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Local cohomology of modular invariant rings

Kriti Goel, Jack Jeffries, Anurag K. Singh

Abstract

For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$ with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let $\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants $R^G$. We study how the local cohomology module $H^n_{\mathfrak{n}}(R^G)$ compares with $H^n_{\mathfrak{n}}(R)^G$. Various results on the $a$-invariant and on the Hilbert series of $H^n_\mathfrak{n}(R^G)$ are obtained as a consequence.

Local cohomology of modular invariant rings

Abstract

For a field, consider a finite subgroup of with its natural action on the polynomial ring . Let denote the homogeneous maximal ideal of the ring of invariants . We study how the local cohomology module compares with . Various results on the -invariant and on the Hilbert series of are obtained as a consequence.
Paper Structure (5 sections, 7 theorems, 85 equations)

This paper contains 5 sections, 7 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Preliminary remarks
  3. Comparing local cohomology
  4. When is the $a$-invariant invariant?
  5. Hilbert series of local cohomology

Key Result

Lemma 2.2

Let $G$ be a finite subgroup of $\operatorname{GL}_n(K)$, without transvections, acting on the polynomial ring $R\colonequals K[x_1,\dots,x_n]$. Then the image of the transfer map ${\operatorname{Tr}}\colon R\longrightarrow R^G$ is an ideal of $R^G$ of height at least two.

Theorems & Definitions (21)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Example 3.4
  • ...and 11 more