Serre Subcategories Research Articles

Local Cohomology Modules, Serre Subcategories and Derived Functors of Torsion Functors

In this research, by using filter regular sequences, we obtain some exact sequences of right or left derived functors of local cohomology modules. Then we use them to gain some conditions under which a right or left derived functor of some special functors over local cohomology modules belongs to a Serre subcategory. These results can conclude some generalizations of previous results in this context or regain some of them.

LOCAL-GLOBAL PRINCIPLE FOR THE FINITENESS AND ARTINIANNESS OF GENERALISED LOCAL COHOMOLOGY MODULES

AbstractLet $\mathcal S$ be a Serre subcategory of the category of $R$-modules, where $R$ is a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $M$ and $N$ be finite $R$-modules. We prove that if $N$ and $H^i_{\mathfrak a}(M,N)$ belong to $\mathcal S$ for all $i\lt n$ and if $n\leq \mathrm {f}$-$\mathrm {grad}({\mathfrak a},{\mathfrak b},N )$, then $\mathrm {Hom}_{R}(R/{\mathfrak b},H^n_{{\mathfrak a}}(M,N))\in \mathcal S$. We deduce that if either $H^i_{\mathfrak a}(M,N)$ is finite or $\mathrm {Supp}\,H^i_{\mathfrak a}(M,N)$ is finite for all $i\lt n$, then $\mathrm {Ass}\,H^n_{\mathfrak a}(M,N)$ is finite. Next we give an affirmative answer, in certain cases, to the following question. If, for each prime ideal ${\mathfrak {p}}$ of $R$, there exists an integer $n_{\mathfrak {p}}$ such that $\mathfrak b^{n_{\mathfrak {p}}} H^i_{\mathfrak a R_{\mathfrak {p}}}({M_{\mathfrak {p}}},{N_{\mathfrak {p}}})=0$ for every $i$ less than a fixed integer $t$, then does there exist an integer $n$ such that $\mathfrak b^nH^i_{\mathfrak a}(M,N)=0$ for all $i\lt t$? A formulation of this question is referred to as the local-global principle for the annihilation of generalised local cohomology modules. Finally, we prove that there are local-global principles for the finiteness and Artinianness of generalised local cohomology modules.

Classifying Serre subcategories via atom spectrum

In this paper, we introduce the atom spectrum of an abelian category as a topological space consisting of all the equivalence classes of monoform objects. In terms of the atom spectrum, we give a classification of Serre subcategories of an arbitrary noetherian abelian category. Moreover we show that the atom spectrum of a locally noetherian Grothendieck category is homeomorphic to its Ziegler spectrum.

Remarks on subcategories of Artinian modules

We study two subcategories of the category of Artinian modules, a wide subcategory and a Serre subcategory. We prove that all wide subcategories of Artinian modules are Serre subcategories. We also provide the bijection between the set of Serre subcategories and the set of specialization closed subsets of the set of closed prime ideals of some completed ring. These results are Artinian analogues of the theorems proved by Takahashi.

Subcategories of extension modules by Serre subcategories

In 1962, P. Gabriel showed the existence of a lattice isomorphism between the set of Serre subcategories of the category consisting of finitely generated modules and the set of specialization closed subsets of the set of prime ideals. In this paper, we consider subcategories consisting of the extensions of modules in two given Serre subcategories to find a method of constructing Serre subcategories of the category of modules. We shall give a criterion for this subcategory to be a Serre subcategory.

Classifying subcategories of finitely generated modules over a Noetherian ring

For R a commutative Noetherian ring, wide and Serre subcategories of finitely generated R -modules have been classified by their support. This paper studies general torsion classes and introduces narrow subcategories. These are closed under fewer operations than wide and Serre subcategories, but still for finitely generated R -modules both narrow subcategories and torsion classes are classified using the same support data. Although for finitely generated R -modules all four kinds of subcategories coincide, they do not coincide in the larger category of all R -modules.

Cofiniteness with respect to a Serre subcategory

Let Φ be a system of ideals in a commutative Noetherian ring R, and let Open image in new window be a Serre subcategory of R-modules. We set $$ H_\Phi ^i ( \cdot , \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Ext_R^i (R/\mathfrak{b}| \otimes R \cdot , \cdot ). $$ . Suppose that a is an ideal of R, and M and N are two R-modules such that M is finitely generated and N ∈ Open image in new window. It is shown that if the functor \( D_\Phi ( \cdot ) = \mathop {\lim }\limits_{\overrightarrow {\mathfrak{b} \in \Phi } } Hom_R (\mathfrak{b}, \cdot ) \) is exact, then, for any \( \mathfrak{b} \in \Phi ,Ext_R^j (R/\mathfrak{b},H_\Phi ^i (M,N)) \) ∈ Open image in new window for all i, j ≥ 0. It is also proved that if there is a nonnegative integer t such that \( H_\mathfrak{a}^i (M,N) \) ∈ Open image in new window for all i < t, then \( Hom_R (R/\mathfrak{a},H_\mathfrak{a}^t (M,N)) \) ∈ Open image in new window, provided that Open image in new window is contained in the class of weakly LaskerianR-modules. Finally, it is shown that if L is an R-module and t is the infimum of the integers i such that \( H_\mathfrak{a}^i (L) \) ∈ Open image in new window, then \( Ext_R^j (R/\mathfrak{a},H_\mathfrak{a}^t (M,L)) \) ∈ Open image in new window if and only if \( Ext_R^j (R/\mathfrak{a},Hom_R (M,H_\mathfrak{a}^t (L))) \) ∈ Open image in new window for all j ≥ 0.

On selfadjoint functors satisfying polynomial relations

We study selfadjoint functors acting on categories of finite dimensional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint functors satisfying several easy relations, in particular, idempotents and square roots of a sum of identity functors, are classified. We also describe various natural constructions for new actions using external direct sums, external tensor products, Serre subcategories, quotients and centralizer subalgebras.

Resolutions for representations of reductive p-adic groups via their buildings

Schneider-Stuhler and Vigneras have used cosheaves on the affine Bruhat-Tits building to construct natural finite type projective resolutions for admissible representations of reductive p-adic groups in characteristic not equal to p. We use a system of idempotent endomorphisms of a representation with certain properties to construct a cosheaf and a sheaf on the building. We establish that these are acyclic and compute homology and cohomology with these coefficients. This implies Bernstein's result that certain subcategories of the category of representations are Serre subcategories. Furthermore, we also get results for convex subcomplexes of the building. Following work of Korman, this leads to trace formulas for admissible representations.

Classifying subcategories of modules over a commutative noetherian ring

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

A Gabriel Theorem for coherent twisted sheaves

The aim of this work is to give a generalization of Gabriel’s Theorem on coherent sheaves to coherent twisted sheaves on noetherian schemes. We start by showing that we can recover a noetherian scheme X from the category Coh(X, α) of coherent α-twisted sheaves over X, where α lies in the cohomological Brauer group of X. This follows from the bijective correspondence between closed subsets of X and Serre subcategories of finite type of Coh(X, α). Moreover, any equivalence between Coh(X, α) and Coh(Y, β), where X and Y are noetherian schemes, and \({\alpha\in Br\,'(X)}\) , β Br ′(Y) induces an isomorphism between X and Y.

Local cohomology and Serre subcategories

The membership of the local cohomology modules H a n ( M ) of a module M in certain Serre subcategories of the category of modules is studied from below ( i < n ) and from above ( i > n ). Generalizations of depth and regular sequences are defined. The relation of these notions to local cohomology are found. It is shown that the membership of the local cohomology modules of a finite module in a Serre subcategory in the upper range just depends on the support of the module.

Reconstructing projective schemes from Serre subcategories

Given a positively graded commutative coherent ring A = ⊕ j ⩾ 0 A j , finitely generated as an A 0 -algebra, a bijection between the tensor Serre subcategories of qgr A and the set of all subsets Y ⊆ Proj A of the form Y = ⋃ i ∈ Ω Y i with quasi-compact open complement Proj A ∖ Y i for all i ∈ Ω is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective graded modules are used in an essential way. Also, there is constructed an isomorphism of ringed spaces ( Proj A , O Proj A ) → ∼ ( Spec ( qgr A ) , O qgr A ) , where ( Spec ( qgr A ) , O qgr A ) is a ringed space associated to the lattice L Serre ( qgr A ) of tensor Serre subcategories of qgr A.

Classifying Serre subcategories of finitely presented modules

Given a commutative coherent ring R R , a bijective correspondence between the thick subcategories of perfect complexes D per ( R ) \mathcal D_{\operatorname {per}}(R) and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of isomorphism classes of indecomposable injective modules are used in an essential way.

Closed Subbifunctors of the Extension Functor

Given a subbifunctor F of Ext1(,), one can ask if one can generalize the construction of the derived category to obtain a relative derived category, where one localizes with respect to F-acyclic sequences. We show that this is possible if and only if F is closed. We also show that for artin algebras the closed subbifunctors correspond to Serre subcategories of a category of finitely presented functors that vanish on projectives, and we use this to find new examples of closed subbifunctors. Using relatively derived categories, we give a relative version of Happel's result on derived equivalences induced by tilting, and we show in an example how this can be used to find ordinary derived equivalences.

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categories is presented. To each locally coherent Grothendieck category C a topological space, the Ziegler spectrum of C, is associated. It is proved that the open subsets of the Ziegler spectrum of C are in bijective correspondence with the Serre subcategories of coh C the subcategory of coherent objects of C. This is a Nullstellensatz for locally coherent Grothendieck categories. If R is a ring, there is a canonical locally coherent Grothendieck category RC (respectively, CR) used for the study of left (respectively, right) R-modules. This category contains the category of R-modules and its Ziegler spectrum is quasi-compact, a property used to construct large (not finitely generated) indecomposable modules over an artin algebra. Two kinds of examples of locally coherent Grothendieck categories are given: the abstract category theoretic examples arising from torsion and localization and the examples that arise from particular modules over the ring R. The duality between coh-(RC) and coh-CR is shown to give an isomorphism between the topologies of the left and right Ziegler spectra of a ring R. The Nullstellensatz is used to give a proof of the result of Crawley-Boevey that every character ξ: K0(coh-C) → Z is uniquely expressible as a Z-linear combination of irreducible characters. 1991 Mathematics Subject Classification: 16D90, 18E15.

The Endomorphism Ring of a Localized Coherent Functor

LetCbe a commutative artinian ring and Λ an artinC-algebra. The category of coherent additive functorsA: mod-Λ→Ab on the finitely presented right Λ-modules will be denoted by Ab(Λ). This category is equivalent to the free abelian category over the ring Λ. If S0⊆Ab(Λ) is the Serre subcategory of the finite length objects of Ab(Λ) andA∈Ab(Λ), it is proved that the endomorphism ring EndAb(Λ)/S0AS0of the localized objectAS0is a locally artinC-algebra. This is used to show that the Krull-Gabriel dimension of the category Ab(Λ) cannot equal 1. In particular, this holds for finite rings.

The spectrum of a locally coherent category

A topology on the spectrum of a locally coherent Grothendieck category is introduced. The closed subsets are related to certain localizing subcategories which are characterized in terms of Serre subcategories of the full subcategory of finitely presented objects.

Perpendicular categories with applications to representations and sheaves

This paper is concerned with the omnipresence of the formation of the subcategories right (left) perpendicular to a subcategory of objects in an abelian category. We encounter these subcategories in various contexts: • • the formation of quotient categories with respect to localizing subcategories (cf. Section 2); • • the deletion of vertices and shrinking of arrows (see [37]) in the representation theory of finite dimensional algebras (cf. Section 5); • • the comparison of the representation theories of different extended Dynkin quivers (cf. Section 10); • • the theory of tilting (cf. Sections 4 and 6); • • the study of homological epimorphisms of rings (cf. Section 4); • • the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme (cf. Section 7 and [21]); • • the study of (maximal) Cohen-Macaulay modules over surface singularities (cf. Sections); • • the comparison of weighted projective lines for different weight sequences (cf. Section 9); • • the formation of affine and local algebras attached to path algebras of extended Dynkin quivers, canonical algebras, and weighted projective lines (cf. Section 11 and [21] and the concept of universal localization in [40]). Formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization. This applies in particular to applications in the domain of finite dimensional algebras and their representations. Several applications of the methods presented in this paper are already in existence, partly published, or appearing in print in the near future (see, for instance, [40, 39, 4, 45, 26, 49, 46]) and have show the versatility of the notion of a perpendicular category. It seems that (right) perpendicular categories first appeared—as the subcategories of so-called closed objects—in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing Serre subcategory (see [18, 47, 34]). Another natural occurrence is encountered in Commutative Algebra, forming the possibly infinitely generated modules of depth ⩾2 (cf. Section 7). The concept and some of the central applications were first presented in a talk given by the first author at the Honnef meeting in January 1985. We also note that the perfectly matching nomination “perpendicular category” was coined by A. Schofield, who discovered independently the usefulness of this concept in dealing with hereditary algebras (see [39], cf. also Section 7). The authors further acknowledge the support of the Deutsche Forschungsgemeinschaft (SPP “Darstellungstheorie von endlichen Gruppen und endlichdimensionalen Algebren”). Throughout this paper rings are associative with unit and modules are unitary right modules. Mod( R) (respectively mod( R)) denotes the category of all (respectively all finitely presented) right R-modules.

The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences

For an Artin Algebra Λ of finite representation type, the category Λ-mod, considered as a ring with several objects, has Krull dimension zero. Contrary, for a wild hereditary Artin Algebra this dimension does not exist. In this paper we show that the Krull dimension of Λ-mod for an Artin Algebra Λ of tame representation type is two. The corresponding Krull-Gabriel filtration by Serre subcategories of the category F of finitely presented contravariant functors on Λ-mod leads to a hierarchy of exact sequences in Λ-mod. Influenced by the functorial approach to almost split sequences by M. Auslander and I. Reiten, we investigate the exact sequences whose corresponding functors become simple in one of the successive quotient categories of the Krull-Gabriel filtration of F.