Serre Subcategories Research Articles

The spectrum of Grothendieck monoid: Classifying Serre subcategories and reconstruction theorem

The Grothendieck monoid of an exact category is a monoid version of the Grothendieck group. We use it to classify Serre subcategories of an exact category and to reconstruct the topology of a noetherian scheme. We first construct bijections between (i) the set of Serre subcategories of an exact category, (ii) the set of faces of its Grothendieck monoid, and (iii) the monoid spectrum of its Grothendieck monoid. By using (ii), we classify Serre subcategories of exact categories related to a finite dimensional algebra and a smooth projective curve. For this, we determine the Grothendieck monoid of the category of coherent sheaves on a smooth projective curve. By using (iii), we introduce a topology on the set of Serre subcategories. As a consequence, we recover the topology of a noetherian scheme from the Grothendieck monoid.

Harish-Chandra bimodules for type A rational Cherednik algebras

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group Sn. In particular, we show that for any parameter c∈C, the category of Harish-Chandra Hc-bimodules admits a fully faithful embedding into the category Oc, and describe the irreducibles in the image. We also construct a duality on the category of Harish-Chandra bimodules, and in fact we do this in a greater generality of quantizations of Nakajima quiver varieties. We use this duality, along with induction and restriction functors, to describe, as an abelian category, the smallest Serre subcategory of the category of Harish-Chandra bimodules containing the regular bimodule, as well as to explicitly describe the tensor products of its irreducible objects.

Reconstruction of a surface from the category of reflexive sheaves

We define a normal surface X to be codim-2-saturated if any open embedding of X into a normal surface with the complement of codimension 2 is an isomorphism. We show that any normal surface X allows a codim-2-saturated model Xˆ together with the canonical open embedding X→Xˆ.Any normal surface which is proper over its affinisation is codim-2-saturated, but the converse does not hold. We give a criterion for a surface to be codim-2-saturated in terms of its Nagata compactification and the boundary divisor.We reconstruct the codim-2-saturated model of a normal surface X from the additive category of reflexive sheaves on X. We show that the category of reflexive sheaves on X is quasi-abelian and we use its canonical exact structure for the reconstruction.In order to deal with categorical issues, we introduce a class of weakly localising Serre subcategories in abelian categories. These are Serre subcategories whose categories of closed objects are quasi-abelian. This general technique might be of independent interest.

-PERPENDICULAR WIDE SUBCATEGORIES

Abstract Let $\Lambda $ be a finite-dimensional algebra. A wide subcategory of $\mathsf {mod}\Lambda $ is called left finite if the smallest torsion class containing it is functorially finite. In this article, we prove that the wide subcategories of $\mathsf {mod}\Lambda $ arising from $\tau $ -tilting reduction are precisely the Serre subcategories of left-finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $\tau $ -tilting reduction. This leads to a natural way to extend the definition of the “ $\tau $ -cluster morphism category” of $\Lambda $ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan–Marsh in the $\tau $ -tilting finite case and by Igusa–Todorov in the hereditary case.

Cofiniteness with respect to Extension of Serre Subcategories at Small Dimensions

Let A denote a commutative Noetherian ring, a be an ideal of A , X be an arbitrary dense subcategory of A -modules, and N be the full subcategory of finite A -modules. In this paper, we study N ⋆ X - c -cofinite modules with respect to the extension subcategory N ⋆ X when dim A / c ≤ 2 . We also study c -cofiniteness with respect to a new dimension.

Melkersson condition for extension of Serre subcategories

Let R be a commutative noetherian ring and let $$\mathfrak {a}$$ be an ideal of R. In this paper, we study a certain condition, namely $$C_{\mathfrak {a}}$$ , introduced by Aghapournahr and Melkersson, on the extension of two subcategories of R-modules. We develop some of the main results of Yoshizawa (Proc Am Math Soc 140(7):2293–2305, 2012; J Commut Algebra 13:137–155, 2021). As an example of extension of subcategories, we study the weakly Laskerian modules and we find some conditions under which the local cohomology modules of a weakly Laskerian module lie in an arbitrary Serre subcategory. Eventually, we investigate the cofiniteness of the local cohomology modules of weakly Laskerian modules.

Cofiniteness with respect to extension of Serre subcategories

Let R be a commutative Noetherian ring, be an ideal of R, be a Serre subcategory of R-modules satisfying the condition , N be a finitely generated R-module and M be an arbitrary R-module with . We show that if M is an --cofinite R-module with , then is --cofinite for each . If with , then M is --cofinite if and only if Moreover, we prove that the subcategory and M is --cofinite of the category of R-modules is abelian. Furthermore, for a nonnegative integer n, we prove that for all if and only if is --cofinite for all and .

Yoneda extensions of abelian quotient categories

Let A be a essentially small abelian category and C be a Serre subcategory of A. Consider the quotient functor q:A→A/C. For an object A∈A and a non-negative integer k we investigate when the natural map qX,Ai:ExtAi(X,A)→ExtA/Ci(q(X),q(A)) is invertible for every X∈A and every i∈{0,1,⋯,k}. In the end we give an application of the main theorem.

Localization, monoid sets and K-theory

We develop the K-theory of sets with an action of a pointed monoid (or monoid scheme), analogous to the K-theory of modules over a ring (or scheme). In order to form localization sequences, we construct the quotient category of a nice regular category by a Serre subcategory.

Coserreness with respect to specialization closed subsets and some Serre subcategories

Let [Formula: see text] be a specialization closed subset of [Formula: see text] and [Formula: see text] be a Serre subcategory of [Formula: see text]. As a generalization of the notion of cofiniteness, we introduce the concept of [Formula: see text]-coserreness with respect to [Formula: see text] (see Definition 4.1). First, as a main result, for some special Serre subcategories [Formula: see text], we show that an [Formula: see text]-module [Formula: see text] with [Formula: see text] is [Formula: see text]-coserre with respect to [Formula: see text] if and only if [Formula: see text] for all ideals [Formula: see text]. Indeed, this result provides a partial answer to a question that was recently raised in [K. Bahmanpour, R. Naghipour and M. Sedghi, Modules cofinite and weakly cofinite with respect to an ideal, J. Algebra Appl. 16 (2018) 1–17]. As an application of this result, we show that the category of [Formula: see text]-coserre [Formula: see text]-modules [Formula: see text] with [Formula: see text] is a full Abelian subcategory of [Formula: see text]. Also, for every homologically bounded [Formula: see text]-complex [Formula: see text] whose homology modules belong to [Formula: see text] we show that the local cohomology modules [Formula: see text] for all [Formula: see text], are [Formula: see text]-coserre in all the cases where [Formula: see text], [Formula: see text] and [Formula: see text].

Exact structures and degeneration of Hall algebras

We study degenerations of the Hall algebras of exact categories induced by degree functions on the set of isomorphism classes of indecomposable objects. We prove that each such degeneration of the Hall algebra H(E) of an exact category E is the Hall algebra of a smaller exact structure E′<E on the same additive category A. When E is admissible in the sense of Enomoto, for any E′<E satisfying suitable finiteness conditions, we prove that H(E′) is a degeneration of H(E) of this kind.In the additively finite case, all such degree functions form a simplicial cone whose face lattice reflects properties of the lattice of exact structures. For the categories of representations of Dynkin quivers, we recover degenerations of the negative part of the corresponding quantum group, as well as the associated polyhedral structure studied by Fourier, Reineke and the first author.Along the way, we give minor improvements to certain results of Enomoto and Brüstle-Langford-Hassoun-Roy concerning the classification of exact structures on an additive category. We prove that for each idempotent complete additive category A, there exists an abelian category whose lattice of Serre subcategories is isomorphic to the lattice of exact structures on A. We show that every Krull-Schmidt category admits a unique maximal admissible exact structure and that the lattice of smaller exact structures of an admissible exact structure is Boolean.

The realization problem for generalized torsion theories

A generalized torsion theory associated with a Serre subcategory was introduced to characterize the Melkersson condition. However, little is known about when generalized torsion theories can be realized using specialization closed subsets of the spectrum of a ring. In this article, we investigate the characterization of generalized torsion theories realized by such a subset.

Cut cotorsion pairs

AbstractWe present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

Reducers and K 0 with support

Let R be a commutative noetherian ring, be a Serre subcategory of the category of finitely generated R-modules and be the category of finitely generated projective R-modules. We define invariants based on the categories of chain complexes from homologically supported in with support between 0 and n () and the K-group with support similar to the stable range in classical K-theory. We introduce a notion called a reducer which we use to express the class of a complex in terms of classes of complexes of smaller amplitude and use these to study and bound the above defined invariants by standard invariants like arithmetic rank, grade and projective dimension. We also give conditions for to be isomorphic to

Some results on formal local cohomology

We study some properties of formal local cohomology modules [Formula: see text] in Serre subcategories. As a consequence, we obtain some results on the minimax modules. We also describe the sets [Formula: see text] and [Formula: see text].

Extension Functors of Generalized Local Cohomology Modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$--module, and $X$ an arbitrary $R$--module. In this paper, for non-negative integers $s, t$ and a finitely generated $R$--module $N$, we study the membership of $\operatorname{Ext}_{R}^{s}(N, \operatorname{H}^{t}_{\mathfrak{a}}(M, X))$ in Serre subcategories of the category of $R$--modules and present some upper bounds for the injective dimension and the Bass numbers of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$. We also give some results on cofiniteness and minimaxness of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$ and finiteness of $\operatorname{Ass}_R(\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$.

Classifying Substructures of Extriangulated Categories via Serre Subcategories

We give a classification of substructures (= closed subbifunctors) of a given skeletally small extriangulated category by using the category of defects, in a similar way to the author's classification of exact structures of a given additive category. More precisely, for an extriangulated category, possible substructures are in bijection with Serre subcategories of an abelian category consisting of defects of conflations. As a byproduct, we prove that for a given skeletally small additive category, the poset of exact structures on it is isomorphic to the poset of Serre subcategories of some abelian category.

Melkersson conditions with respect to a prime ideal

Aghapournahr and Melkersson introduced the notion of Melkersson condition on a Serre subcategory of the category of modules over a commutative Noetherian ring. This paper investigates the structure of a set of prime ideals satisfying the Melkersson condition on a Serre subcategory. We try to calculate members of this set for a subcategory consisting of extension modules in two given Serre subcategories. Moreover, we classify the structure of sets consisting of prime ideals satisfying the Melkersson condition over rings with a small dimension.

English

Let R be a Noetherian ring, and I and J be two ideals of R. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI and M be a ZD-module. As a generalization of the S-depth(I, M) and depth(I, J, M), the S-depth of (I, J) on M is defined as $$S{\rm{ - depth}}\left( {I,J,M} \right) = \inf \left\{ {S{\rm{ - depth}}\left( {\mathfrak{a},M} \right): \mathfrak{a}\in \widetilde{W}\left( {I,J} \right)} \right\}$$ , and some properties of this concept are investigated. The relations between S-depth(I, J, M) and H (M) are studied, and it is proved that S-depth(I, J, M) = inf{i: H (M) ∉ S}, where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with respect to a pair of ideals coincide with ordinary local cohomology modules under these conditions. Let SuppRH (M) be a finite subset of Max(R) for all i < t, where M is an arbitrary R-module and t is an integer. It is shown that there are distinct maximal ideals $$\mathfrak{m}_1,\mathfrak{m}_2,...\mathfrak{m}_k\in{W(I,J)}$$ such that $$H_{I,J}^i\left( M \right) \cong H_{\mathfrak{m}{_1}}^i\left( M \right) \oplus H_{\mathfrak{m}{_2}}^i\left( M \right) \oplus \ldots \oplus H_{\mathfrak{m}_{_k}}^i\left( M \right)$$ for all i < t.

Cotorsion pairs and a K-theory localization theorem

We show that a complete hereditary cotorsion pair (C,C⊥) in an exact category E, together with a subcategory Z⊆E containing C⊥, determines a Waldhausen category structure on the exact category C, in which Z is the class of acyclic objects.This allows us to prove a new version of Quillen's Localization Theorem, relating the K-theory of exact categories A⊆B to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require A to be a Serre subcategory, which produces new examples.Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences.