Gorenstein Homological Dimension of Groups Through Flat-Cotorsion Modules

The representation theory of groups is one of the most interesting exam- ples of the interaction between physics and pure mathematics, where group rings play the main role. The group ring RΓ is actually an associative ring that inherits the properties of the group Γ 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 topol- ogy, 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 RΓ, where Γ is a group and R is a commutative ring. It will be shown that if Γ, :( Γ is a finite-index subgroup, then the restriction of scalars along the ring homomorphism RΓ, → RΓ as well as its right adjoint RΓ RΓI , preserve the class of Gorenstein flat-cotorsion mod- ules. Then, as a result, Serre’s Theorem is proved for the invariant GhdRΓ, which refines the Gorenstein homological dimension of Γ over R, GhdRΓ, and is defined using flat-cotorsion modules. Moreover, we show that the in- equality GFCD(RΓ) :( GFCD(R) +cdRΓ holds for the group ring RΓ, where GFCD(R) denotes the supremum of Gorenstein flat-cotorsion dimensions of all R-modules and cdRΓ is the cohomological dimension of Γ over R.