Representation and cohomologies of groups and semigroups

Abstract

This article surveys results related to the cohomology of lattices over group rings, particularly over the group ring of the Klein four--group. It is shown that in this setting, syzygies coincide with the Auslander-Reiten translation, enabling the full computation of Tate cohomology for all lattices and the explicit description of the action of group automorphisms. Special attention is paid to regular lattices and their applications to the classification of multidimensional crystallographic and Chernikov groups. The article also explores the connection between the cohomology and representations of crossed group rings and those of coefficient rings, especially for semidirect product group rings. The approach is illustrated by a new classification of integral representations of the alternating group A₄, leading to explicit cohomological invariants. The results are of interdisciplinary interest for group theory, homological algebra, crystallography, and coding theory.