Let A be a commutative ring, and let a be a finitely generated Let A be a commutative ring, and let a be a finitely generated ideal in A. The category of A-modules is denoted by M(A). There are two functors on the category M(A) that the ideal a determines: the a-torsion functor Γ a , and the aadic completion functor Λ a . These are idempotent additive functors: Γ a • Γ a ∼ = Γ a and Λ a • Λ a ∼ = Λ a .The functors Γ a and Λ a seem as though they could be adjoint to each other. This is false however. What is true is that under suitable assumptions, the derived functors RΓ a , LΛ a :are adjoint to each other. Here D(A) is the (unbounded) derived category of Amodules. The most general condition under which this is known to hold is when the ideal a is weakly proregular. By definition, the ideal a is weakly proregular if it can be generated by a weakly proregular sequence. A sequence of elements a = (a 1 , . . . , a n ) in A is called weakly proregular if a rather complicated condition is satisfied by the Koszul complexes associated to powers of a; see Definition 4.1. This condition was first stated, without a name, by A. Grothendieck (see [LC] and [SGA2]); the name was only given by L. Alonso, A. Jeremias and J. Lipman in [AJL, Correction]. If A is noetherian, then every ideal in it is weakly proregular; but there are many non-noetherian examples (see Examples 4.10-4.12). The next theorem is the culmination of results by Grothendieck [LC], [SGA2]; E. Matlis [Ma]; J.P.C. Greenlees and J.P. May [GM]; Alonso, Jeremias and Lipman [AJL]; P. Schenzel [Sn]; M. Kashiwara and P. Schapira [KS3]; and M. Porta, L. Shaul and Yekutieli [PSY1]. There is parallel recent work on these matters by L. Positselski [Po]. Let a ⊆ A be a finitely generated ideal. A complex M ∈ D(A) is called derived a-torsion if the canonical morphism RΓ a (M ) → M is an isomorphism. Similarly, a complex M ∈ D(A) is called derived a-adically complete if the canonical morphism M → LΛ a (M ) is an isomorphism. (In [PSY1] these complexes were called cohomologically complete and cohomologically torsion, respectively; but now we realize that the adjective "derived" is better suited than "cohomologically". See Definition 3.7 below.) We denote by D(A) a-tor and D(A) a-com the full subcategory of D(A) on the derived a-torsion and the derived a-adically complete complexes, respectively. These are full triangulated subcategories.Theorem 0.1 (MGM Equivalence, [PSY1]). Let a be a weakly proregular ideal in a commutative ring A.(1) The functor LΛ a is right adjoint to the functor RΓ a .(2) The functors RΓ a and LΛ a are idempotent. (3) The categories D(A) a-tor and D(A) a-com are the essential images of the functors RΓ a and LΛ a , respectively. (4) The functor LΛ a : D(A) a-tor → D(A) a-comis an equivalence of triangulated categories, with quasi-inverse RΓ a .Actually, item (1) of this theorem is usually called GM Duality, and it is [PSY1, Erratum, Theorem 9]. Item (2) means that RΓ a •RΓ a ∼ = RΓ a and LΛ a •LΛ a ∼ = LΛ a .
We construct a 2-generator recursively presented group with infinite torsion length. We also explore the construction in the context of solvable and word-hyperbolic groups.
Weakly stable torsion classes were introduced by the author and Yekutieli to provide a torsion theoretic characterisation of the notion of weak proregularity from commutative algebra. In this paper we investigate weakly stable torsion classes, with a focus on aspects related to localisation and completion. We characterise when torsion classes arising from left denominator sets and idempotent ideals are weakly stable. We show that every weakly stable torsion class T can be associated with a dg ring A T ; in well behaved situations there is a homological epimorphism A → A T . We end by studying torsion and completion with respect to a single regular and normal element.
We prove an equivalent condition for the existence of a link between prime ideals in terms of the structure of a certain cohomology module. We use this formulation to answer an open question regarding the nature of module extensions over one sided noetherian rings. We apply the techniques developed in this paper to the local link structure of prime ideals of small homological height and examine when certain noetherian rings satisfy the density condition.Comment: 19 pages, Journal reference and DOI adde
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 ...
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