Gorenstein Homological Dimension of Groups Through Flat-Cotorsion Modules

‎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 $R$‎.