Ore localization and minimal injective resolutions

Abstract

Abstract. In this paper, we describe the structure of the localization of Ext i R (R/P, M ), where P is a prime ideal and M is a module, at certain Ore sets X. We first study the situation for FBN rings, and then consider matters for a large class of Auslander-Gorenstein rings. We need to impose suitable homological regularity conditions to get results in the more general situation. The results obtained are then used to study the shape of minimal injective resolutions of modules over noetherian rings.If R is a commutative noetherian ring, S ⊆ R a multiplicatively closed set containing 1, and M a finitely generated module, then for any R-module N and any integer i, there is an isomorphism:In general, the question of whether there is an analogous result for noncommutative noetherian rings does not make sense; Hom R (M, N ) need not carry an R-module structure for arbitrary modules M and N . However, if M or N is a bimodule, then such questions become valid and can be quite subtle to answer.In this paper, we will show that for some rings R at certain Ore sets X, given a finitely generated module M and a prime ideal P , there is an isomorphism. This is enough for some applications. Given a module M over a noetherian ring R, what can one say about the 'shape' of a minimal injective resolution of M ? For a commutative noetherian ring A, this is known (see [23], [10], [9]): the minimal injective resolution of a module M is determined by geometric data corresponding to M , i.e. the support of M , Supp(M ), in Spec(A). Using the results in this paper, we will show that matters behave similarly for some classes of noncommutative noetherian rings.In §1, we gather some preliminary results and definitions. In §2, we make our first attempt to localize Ext. We show that if X is an Ore set such that essentiality is preserved under localization (such Ore sets are characterized in [12]), then matters behave well without additional homological hypotheses. Over an FBN ring, localization at every Ore set preserves essentiality -this was first proved in [11], and we provide a proof of a slightly more general result in Proposition 2.3. Corollary 2.4, in particular, proves the following:Theorem. Let R be a noetherian right FBN ring. Let X be an Ore set in R. Let M be an R-module. Then, given a prime ideal P , there is an isomorphism of R X -modules for all i:If, in addition, S is a ring and M is an (S, R)-bimodule, then the above isomorphism is one of bimodules.In [5, Lemma 3.2], this was proved, with a mild homological hypothesis, for bimodules. We reprove this result without the homological restrictions on R.Recall that over a right noetherian ring, indecomposable injective modules come in two distinct flavours, tame and wild (see §1.1 for details). Fully Bounded Noetherian (FBN) rings are precisely the noetherian rings for which all indecomposable injectives are tame, and this fact allows us to work directly with injective resolutions to prove localization results. Injective modules over rings which are not FBN, however, can be far ...

A homological reformulation of the link condition

“…Proof. This is [20 We note that there is a symmetric result for left modules, with the map D replaced by the map D ′ .…”