The Gorenstein property for modular binary forms invariants

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• 作者：Amiram Braun ^{abraun@math.haifa.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Gorenstein ; Modular invariants
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：451
• 期：Complete
• 页码：232-247
• 全文大小：379 K

文摘

Let G⊆SL(2,F)$G\subseteq \mathit{SL}(2,F)$ be a finite group, V=F²$V=F^2$ the natural SL(2,F)$\mathit{SL}(2,F)$-module, and charF=p>0$\mathit{char}F=p>0$. Let S(V)$S(V)$ be the symmetric algebra of V   and 426d22c88b37768048b3f3daa9c2c0b" title="Click to view the MathML source">S(V)^G$S{(V)}^G$ the ring of G-invariants. We provide examples of groups G  , where 426d22c88b37768048b3f3daa9c2c0b" title="Click to view the MathML source">S(V)^G$S{(V)}^G$ is Cohen–Macaulay, but is not Gorenstein. This refutes a natural conjecture due to Kemper, Körding, Malle, Matzat, Vogel and Wiese. Let 81bc841770d5a04dac4d604a33614325" title="Click to view the MathML source">T(G)$T(G)$ denote the subgroup generated by all transvections of G  . We show that 426d22c88b37768048b3f3daa9c2c0b" title="Click to view the MathML source">S(V)^G$S{(V)}^G$ is Gorenstein if and only if one of the following cases holds:

(1)

T(G)={1_G}$T(G)=\{1_G\}$,

(2)

V   is an irreducible 81bc841770d5a04dac4d604a33614325" title="Click to view the MathML source">T(G)$T(G)$-module,

(3)

V   is a reducible 81bc841770d5a04dac4d604a33614325" title="Click to view the MathML source">T(G)$T(G)$-module and 428b525a6074" title="Click to view the MathML source">|G|$|G|$ divides |T(G)|(|T(G)|−1)$|T(G)|(|T(G)|-1)$.