Gorenstein flat covers and Gorenstein cotorsion modules over integral group rings. (Q2644168)

Recall that a ring \(R\) is said to be Iwanaga-Gorenstein if \(R\) is left and right Noetherian and \(R\) has finite self-injective dimension on both sides [see \textit{J. Xu}, Flat covers of modules, Lect. Notes Math. 1634 (1996; Zbl 0860.16002]). Moreover, every module over an Iwanaga-Gorenstein ring has a Gorenstein flat cover [\textit{E. Enochs} and \textit{J. Xu}, J. Algebra 181, No. 1, 288-313 (1996; Zbl 0847.16003)]. Since every integral group ring \(\mathbb{Z} G\) of a polycyclic-by-finite group is Iwanaga-Gorenstein [\textit{E. E. Enochs} and \textit{O. M. G. Jenda}, Tsukuba J. Math. 19, No. 1, 1-13 (1995; Zbl 0839.16012)], it follows that every module over the integral group ring of a finite group \(G\) has a Gorenstein flat cover. The purpose of this paper is to study the Gorenstein flat covers of the integral group rings \(\mathbb{Z} G\) of finite groups \(G\). With appropriate modifications of some results of \textit{D. K. Harrison} [Ann. Math. (2) 69, 366-391 (1959; Zbl 0100.02901)], the authors give a complete characterization of torsion-free cotorsion \(\mathbb{Z} G\)-modules, when \(G\) is finite. So they classify the Gorenstein cotorsion modules which are also Gorenstein flat over these integral group rings. With these results the authors classify the modules that can be the kernels of Gorenstein flat covers of integral group rings of finite groups. Using this tool, the authors associate an integer invariant \(n\) with every finite group \(G\) and prime \(p\). They show that \(1\leq n\leq[G: P]\), where \(P\) is a Sylow \(p\)-subgroup of \(G\), and give some indication of the significance of this invariant. These results are used to describe the co-Galois groups which are associated to the Gorenstein flat cover of a \(\mathbb{Z} G\)-module.