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Group class operations and homological conditions

Ioannis Emmanouil, Wei Ren

TL;DR

This work develops a systematic framework for expanding families of groups whose group algebras exhibit favorable homological behavior by iterating the group-class operations $LH$ and $Φ$. By analyzing Ext^1-orthogonality and stable properties of Gorenstein modules under these operations, the authors construct cogenerating sets for Gorenstein cotorsion pairs and prove two main results: (i) the classes generated from weakly Gorenstein regular groups yield virtually Gorenstein group algebras (Theorem A), and (ii) the classes obtained from finite groups through $LH$, $Φ$, and $Φ_{flat}$ satisfy Moore's conjecture over $k$ (Theorem B). These results provide a broad source of examples and unify previous cases under a common hierarchical methodology, linking homological properties of $kG$-modules to geometric group-theoretic constructions. The work has implications for cotorsion theory, relative projectivity, and the interplay between restriction/induction and Gorenstein homological algebra in group rings.

Abstract

Kropholler's operation ${\scriptstyle{\bf LH}}$ and Talelli's operation $Φ$ can be often used to formally enlarge the class of available examples of groups that satisfy certain homological conditions. In this paper, we employ this enlargement technique regarding two specific homological conditions. We thereby demonstrate the abundance of groups that (a) have virtually Gorenstein group algebras, as defined by Beligiannis and Reiten, and (b) satisfy Moore's conjecture on the relation between projectivity and relative projectivity, that was studied by Chouinard, Aljadeff, Cornick, Ginosar, Kropholler and Meir.

Group class operations and homological conditions

TL;DR

This work develops a systematic framework for expanding families of groups whose group algebras exhibit favorable homological behavior by iterating the group-class operations and . By analyzing Ext^1-orthogonality and stable properties of Gorenstein modules under these operations, the authors construct cogenerating sets for Gorenstein cotorsion pairs and prove two main results: (i) the classes generated from weakly Gorenstein regular groups yield virtually Gorenstein group algebras (Theorem A), and (ii) the classes obtained from finite groups through , , and satisfy Moore's conjecture over (Theorem B). These results provide a broad source of examples and unify previous cases under a common hierarchical methodology, linking homological properties of -modules to geometric group-theoretic constructions. The work has implications for cotorsion theory, relative projectivity, and the interplay between restriction/induction and Gorenstein homological algebra in group rings.

Abstract

Kropholler's operation and Talelli's operation can be often used to formally enlarge the class of available examples of groups that satisfy certain homological conditions. In this paper, we employ this enlargement technique regarding two specific homological conditions. We thereby demonstrate the abundance of groups that (a) have virtually Gorenstein group algebras, as defined by Beligiannis and Reiten, and (b) satisfy Moore's conjecture on the relation between projectivity and relative projectivity, that was studied by Chouinard, Aljadeff, Cornick, Ginosar, Kropholler and Meir.
Paper Structure (6 sections, 19 theorems, 38 equations)

This paper contains 6 sections, 19 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A convenient class of groups for studying Gorenstein modules
  4. $\hbox{Ext}^1$-orthogonality and group class operations
  5. Virtually Gorenstein group algebras
  6. Groups satisfying Moore's conjecture

Key Result

Theorem 2.1

The classes $\mathfrak{X}$, $\mathfrak{X}'$ and $\mathfrak{Y}$ are subgroup-closed and have the following properties: (i) $\mathfrak{X} = \mathfrak{X}' \supseteq \mathfrak{Y}$; cf. EmmT2. (ii) $\mathfrak{X}$ is closed under the operations ${{{\bf LH}}}$, $\Phi$ and $\Phi_{flat}$; cf. EmmT2 (iii) $\m

Theorems & Definitions (19)

  • Theorem 2.1
  • Corollary 2.2
  • Theorem 3.1
  • Corollary 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Corollary 3.5
  • Lemma 4.1
  • Theorem 4.2
  • Proposition 4.3
  • ...and 9 more