%0 Journal Article
%T Gorenstein Homological Dimension of Groups Through Flat-Cotorsion Modules
%J Mathematics Interdisciplinary Research
%I University of Kashan
%Z 2538-3639
%A Hajizamani, Ali
%D 2024
%\ 03/01/2024
%V 9
%N 1
%P 23-43
%! Gorenstein Homological Dimension of Groups Through Flat-Cotorsion Modules
%K Group ring
%K Flat-cotorsion module
%K Gorenstein flat-cotorsion module
%K Gorenstein flat-cotorsion dimension
%R 10.22052/mir.2023.253090.1414
%X The representation theory of groups is one of the most interesting examples of the interaction between physics and pure mathematics, where group rings play the main role. The group ring $\rga$ is actually an associative ring that inherits the properties of the group $\ga$ and the ring of coefficients $R$. In addition to the fact that the theory of group rings is clearly the meeting point of group theory and ring theory, it also has applications in algebraic topology, homological algebra, algebraic K-theory and algebraic coding theory.In this article, we provide a complete description of Gorenstein flat-cotorsion modules over the group ring $\rga$,where $\ga$ is a group and $R$ is a commutative ring. It will be shown that if $\ga'\leqslant \ga$ is a finite-index subgroup, then the restriction of scalars along the ring homomorphism $\rga'\rt\rga$ as well as its right adjoint $\rga\otimes_{\rga'}-$, preserve the class of Gorenstein flat-cotorsion modules. Then, as a result, Serre's Theorem is proved for the invariant $\Ghcd_{R}\ga$, which refines the Gorenstein homological dimension of $\ga$ over $R$, $\Ghd_{R}\ga$, and is defined using flat-cotorsion modules. Moreover, we show that the inequality $\GF (\rga)\leqslant \GF (R)+{\cd_{R}\ga}$ holds for the group ring $\rga$, where $\GF (R)$ denotes the supremum of Gorenstein flat-cotorsion dimensions of all $R$-modules and $\cd_{R}\ga$ is the cohomological dimension of $\ga$ over $R$.
%U https://mir.kashanu.ac.ir/article_114195_d85625e33cc13cf16c9558e8e3a31386.pdf