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Permutation modules and cohomological singularity

Paul Balmer, Martin Gallauer

Abstract

We define a new invariant of finitely generated representations of a finite group, with coefficients in a commutative noetherian ring. This invariant uses group cohomology and takes values in the singularity category of the coefficient ring. It detects which representations are controlled by permutation modules.

Permutation modules and cohomological singularity

Abstract

We define a new invariant of finitely generated representations of a finite group, with coefficients in a commutative noetherian ring. This invariant uses group cohomology and takes values in the singularity category of the coefficient ring. It detects which representations are controlled by permutation modules.

Paper Structure

This paper contains 4 sections, 13 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Recollections and preparations
  3. Cohomological singularity
  4. Main result

Key Result

Theorem 1.4

The subcategory $\mathop{\mathrm{D}}\nolimits_{\mathop{\mathrm{perm}}\nolimits}(G;R)$ of $\mathop{\mathrm{D}}\nolimits_{\mathop{\mathrm{b}}\nolimits}(R G)$, given in eq:Dperm, consists of those complexes $X\in\mathop{\mathrm{D}}\nolimits_{\mathop{\mathrm{b}}\nolimits}(R G)$ such that the invariants vanish in the big singularity category $\mathop{\mathrm{D}}\nolimits^{\mathop{\mathrm{sing}}\nolimi

Theorems & Definitions (41)

  • Theorem 1.4: \ref{['Thm:main']}
  • Lemma 2.10
  • proof
  • Lemma 2.12
  • proof
  • Remark 2.13
  • Definition 3.4
  • Example 3.5
  • Remark 3.6
  • Remark 3.7
  • ...and 31 more