Grothendieck Category Research Articles

Strongly Rickart objects in abelian categories: Applications to strongly regular and strongly Baer objects

ABSTRACTWe show how the theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories. For each of them, we prove general properties, we analyze the behavior with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories.

Strongly Rickart objects in abelian categories

ABSTRACTWe introduce and study (dual) strongly relative Rickart objects in abelian categories. We prove general properties, we analyze the behaviour with respect to (co)products, and we study the transfer via functors. We also give applications to Grothendieck categories, (graded) module categories and comodule categories. Our theory of (dual) strongly relative Rickart objects may be employed in order to study strongly relative regular objects and (dual) strongly relative Baer objects in abelian categories.

Ordered Tensor Categories and Representations of the Mackey Lie Algebra of Infinite Matrices

We introduce (partially) ordered Grothendieck categories and apply results on their structure to the study of categories of representations of the Mackey Lie algebra of infinite matrices $\mathfrak {gl}^{M}\left (V,V_{*}\right )$ . Here $\mathfrak {gl}^{M}\left (V,V_{*}\right )$ is the Lie algebra of endomorphisms of a nondegenerate pairing of countably infinite-dimensional vector spaces $V_{*}\otimes V\to \mathbb {K}$ , where $\mathbb {K}$ is the base field. Tensor representations of $\mathfrak {gl}^{M}\left (V,V_{*}\right )$ are defined as arbitrary subquotients of finite direct sums of tensor products (V∗)⊗m ⊗ (V∗)⊗n ⊗ V⊗p where V∗ denotes the algebraic dual of V. The category $\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ which they comprise, extends a category $\mathbb {T}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ previously studied in Dan-Cohen et al. Adv. Math. 289, 205–278, (2016), Penkov and Serganova (2014) and Sam and Snowden Forum Math. Sigma 3(e11):108, (2015) . Our main result is that $\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ is a finite-length, Koszul self-dual, tensor category with a certain universal property that makes it into a “categorified algebra” defined by means of a handful of generators and relations. This result uses essentially the general properties of ordered Grothendieck categories, which yield also simpler proofs of some facts about the category $\mathbb {T}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ established in Penkov and Serganova (2014). Finally, we discuss the extension of $\mathbb {T}^{3}_{\mathfrak {gl}^{M}\left (V,V_{*}\right )}$ obtained by adjoining the algebraic dual (V∗)∗ of V∗.

Grothendieck Categories as a Bilocalization of Linear Sites

We prove that the 2-category Grt of Grothendieck abelian categories with colimit preserving functors and natural transformations is a bicategory of fractions in the sense of Pronk of the 2-category Site of linear sites with continuous morphisms of sites and natural transformations. This result can potentially be used to make the tensor product of Grothendieck categories from earlier work by Lowen, Shoikhet and the author into a bi-monoidal structure on Grt.

Rickart and Dual Rickart Objects in Abelian Categories: Transfer via Functors

We study the transfer of (dual) relative Rickart properties via functors between abelian categories, and we deduce the transfer of (dual) relative Baer property. We also give applications to Grothendieck categories, comodule categories and (graded) module categories, with emphasis on endomorphism rings.

Torsion pairs in silting theory

In the setting of compactly generated triangulated categories, we show that the heart of a (co)silting t-structure is a Grothendieck category if and only if the (co)silting object satisfies a purity assumption. Moreover, in the cosilting case the previous conditions are related to the coaisle of the t-structure being a definable subcategory. If we further assume our triangulated category to be algebraic, it follows that the heart of any nondegenerate compactly generated t-structure is a Grothendieck category.

Projective and injective dimensions in the category σ[M

ABSTRACTFor any module M, the smallest subcategory of R-modules which is a Grothendieck category and contains M is the category σ[M]. In this article, by introducing two classes of R-modules under the titles -modules and Σn-modules, the notions σ-projective and σ-injective dimension in the category σ[M] have been defined and studied. The relationships between these notions and the corresponding notions defined in the category of R-modules have been investigated. Also, σ-global dimension is defined and some of its properties have been studied.

On the Tensor Product of Linear Sites and Grothendieck Categories

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.

Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories

We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.

Modular C11 lattices and lattice preradicals

This paper deals with properties of modular [Formula: see text] lattices involving hereditary preradicals on hereditary classes of modular lattices. Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.

On the isomorphism of injective objects in Grothendieck categories

Recently, some of the authors of the present note provided a generalization of Bumby’s theorem to the injectivity of modules over some classes of monomor-phisms. The question of whether Bumby’s criterion could be extended to more general classes of categories was left without an affirmative response, though. However, the present note provides an extension of that theorem to Grothendieck categories. More concretely, we establish that any two injective objects in a Grothendick category are isomorphic when they are isomorphic to subobjects of each other. As a corollary, we prove that two objects have isomorphic injective envelops (if they exist) whenever each of them is isomorphic to a subobject of the other. In the way, it will be indispensable to note that the main results of our previous work are still valid under weaker conditions.

Flat complexes, pure periodicity and pure acyclic complexes

The paper can be viewed as an addition and extension of the recent paper of Emmanouil (2016) [5]. Among others, an alternative functor category approach to Emmanouil's results is presented. By applying an idea of Neeman (2008) [16] and the main results obtained there, we prove that a chain complex F in a locally finitely presented Grothendieck category A is pure acyclic if and only if any chain map f:P→F from a complex P of pure-projective objects in A to F is null-homotopic. As a consequence we prove that any pure periodic object in A is pure-projective. Moreover, we show that A is pure semisimple if and only if A has the pure QF-property, that is, every pure-injective object in A is pure-projective.

On locally coherent hearts

We show that, under particular conditions, if a t-structure in the unbounded derived category of a locally coherent Grothendieck category restricts to the bounded derived category of its category of finitely presented objects, then its heart is itself a locally coherent Grothendieck category. Those particular conditions are always satisfied when the Grothendieck category is arbitrary and one considers the t-structure associated to a torsion pair in the category of finitely presented objects. They are also satisfied when one takes any compactly generated t-structure in the derived category of a commutative noetherian ring which restricts to the bounded derived category of finitely generated modules. As a consequence, any t-structure in this latter bounded derived category has a heart which is equivalent to the category of finitely presented objects of some locally coherent Grothendieck category.

Pure exact structures and the pure derived category of a scheme

AbstractLet$\mathcal{C}$be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the categoryC($\mathcal{C}$) of unbounded chain complexes in$\mathcal{C}$. We use λ-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasi–coherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi–coherent sheaves.

Attached atoms in Grothendieck categories

ABSTRACTLet 𝒜 be a Grothendieck category. In this paper we define attached atoms of nonzero objects of 𝒜 as a dual of the associated atoms of nonzero objects introduced by Kanda [2]. If R is a (non)commutative ring, then we show that this new notion can be corresponded to the attached prime ideals of nonzero modules introduced by Macdonald [5] and Annin [1].

HEREDITARY TORSION THEORIES OF A LOCALLY NOETHERIAN GROTHENDIECK CATEGORY

Let${\mathcal{A}}$be a locally noetherian Grothendieck category. We construct closure operators on the lattice of subcategories of${\mathcal{A}}$and the lattice of subsets of$\text{ASpec}\,{\mathcal{A}}$in terms of associated atoms. This establishes a one-to-one correspondence between hereditary torsion theories of${\mathcal{A}}$and closed subsets of$\text{ASpec}\,{\mathcal{A}}$. If${\mathcal{A}}$is locally stable, then the hereditary torsion theories can be studied locally. In this case, we show that the topological space$\text{ASpec}\,{\mathcal{A}}$is Alexandroff.

On the exactness of products in the localization of (Ab.4⁎) Grothendieck categories

Let C be an (Ab.4⁎) Grothendieck category, that is, products are exact in C. Given a hereditary torsion class T⊆C, we study the exactness of products in the Gabriel localization C/T of C. We show that, under suitable assumptions on C, the k+1-th derived functor of the product vanishes, provided the Gabriel dimension of C/T is smaller than k. As a consequence, we deduce that, under suitable hypotheses, the derived category D(C/T) is left-complete.

New results on C 11 and C 12 lattices with applications to Grothendieck categories and torsion theories

In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions (C i ) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions (C 11) or (C 12). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.

A classification theorem for t-structures

We give a classification theorem for a relevant class of t-structures in triangulated categories, which includes, in the case of the derived category of a Grothendieck category, a large class of t-structures whose hearts have at most n fixed consecutive non-zero cohomologies. Moreover, by this classification theorem, we deduce the construction of the t-tree, a new technique which generalizes the filtration induced by a torsion pair. At last we apply our results in the tilting context generalizing the 1-tilting equivalence proved by Happel, Reiten and Smalø [16]. The last section provides applications to classical n-tilting objects, examples of t-trees for modules over a path algebra, and new developments on compatible t-structures [22,20].

Some remarks on a theorem of Bergman

We extend a result of Bergman to show that any object in an arbitrary Grothendieck category may be expressed as an inverse limit of injectives. We also study inverse systems of κ-injective objects, where κ is an infinite regular cardinal.