Classifying subcategories of modules over a commutative noetherian ring

Let R be a quotient ring of a commutative coherent regular ringby a finitely generated ideal. Hovey gave a bijection betweenthe set of coherent subcategories of the category of finitelypresented R-modules and the set of thick subcategories of thederived category of perfect R-complexes. Using this bijection,he proved that every coherent subcategory of finitely presentedR-modules is a Serre subcategory. In this paper, it is provedthat this holds whenever R is a commutative noetherian ring.This paper also yields a module version of the bijection betweenthe set of localizing subcategories of the derived categoryof R-modules and the set of subsets of Spec R which was givenby Neeman.     

Classifying resolving subcategories over a Cohen–Macaulay local ring

Let R be a Cohen–Macaulay local ring. Denote by mod R the category of finitely generated R-modules. In this paper, we consider the classification problem of resolving subcategories of mod R in terms of specialization-closed subsets of Spec R. We give a classification of the resolving subcategories closed under tensor products and transposes. Under restrictive hypotheses, we also give better classification results.     

Local bijective gabriel correspondence and relative fully boundedness in τ-noetherian rings

Let Rbe a right τ-noetherian ring, where τ denotes a hereditary torsion theory on the category of right R-modules. It is shown that every essential τ-closed right ideal of every prime homomorphic image of Rcontains a nonzero two-sided ideal if and only if any two τ-torsionfree injective indecomposable right R-modules with identical associated prime ideals are isomorphic, and for any τ-closed prime ideal Pthe annhilator of a finitely generated P-tame right R-module cannot be a prime ideal properly contained in P. Furthermore, if in the last condition finitely generated is replaced by r-noetherian, then all τ-noetherian τ-torsionfree modules turn out to be finitely annihilated.     

On the closedness of taking injective hulls of several Serre subcategories

Belshoff and Xu showed that every Matlis reflexive module has a Matlis reflexive injective hull if and only if R is complete and has dimension less than or equal to 1. In this paper, we give a characterization of the closedness of taking injective hulls for a Serre subcategory consisting of Minimax modules. In addition, the closedness of taking injective hulls for a Serre subcategory consisting of extension modules of finitely generated modules by modules with finite support is characterized by the number of prime ideals. Our results provide a negative answer to Aghapournahr and Melkersson’s question concerning Melkersson subcategories.     

Some morphisms of modules over the ring of pseudorational numbers

We study various morphisms of modules over the ring of pseudorational numbers R. We obtain a criterion for a quasi-isomorphism between finitely generated R-modules, introduce the concept of a pseudohomomorphism, and prove that the Krull-Remak-Schmidt theorem holds in the category of pseudohomomorphisms of finitely generated R-modules.     

V-Categories: Applications to Graded Rings

Motivated by the study of V-rings, we introduce the concept of V-category, as a Grothendieck category with the property that any simple object is injective. We present basic properties of V-categories, and we study this concept in the special case of locally finitely generated categories, for instance the category R-gr of all graded left R-modules, where R is a graded ring. We use the characterizations of V-categories in the study of graded V-rings. Since V-rings are closely related to Von Neumann regular rings (in the commutative case these classes of rings coincide), the last part of the article is devoted to graded regular rings.     

Self-injective Right Artinian Rings and Igusa Todorov Functions

We show that a right artinian ring R is right self-injective if and only if ψ(M)?=?0 (or equivalently ?(M)?=?0) for all finitely generated right R-modules M, where ψ, $\phi :\!\!\!\! \mod R \to \mathbb N$ are functions defined by Igusa and Todorov. In particular, an artin algebra Λ is self-injective if and only if ?(M)?=?0 for all finitely generated right Λ-modules M.     

Preinjective modules over pure semisimple rings

For a left pure semisimple ring R, it is shown that the local duality establishes a bijection between the preinjective left R-modules and the preprojective right R-modules, and any preinjective left R-module is the source of a left almost split morphism. Moreover, if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R-mod, the direct sum of all non-preinjective indecomposable direct summands of products of preinjective left R-modules is a finitely generated product-complete module. This generalizes a recent theorem of Angeleri Hügel [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007) 361-376] for hereditary rings.     

Artinian cofinite modules over complete Noetherian local rings

Let (R, m) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R-module. In this paper it is shown that if p is a prime ideal of R such that dim R/p = 1 and (0:M p) is not finitely generated and for each i ? 2 the R-module Ext _R ⁱ (M,R/p) is of finite length, then the R-module Ext _R ¹ (M, R/p) is not of finite length. Using this result, it is shown that for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext _R ⁱ (N,M) are of finite length, if and only if, for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext _R ⁱ (M,N) are of finite length.     

The hereditary monocoreflective subcategories of Abelian groups and R-modules

The non-trivial hereditary monocoreflective subcategories of the Abelian groups are the following ones: {G ?? Ob Ab | G is a torsion group, and for all g ?? G the exponent of any prime p in the prime factorization of o(g) is at most E(p)}, where E(·) is an arbitrary function from the prime numbers to {0, 1, 2, ??,??}. (o(·) means the order of an element, and n ?? ?? means n < ??.) This result is dualized to the category of compact Hausdorff Abelian groups (the respective subcategories are {G ?? Ob CompAb | G has a neighbourhood subbase {G _?? } at 0, consisting of open subgroups, such that G/G _?? is cyclic, of order like o(g) above}), and is generalized to categories of unitary R-modules for R an integral domain that is a principal ideal domain. For general rings R with 1, an analogous theorem holds, where the hereditary monocoreflective subcategories of unitary left R-modules are described with the help of filters L in the lattice of the left ideals of the ring R. These subcategories consist of those left R-modules, for which the annihilators of all elements belong to L. If R is commutative, then this correspondence between these subcategories and these filters L is bijective.     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript> and rings over which the class of finitely generated strongly gorenstein projective modules is closed under extensions

In this paper, we consider the rings over which the class of finitely generated strongly Gorenstein projective modules is closed under extensions (called fs-closed rings). We give a characterization about the Grothendieck groups of the category of the finitely generated strongly Gorenstein projective R-modules and the category of the finitely generated R-modules with finite strongly Gorenstein projective dimensions for any left Noetherian fs-closed ring R.     

Localization of Injective Modules Over Arithmetical Rings

It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P, R _P is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreover, if R is a Prüfer domain of finite character, localizations of injective R-modules are injective.     

Universal Deformation Rings of Modules for Algebras of Dihedral Type of Polynomial Growth

Let k be an algebraically closed field, and let Λ be an algebra of dihedral type of polynomial growth as classified by Erdmann and Skowroński. We describe all finitely generated Λ-modules V whose stable endomorphism rings are isomorphic to k and determine their universal deformation rings R(Λ, V). We prove that only three isomorphism types occur for R(Λ, V): k, k[[t]]/(t ²) and k[[t]].     

Completions of commutative rings and modules

In this work we continue studying the notion of completion ofR-modules, over a commutative ringR, relative to a torsion theoryϑ. We develop some techniques relative to localization at prime ideals and give structural results on the completion of finitely generatedR-modules, describing it as the product of classical completions on local noetherian rings. The authors acknowledge partial support from the D.G.I.C.Y T.     

<Emphasis Type="Italic">G</Emphasis>-stable support <Emphasis Type="Italic">τ</Emphasis>-tilting modules

Motivated by τ-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra Λ with action by a finite group G; we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over Λ; G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective Λ-modules, and G-stable functorially finite torsion classes in the category of finitely generated left Λ-modules. In the case when Λ is the endomorphism of a G-stable cluster-tilting object T over a Hom-finite 2-Calabi-Yau triangulated category C with a G-action, these are also in bijection with G-stable cluster-tilting objects in C. Moreover, we investigate the relationship between stable support τ-tilitng modules over Λ and the skew group algebra ΛG     

Thick subcategories of modules over commutative noetherian rings (with an appendix by Srikanth Iyengar)

For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking kernels, cokernels, and extensions) and closed under taking arbitrary direct sums. In addition, subcategories of A-modules that are closed under taking submodules, extensions, and direct unions are classified via associated prime ideals.     

When Cyclic Modules Have Σ-Injective Hulls

Abstract

A theorem of Cartan-Eilenberg (Cartan, H., Eilenberg, S. (1956). Homological Algebra. Princeton: Princeton University Press, pp. 390.) states that a ring Ris right Noetherian iff every injective right module is Σ-incentive. The purpose of this paper is to study rings with the property, called right CSI, that, all cyclic right R-modules have Σ-injective hulls, i.e., injective hulls of cyclic right R-modules are Σ-injective. In this case, all finitely generated right R-modules have Σ-injective hulls, and this implies that Ris right Noetherian for a lengthy list of rings, most notably, for Rcommutative, or when Rhas at most finitely many simple right R-modules, e.g., when Ris semilocal. Whether all right CSIrings are Noetherian is an open question. However, if in addition, R/rad Ris either right Kasch or von Neuman regular (=VNR), or if all countably generated (sermisimple) right R-modules have Σ-injective hulls then the answer is affirmative. (See Theorem A.) We also prove the dual theorems for Δ-injective modules.     

On the codimension of modules over skew power series rings with applications to Iwasawa algebras

The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second goal is the investigation of the class of S-modules which are finitely generated as R-modules. In the case that S and R are Auslander regular we show in particular that the codimension of M as S-module is one higher than the codimension of M as R-module. Furthermore its class in the Grothendieck group of S-modules of codimension at most one less vanishes, which is in the spirit of the Gersten conjecture for commutative regular local rings. Finally we apply these results to Iwasawa algebras of p-adic Lie groups.     

On Multigraded Generalizations of Kirillov–Reshetikhin Modules

We study the category of $\mathbb Z^\ell$ -graded modules with finite-dimensional graded pieces for certain $\mathbb Z^\ell_+$ -graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov–Reshetikhin modules and give a recursive formula for computing their graded characters.     

Every finitely generated group is weakly exact

We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf G-modules are relatively injective, which implies that bounded cohomology groups with coefficients in all Hopf G-modules vanish in all positive degrees. We also prove a general fixed point theorem for actions of finitely generated groups on ?_∞-type spaces. Finally, we define the notion of weak exactness for certain Banach algebras.