arXiv subjects

Ioannis Emmanouil

Publications and source records attributed to Ioannis Emmanouil.

10 recordsLinked to original sources

arXiv2026

Gorenstein dimensions and Hirsch length of groups

We study the relation between the Gorenstein homological and the Gorenstein cohomological dimension of groups. We prove that for every module of type $FP_\infty$ over any ring the Gorenstein projective and the Gorenstein flat dimensions are always equal to each other. In particular, we have an equality $\mathrm{Gcd}_kG=\mathrm{Ghd}_kG$ for every group $G$ of type $FP_\infty$ over a commutative coefficient ring $k$. For non-locally-finite groups in Kropholler's class ${\scriptstyle\mathbf{LH}}\mathfrak F$, we characterize the groups of Gorenstein (co)homological dimension one, in terms of actions on trees with finite stabilizers. The Hirsch length of a virtually soluble group $G$ determines its Gorenstein (co)homological dimension, even when $G$ has torsion: We show that $\mathrm{Ghd}_\mathbb{Z}G =h(G)$ and, if $G$ is countable, $h(G)\leq\mathrm{Gcd}_\mathbb{Z}G\leq h(G)+1$. Some consequences for elementary amenable groups of finite Hirsch length are also obtained. Finally, we investigate the Gorenstein dimension of modules over certain algebras of groups with torsion and discuss applications to classifying spaces for proper actions.

math.GR↗

arXiv2026

The structure of $\lim^1$-groups

If $(A_n)_n$ is a decreasing filtration of a module $A$ and $\widehat{A} = \lim_n A/A_n$, then $\lim^1_n A_n$ is identified with the cokernel of the canonical map $A \longrightarrow \widehat{A}$. In this note, we show that any $\lim^1$-group is canonically of that form: For any inverse sequence of modules $(X_n)_n$ there exists an inverse sequence $(A_n)_n$ as above and a morphism $(A_n)_n \longrightarrow (X_n)_n$, depending functorially on $(X_n)_n$, that induces an isomorphism on $\lim^1$. The proof is based on Quillen's small object argument, as formulated by Eklof and Trlifaj in their investigation of the existence of enough injective objects in certain cotorsion pairs, and also uses a construction by Salce that provides enough projective objects therein.

math.RA↗

arXiv2025

On some triangulated categories over group algebras

In this paper, we introduce the cofibrant derived category of a group algebra $kG$ and study its relation to the derived category of $kG$. We also define the cofibrant singularity category of $kG$, whose triviality characterizes the regularity of $kG$ with respect to the cofibrant dimension, and examine its significance as a measure of the obstruction to the equality between the classes of Gorenstein projective and cofibrant modules. We show that the same obstruction can be measured by certain localization sequences between stable categories.

math.CT↗

arXiv2025

Monoidal model structures over infinite groups

The stable category of modules over the algebra of a finite group with coefficients in a field is a compactly generated tensor triangulated category, that has been studied extensively in representation theory. In this paper, we provide a plethora of infinite groups G, for which the category of kG-modules (where k is a commutative coherent ring of finite global dimension) admits a monoidal model structure, in the sense of Hovey, whose associated homotopy category is a compactly generated tensor triangulated category. To that end, we use a technique recently introduced by the authors, which is based on Kropholler's operation LH and the second author's operation Φ.

math.RT↗

arXiv2025

Group class operations and homological conditions

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.

math.GR↗

arXiv2025

Total acyclicity of complexes over group algebras

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.

math.RT↗

arXiv2025

On the class of Benson's cofibrant modules

In this paper, we study the class of cofibrant modules over a group algebra $kG$, that were introduced by Benson. We prove that this class is always the left-hand side of a complete hereditary and projective cotorsion pair. We also examine the relation between cofibrant and Gorenstein projective modules and the behaviour of cofibrant modules with respect to the induction functor from subgroups. As an application, we show that the cofibrant cotorsion pair induces a monoidal model structure on the category of $kG$-modules over {\em any} commutative coefficient ring $k$, provided that the group $G$ is obtained from the class of finite groups by iterated applications of Kropholler's operation ${\scriptstyle{{\bf LH}}}$ and Talelli's operation $\Phi$. The corresponding symmetric monoidal homotopy category is equivalent to the stable category of cofibrant $kG$-modules.

math.KT↗

arXiv2025

On certain classes of modules over group algebras

In this paper, we examine the relation between certain subclasses of the classes of Gorenstein projective, Gorenstein flat and Gorenstein injective modules over a group algebra, which consist of the cofibrant, cofibrant-flat and fibrant modules respectively. These subclasses have all structural properties that the Gorenstein classes are known to have, regarding the existence of complete cotorsion pairs and various approximations between them. Furthermore, these subclasses have certain properties, which are conjecturally anticipated to hold for the Gorenstein classes as well. Since the Gorenstein classes are actually equal to the corresponding subclasses for large families of groups, we thus obtain (indirectly) some information about the Gorenstein classes over these groups.

math.KT↗

arXiv2023

Homological dimension based on a class of Gorenstein flat modules

In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying the finiteness of the PGF global dimension reveals a connection between classical homological invariants of left and right modules over the ring, that leads to generalizations of certain results by Jensen, Gedrich and Gruenberg that were originally proved in the realm of commutative Noetherian rings.

math.RA↗

arXiv2008

A limit approach to group homology

In this paper, we consider for any free presentation $G = F/R$ of a group $G$ the coinvariance $H_{0}(G,R_{ab}^{\otimes n})$ of the $n$-th tensor power of the relation module $R_{ab}$ and show that the homology group $H_{2n}(G,{\mathbb Z})$ may be identified with the limit of the groups $H_{0}(G,R_{ab}^{\otimes n})$, where the limit is taken over the category of these presentations of $G$. We also consider the free Lie ring generated by the relation module $R_{ab}$, in order to relate the limit of the groups $γ_{n}R/[γ_{n}R,F]$ to the $n$-torsion subgroup of $H_{2n}(G,{\mathbb Z})$.

math.GR↗