Generalized $F$-signatures of the rings of invariants of finite group schemes

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 $S=\mathrm{Sym}V$ be the symmetric algebra with the standard grading. Let $M$ be a $\mathbb{Q}$-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 $V\setminus F$ is free. Generalizing the result of P. Symonds and the first author, we describe the Frobenius limit $\mathrm{FL}(L^G)$ 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).

arXiv comment19 pages.In the 2nd version of this paper, the statements of Theorem 3.16 in the introduction and the same theorem in the text were slightly different. We modified the statement in the introduction so that it is exactly the same as that in the text