Covers and envelopes over Gorenstein rings

Abstract

A module over a Gorenstein ring is said to be Gorenstein injective if it splits under all modules of finite projective dimension. We show that over a Gorenstein ring every module has a Gorenstein injective envelope. We apply this result to the group algebra $\hat{Z}_{\rho}G$ (with $G$ a finite group and $\hat{Z}_{p}$ the ring of $p$-adic integers for some prime $p$) and show that ever finitely generated $\hat{Z}_{p}G$-module has a cover by a lattice. This gives a way of lifting finite dimensional representations of $G$ over $Z/(p)$ to modular representations of $G$ over $\hat{Z}_{p}$.