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- W4367000377 abstract "Let $k$ be a perfect field of prime characteristic $p$, $G$ a finite group scheme over $k$, and $V$ a finite-dimensional $G$-module. Let $S=mathop{mathrm{Sym}}V$ be the symmetric algebra with the standard grading. Let $M$ be a $Bbb Q$-graded $S$-finite $S$-free $(G,S)$-module, and $L$ be its $S$-reflexive graded $(G,S)$-submodule. Assume that the action of $G$ on $V$ is small in the sense that there exists some $G$-stable Zariski closed subset $F$ of $V$ of codimension two or more such that the action of $G$ on $Vsetminus F$ is free. Generalizing the result of P. Symonds and the first author, we describe the Frobenius limit $mathop{mathrm{FL}}(L^G)$ of the $S^G$-module $L^G$. In particular, we determine the generalized $F$-signature $s(M,S^G)$ for each indecomposable gradable reflexive $S^G$-module $M$. In particular, we prove the fact that the $F$-signature $s(S^G)=s(S^G,S^G)$ equals $1/dim k[G]$ if $G$ is linearly reductive (already proved by Watanabe--Yoshida, Carvajal-Rojas--Schwede--Tucker, and Carvajal-Rojas) and $0$ otherwise (some important cases has already been proved by Broer, Yasuda, Liedtke--Martin--Matsumoto)." @default.
- W4367000377 created "2023-04-27" @default.
- W4367000377 creator A5003002171 @default.
- W4367000377 creator A5004207681 @default.
- W4367000377 date "2023-04-24" @default.
- W4367000377 modified "2023-10-17" @default.
- W4367000377 title "Generalized $F$-signatures of the rings of invariants of finite group schemes" @default.
- W4367000377 doi "https://doi.org/10.48550/arxiv.2304.12138" @default.
- W4367000377 hasPublicationYear "2023" @default.
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