Total acyclicity of complexes over group algebras

Ioannis Emmanouil; Olympia Talelli

Home/Archive/arXiv

arXiv · 2505.10920

Total acyclicity of complexes over group algebras

Ioannis Emmanouil·Olympia Talelli

View original source ↗Read source document ↗Export metadata ↓

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 {\Phi} 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.

Explore related subjects

math.RT math.GR math.KT

Keep this discovery

BibTeX RIS

Ioannis Emmanouil, Olympia Talelli. 2025-05-16. Total acyclicity of complexes over group algebras. https://arxiv.org/abs/2505.10920

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Whittaker supermodules over Lie superalgebras

In this paper, we develop a general theory of quasi-Whittaker supermodules over Lie superalgebras induced from an arbitrary ideal. We determine the quasi-Whittaker vectors in universal supermodules, establish an irreducibility criterion, and classify several families of irreducible supermodules. The odd part produces a new irreducibility phenomenon absent from the Lie algebra setting. As applications, we determine all irreducible quasi-Whittaker supermodules over the $N=1$ super Schr\"odinger algebra and the $N=1$ $\frac{3}{2}$-conformal Galilei superalgebra, and over the complete spectrum-generating superalgebra in a special case.

math.RT↗

arXiv2026

Rankin--Selberg integrals of opposite conductor--one newforms

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni--Zhang's conjecture on the nonvanishing of local relative characters.

math.RT↗

arXiv2026

Obstructions to Jacobi-Finiteness of Quivers with Potentials

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod--Shafarevich--Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

math.RT↗