Injective modules for group rings and Gorenstein orders

Abstract

Commutative noetherian Gorenstein rings (i.e., rings of finite selfinjective dimension) were first studied by Bass in 1963 [2]. S is a Gorenstein ring of injective dimension one if and only if the total ring of quotients Q of S is Gorenstein and Q S is injective. In this case, Q S is the direct sum of the injective envelopes of S p , where p runs over all height one prime ideals of S. Drozd-Kirichenko-Roiter 1967 [5] have shown that Gorenstein orders behave nearly as nice as commutative noetherian Gorenstein rings. Levy ([8], unpublished) has proved a similar result as above for a special class of commutative Gorenstein orders. In the present paper we classify injective modules over Gorenstein orders. Since integral group rings of finite groups are Gorenstein orders, this gives the injective modules over integral group rings. To be more precise, let R be a Dedekind domain with quotient field K and Λ and R-order in the finite dimensional separable K-algebra A. Λ is said to be a Gorenstein order if Λ has selfinjective dimension one. For a Gorenstein order Λ, all indecomposable injective modules occur as direct summands of A ⊕ A Λ . The injective envelope of an R-torsionfree left Λ-module M is KM, and the injective envelopes of the simple left Λ-modules are of the form A ̂ p ϵ \\ ̂ gL p ϵ , where ≏ denotes the p-adic completion at a maximal ideal p of R and ϵ is a primitive idempotent of \\ ̂ gL p . We also give the multiplicity with which an indecomposable injective Λ-module occurs in A ⊕ A Λ . In case A is commutative, this multiplicity is one, and we get an extension of Levy's result [8].