Serre Class Research Articles

A characterization of cofinite local cohomology modules in a certain Serre class

A characterization of cofinite local cohomology modules in a certain Serre class

Local homology, Koszul homology and Serre classes

Given a Serre class $\mathcal {S}$ of modules, we compare the containment of Koszul homology, Ext modules, Tor modules, local homology and local cohomology in $\mathcal {S}$ up to a given bound $s \geq 0$. As applications, we give a full characterization of Noetherian local homology modules. Furthermore, we establish a comprehensive vanishing result which readily leads to formerly known descriptions of the numerical invariants' width and depth in terms of Koszul homology, local homology and local cohomology. In addition, we recover a few renowned vanishing criteria scattered throughout the literature.

Some Results on the Finiteness of Generalized Local Cohomology Modules Using Serre Classes

Let R be a commutative Noetherian ring with identity and \(\mathfrak{a}\) be an ideal of R. In this paper, we investigate the conditions under which the set of associated primes of generalized local cohomology modules is finite. For this aim, we study some relations between generalized local cohomology modules, ideal transform, Ext and ordinary local cohomology modules in the Serre classes of R-modules. Such relations provided a common language for expressing some results about the generalized local cohomology R-modules, that has appeared in some papers.

A Unified Approach to Local Cohomology Modules using Serre Classes

AbstractThis paper discusses the connection between the local cohomology modules and the Serre classes of R-modules. This connection has provided a common language for expressing some results regarding the local cohomology R-modules that have appeared in different papers.

Balance in Stable Categories

We study when the stable category of an abelian category modulo a full additive subcategory is balanced and, in case the subcategory is functorially finite, we study a weak version of balance. Precise necessary and sufficient conditions are given in case the subcategory is either a Serre class or a class consisting of projective objects. The results in this second case apply very neatly to (generalizations of) hereditary abelian categories.

Cohen–Macaulayness with respect to Serre classes

Let $R$ be a commutative Noetherian ring. The notion of regular sequences with respect to a Serre class of $R$-modules is introduced and some of their essential properties are given. Then in the local case, we explore a theory of Cohen–Macaulayness with respect to Serre classes.

A Morita type theorem for a sort of quotient categories

We consider the quotient categories of two categories of modules relative to the Serre classes of modules which are bounded as abelian groups and we prove a Morita type theorem for some equivalences between these quotient categories.

Serre's contribution to the development of algebraic topology

We describe in detail Serre's application of spectral sequence theory to the study of the relations between the homology of total space, base space and fibre in a Serre fibration; and we apply the results to establish that a 1-connected space X has homology groups (in positive dimension) in a Serre class C if and only if its homotopy groups are in C. We include in this paper some personal reflections on the contact the author had with Serre during the decade of the 1950's when Serre's revolutionary work in homotopy theory was completely changing the face of algebraic topology.

On almost finitely generated nilpotent groups

A nilpotent group G is fgp if Gp, is finitely generated (fg) as a p‐local group for all primes p; it is fg‐like if there exists a nilpotent fg group H such that Gp≃Hp for all primes p. The fgp nilpotent groups form a (generalized) Serre class; the fg‐like nilpotent groups do not. However, for abelian groups, a subgroup of an fg‐like group is fg‐like, and an extension of an fg‐like group by an fg‐like group is fg‐like. These properties persist for nilpotent groups with finite commutator subgroup, but fail in general.

On a family of Serre classes of nilpotent groups

Let N be a P-local nilpotent group. We say that N is finitely generated (fg) as P-local group if there is a finite subset S of N such that the smallest P-local subgroup of N containing S is N itself; we could also say that S generates N as P-local group and write N=〈 S〉 p . If G is nilpotent, we say that G is finitely generated at every prime (fgp) if G p is fg as p-local group for all primes p. Such groups share with fg nilpotent groups many important properties (e.g., they are Hopfian). We show that they satisfy the axioms for a Serre class, as extended by Hilton and Roitberg to nilpotent groups.

Some remarks on almost finitely generated nilpotent groups

We identify two generalizations of the notion of a finitely generated nilpotent. Thus a nilpotent group G is fgp if Gp is fg as p-local group for each p; and G is fg-like if there exists a fg nilpotent group H such that Gp @ Hp for all p. The we have proper set-inclusions: {fg} I {fg-like} I {fgp}. We examine the extent to which fg-like nilpotent groups satisfy the axioms for a Serre class. We obtain a complete answer only in the case that [G, G] is finite. (The collection of fgp nilpotent groups is known to form a Serre class in the extended sense).

Localization of triangulated categories and derived categories

The notion of quotient and localization of abelian categories by dense subcategories (i.e., Serre classes) was introduced by Gabriel, and plays an important role in ring theory [6, 131. The notion of triangulated categories was introduced by Grothendieck and developed by Verdier [9, 16) and is recently useful in representation theory [8,4, 143. The quotient of triangulated categories by epaisse subcategories is constructed in [16]. Both quotients were indicated by Grothendieck, and they resemble each other. In this paper, we will consider triangulated categories and derived categories from the point of view of localization of abelian categories. Verdier gave a condition which is equivalent to the one that a quotient functor has a right adjoint, and considered a relation between tpaisse subcategories [16]. We show that localization of triangulated categories is similarly defined, and have a relation between localizations and Cpaisse subcategories. Beilinson, Bernstein, and Deligne introduced the notion of t-structure which is similar to torsion theory in abelian categories [2]. We, in particular, consider a stable t-structure, which is an epaisse subcategory, and deal with a correspondence between localizations of triangulated categories and stable r-structures. And then recollement, in the sense of [2], is equivalent to bilocalization. Next, we show that quotient and localization of abelian categories induce quotient and localization of its derived categories. In Section 1, we recall standard notations and terminologies of quotient and localization of abelian categories. In Section 2, we define localization of triangulated categories, and consider a relation between localizations and stable r-structures (Theorem 2.6). In Section 3, we show that if A + A/C is a quotient of abelian categories, then D*(A) -+ D*(A/C) is a quotient of triangulated categories, where * = + , -, or b (Theorem 3.2). Moreover,

MOD-C Postnikov Approximation of a 1-Connected Space

Deleanu, Frei and Hilton have developed the notion of generalized Adams completion in a categorical context [4]. They have also shown that if the set of morphisms is saturated then the Adams completion of an object is characterized by a certain couniversai property. We want to prove a stronger version of this result by dropping the saturation assumption on the set of morphisms; we also prove that the canonical map from an object to its Adams completion is an element of the set of morphisms under very moderate assumptions. These two results are fairly general in nature and are applicable to most cases of interest. Further using these two results and introducing “modulo a Serre class C of abelian groups” [9] we have obtained the mod-C Postnikov approximation of a 1-connected based CW-complex, with the help of a suitable set of morphisms.

Serre classes for toposes

We prove first that a logical fraction functor from a topos to a topos must be a filter-power functor, then we prove that such functors can have adjoints only when the filter is principal. Finally we refine this so that we are able to prove that the filter-power of a Grothendieck topos is Grothendieck if and only if the filter is principal.

The Frobenius rank equality for morphisms

Let U ( ▪,ε) be an abelian category with Serre class ▪ and Euler-Poincaré mapping ε. For η a morphism in U ( ▪,ε) with Imη a member of ▪, let rank εη =ε(Imη). A proof is given of the Frobenius rank equality: if αβγ is a composition of three morphisms in U ( ▪,ε) and Imβ is a member of ▪, then rank εαβ+rank εβγ+rank ε(kerβγ)β(cokαβ)=rank εβ+rank εαβγ.

Generalized homotopy excision theorems modulo a Serre class of nilpotent groups

The main results are stated in the context of Serre classes of nilpotent groups. Our proof of the Blakers-Massey theorem appears to be new, and applies moreover to any map of nilpotent spaces which induces an epimorphism of fundamental groups (cf. [18, Corollary 6, p. 487]). We close with a very general Freudenthal Suspension Theorem.

Extensions relative to a Serre class

Consider a class C of projective R-modules, where R is a commutative ring with identity, which satisfies the conditions of (2), namely that C is closed under the operations of direct sum and isomorphism and C contains the zero module. Following (2) a module M is said to have C-cotype n (respectively C-type n) if it has a projective resolution … → Pn → P0 → M → 0 with Pi ∈ C for i>n (respectively Pi ∈ C for i≦n). Let S be the class of modules of C-cotype −1, equivalently of C-type infinity. It is assumed throughout that S is a Serre Class. We define an abelian category of modules with the property that C-cotype is homological dimension in while in the case C = 0, S is just the category of R-modules. It follows that all categorical results on homological dimension also hold for cotype.

When Flats are Torsion Free

Given a complete Serre class τ this determines a torsion theory with T the class of torsion modules. It also determines the torsion free modules. For the classical torsion in the category of abelian groups the torsion free modules are flat and visa-versa. Which rings are characterized by this property? More precisely: Which rings admit a torsion theory for which the concepts of torsion free and flat are equivalent? We also dispose of the cases when R admits a toision theory for which torsion free is equivalent to injective and when projective is equivalent to torsion free.

Quotient categories and rings of quotient

A Serre class in an abelian category cA is a nonempty subclass : R —» R^, cRs fi S (the class of Rs modules which belong to <£ when considered as R modules via $) is a Serre class of cR^, and in [3] , a

Triples and Localizations(1)

Let A be a ring (associative) with unity, and let denote the category of unital left A-modules. If is a strongly complete Serre class in then (see [3], and also [6]) there is an exact functor S: , where is the quotient category , and is an abelian category.