Total acyclicity of complexes over group algebras
Ioannis Emmanouil, Olympia Talelli
TL;DR
The paper develops a unified framework for when group algebras $kG$ exhibit controlled Gorenstein homological behavior, linking Gorenstein projective/flat/injective modules and their totally acyclic complexes via cotorsion pairs and duality. It introduces group-classes ${\mathfrak Z}$, ${\mathfrak Y}$, ${\mathfrak X}$ (and their finite counterparts) closed under Kropholler’s ${\bf LH}$ and various ${\Phi}$-type operations, proving that these closures preserve desirable properties such as completeness of cotorsion pairs and total-acyclicity of acyclic complexes. The work provides a common framework that recovers and extends numerous earlier results about acyclic and totally acyclic complexes over group algebras, and it delineates precise hierarchical families (e.g., ${\mathfrak Z}_{Gor}$, ${\mathfrak X}_{Gor}$) where all the stated Gorenstein conditions hold. The approach relies on a blend of homological algebra, duality considerations, and group-class techniques to yield broad, transferable criteria applicable to a wide range of coefficient rings, including $\mathbb Z$ and more general Noetherian or Gorenstein-regular rings. Overall, the results significantly broaden the class of groups for which the Gorenstein homological theory behaves in a controlled, predictable way, with implications for cotorsion theory and the structure of acyclic complexes in group-graded settings.
Abstract
In this paper, we study group algebras over which modules have a controlled behaviour with respect to the notions of Gorenstein homological algebra, namely: (a) Gorenstein projective modules are Gorenstein flat, (b) any module whose dual is Gorenstein injective is necessarily Gorentein flat, (c) the Gorenstein projective cotorsion pair is complete and (d) any acyclic complex of projective, injective or flat modules is totally acyclic (in the respective sense). We consider a certain class of groups satisfying all of these properties and show that it is closed under the operation LH defined by Kropholler and the operation Φ defined by the second author. We thus generalize all previously known results regarding these properties over group algebras and place these results in an appropriate framework.