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Generalized $F$-signatures of the rings of invariants of finite group schemes
Commutative Algebra · Submitted 24 Apr 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In particular, we determine the generalized $F$-signature $s(M,S^G)$ for each indecomposable gradable reflexive $S^G$-module $M$.
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The abstract, sentence by sentence
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 $V\setminus F$ 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).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $k$ be a perfect field of prime characteristic $p$, $G$ a finite group scheme over $k$, and $V$ a finite-dimensional $G$-module. | background 0.99 | 0.07 | 0.01 | asserted 0.62 |
| s1 | Let $S=\mathop{\mathrm{Sym}}V$ be the symmetric algebra with the standard grading. | background 0.98 | 0.21 | 0.01 | asserted 0.86 |
| s2 | Let $M$ be a $\Bbb Q$-graded $S$-finite $S$-free $(G,S)$-module, and $L$ be its $S$-reflexive graded $(G,S)$-submodule. | background 0.93 | 0.16 | 0.02 | asserted 0.81 |
| s3 | 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. | background 0.70 | 0.15 | 0.02 | asserted 0.94 |
| s4 | Generalizing the result of P. | contribution 0.64 | 0.22 | 0.02 | unclear 0.87 |
| s5 | Symonds and the first author, we describe the Frobenius limit $\mathop{\mathrm{FL}}(L^G)$ of the $S^G$-module $L^G$. | contribution 0.86 | 0.81 | 0.23 | asserted 0.99 |
| s6 | In particular, we determine the generalized $F$-signature $s(M,S^G)$ for each indecomposable gradable reflexive $S^G$-module $M$. | result 0.87 | 0.94 | 0.45 | asserted 1.00 |
| s7 | 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). | result 0.96 | 0.73 | 0.48 | asserted 1.00 |
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The paper
Original 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 $V\setminus F$ 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).
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2 arXiv papers cite this in 2 sentences
Papers citing it · 1 sentence without a confident label
Compares with it 1 paper
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After finishing the first draft of this paper, we noticed that Hashimoto and Kobayashi [ref] independently obtained a result similar to our Theorems [ref] and [ref].
Non-commutative resolutions of linearly reductive quotient singularities · Examples · 1.00
What it cites · 3 sentences without a confident label
Compares with 1 paper
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After the first version of this paper put on arXiv, we got aware that [ref] by Liedtke and Yasuda has some overlap with Corollary [ref].
Non-commutative resolutions of linearly reductive quotient singularities · Introduction · 1.00
Cites as background 1 paper
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For basics on almost principal $G$-bundles, see [ref] .
Equivariant class group. III. Almost principal fibrer bundles · Preliminaries · 1.00
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