology Homotopy and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/hha.2023.v25.n1.a18","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
This paper generalizes a result of Lynn on the"degree"of an equivariant cohomology ring $H^*_G(X)$. The degree of a graded module is a certain coefficient of its Poincar\'{e} series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree: $$deg(H^*_G(X)) = \sum_{[A,c] \in \mathcal{Q'}_{max}(G,X)}\frac{1}{|W_G(A,c)|} \deg(H^*_{C_G(A,c)}(c)).$$ We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algebraic and geometric character of the degree invariant.
期刊介绍：
Homology, Homotopy and Applications is a refereed journal which publishes high-quality papers in the general area of homotopy theory and algebraic topology, as well as applications of the ideas and results in this area. This means applications in the broadest possible sense, i.e. applications to other parts of mathematics such as number theory and algebraic geometry, as well as to areas outside of mathematics, such as computer science, physics, and statistics. Homotopy theory is also intended to be interpreted broadly, including algebraic K-theory, model categories, homotopy theory of varieties, etc. We particularly encourage innovative papers which point the way toward new applications of the subject.
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