In the present paper, we study the relationships of [Formula: see text]-cotorsion pairs among three abelian categories in a recollement. Under certain conditions, we present an explicit construction of gluing of [Formula: see text]-cotorsion pairs in an abelian category [Formula: see text] with respect to [Formula: see text]-cotorsion pairs in abelian categories [Formula: see text], [Formula: see text] respectively. On the other hand, we study the construction of [Formula: see text]-cotorsion pairs in abelian categories [Formula: see text], [Formula: see text] obtained from [Formula: see text]-cotorsion pairs in an abelian category [Formula: see text].

Abstract Let $$\Lambda $$ Λ be a $$\mathbb {k}$$ k -algebra where $$\mathbb {k}$$ k is a field of arbitrary characteristic, and let $$\mathscr {A}_\mathbb {k}$$ A k be a full subcategory of $$\Lambda $$ Λ -Mod, the abelian category of left $$\Lambda $$ Λ -modules. In particular, $$\mathscr {A}_\mathbb {k}$$ A k is a $$\mathbb {k}$$ k -category, i.e. the set of morphisms between objects in $$\mathscr {A}_\mathbb {k}$$ A k is a vector space over $$\mathbb {k}$$ k and the composition of morphisms is $$\mathbb {k}$$ k -bilinear. Following M. Kleiner and I. Reiten, $$\mathscr {A}_\mathbb {k}$$ A k is Hom-finite if the hom-space between any two objects in $$\mathscr {A}_\mathbb {k}$$ A k is finite-dimensional over $$\mathbb {k}$$ k . We further say that $$\mathscr {A}_\mathbb {k}$$ A k is Ext-finite if $$\dim _\mathbb {k}\textrm{Ext}^i_\Lambda (X,Y)<\infty $$ dim k Ext Λ i ( X , Y ) < ∞ for all objects X and Y in $$\mathscr {A}_\mathbb {k}$$ A k . Let V be an object in $$\mathscr {A}_\mathbb {k}$$ A k . In this note we prove that if $$\textrm{End}_\Lambda (V)$$ End Λ ( V ) is isomorphic to $$\mathbb {k}$$ k , then V has a universal deformation ring $$R(\Lambda ,V)$$ R ( Λ , V ) , which is a local complete Noetherian commutative $$\mathbb {k}$$ k -algebra whose residue field is also isomorphic to $$\mathbb {k}$$ k . We use this result to prove that if $$\Lambda $$ Λ is a two-point infinite-dimensional gentle $$\mathbb {k}$$ k -algebra (in the sense of V. Bekkert et al), then $$R(\Lambda ,V)$$ R ( Λ , V ) is isomorphic either to $$\mathbb {k}$$ k , to $$\mathbb {k}[\![t]\!]/(t^2)$$ k [ [ t ] ] / ( t 2 ) or to $$\mathbb {k}[\![t]\!]$$ k [ [ t ] ] .

Abstract In this paper, we study triangular matrix categories by using the theory of recollements of abelian categories. Given a triangular matrix category, we construct two canonical recollements. We show that if certain functors of these recollements are exact, then the category appearing in the middle term is actually a category of modules over a triangular matrix category. This result is a generalization of one given by Li (Commun Algebra 46(2):615–628, 2017. https://doi.org/10.1080/00927872.2017.1327051 ). Finally, we show that if $$\textrm{Mod}(\mathcal {C})$$ Mod ( C ) admits a nontrivial torsion pair by abelian categories, then $$\mathcal {C}$$ C is equivalent to a triangular matrix category.

Abstract In this paper, we define operations on preradicals in arbitrary abelian categories. We define idempotent preradicals and radicals. We prove that every adjoint pair between abelian categories induces a Galois connection between the corresponding ordered collections of preradicals. If the abelian categories are bicomplete, we construct alpha and omega preradicals and study their respective preservation under the Galois connection. If a bicomplete abelian category is in addition locally small, then the corresponding collection of preradicals is a complete lattice. The Galois connection induced by an adjoint pair between locally small bicomplete abelian categories preserves, respectively, idempotent preradicals and radicals.

Abstract We study the spectrum of closed subcategories in a quasi-scheme, i.e. a Grothendieck category X . The closed subcategories are the direct analogs of closed subschemes in the commutative case, in the sense that when X is the category of quasi-coherent sheaves on a quasi-projective scheme S , then the closed subschemes of S correspond bijectively to the closed subcategories of X . Many interesting quasi-schemes, such as the noncommutative projective scheme $$\operatorname {Qgr-}\hspace{-2.0pt}B = \operatorname {Gr-}\hspace{-2.0pt}B/\operatorname {Tors-}\hspace{-2.0pt}B$$ Qgr- B = Gr- B / Tors- B associated to a graded algebra B , arise as quotient categories of simpler abelian categories. In this paper, we will show how to describe the closed subcategories of any quotient category X / Y in terms of closed subcategories of X with special properties, when X is a category with a set of compact projective generators.

The notion of a pseudo cluster tilting subcategory in an extriangulated category is defined in this article. We prove that the quotient category [Formula: see text], obtained by factoring an extriangulated category [Formula: see text] by a pseudo cluster tilting subcategory [Formula: see text], is a semi-abelian category. Furthermore, we also show that the quotient category [Formula: see text] is an abelian category if and only if certain self-orthogonal conditions are satisfied. As an application, these results generalize the work of Xu and Zheng in the exact category.

Abstract We continue the study of thick triangulated subcategories, started by Valery Lunts and the author in “Thick subcategories on curves”, and consider thick subcategories in the derived category of coherent sheaves on a weighted projective curve and the corresponding abelian thick subcategories. Our main result is that any thick subcategory on a weighted projective curve either is equivalent to the derived category of nilpotent representations of some quiver (we call such categories quiver-like) or is the orthogonal subcategory to an exceptional collection of torsion sheaves (we call such subcategories big). We examine the structure of thick subcategories: in particular, for weighted projective lines, we prove that any admissible subcategory is generated by an exceptional collection and any exceptional collection is a part of a full one. We show that the derived categories of weighted projective curves satisfy the Jordan–Hölder property and do not contain phantoms. Finally, we extend and simplify results from loc. cit., providing sufficient criteria for a triangulated or abelian category to be quiver-like.

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an arbitrary $R$-module. We show that if the set $X=\cup_{i\geq 2}{\rm Supp}_R({\rm H}^{i}_{I}(R))$ is finite, then the category of all weakly cofinite modules with respect to the ideal $I$ of $R$ is an Abelian subcategory of the category of $R$-modules. In particular, it is true whenever ${\rm q}(I,R)\leq 1$.

Cotorsion Pairs in Extensions of Abelian Categories

This paper introduces and develops Möbius homology , a homology theory for representations of finite posets into abelian categories. Although the connection between poset topology and Möbius functions is classical, we go further by establishing a direct connection between poset topology and Möbius inversions. In particular, we show that Möbius homology categorifies the Möbius inversion, as its Euler characteristic coincides with the Möbius inversion applied to the dimension function of the representation. We also present a homological version of Rota’s Galois Connection Theorem, relating the Möbius homologies of two posets connected by a Galois connection. Our main application concerns persistent homology over general posets. We prove that, under a suitable definition, the persistence diagram arises as an Euler characteristic over a poset of intervals, and thus Möbius homology provides a categorification of the persistence diagram. This furnishes a new invariant for persistent homology over arbitrary finite posets. Finally, leveraging our homological variant of Rota’s Galois Connection Theorem, we establish several results about the persistence diagram.

We construct a new Weil cohomology for smooth projective varieties over a field, universal among Weil cohomologies with values in rigid additive tensor categories. A similar universal problem for Weil cohomologies with values in rigid abelian tensor categories also has a solution. We give a variant for Weil cohomologies satisfying more axioms, like weak and strong Lefschetz. As a consequence, we get a different construction of André’s category of motives for motivated correspondences and show that it has a universal property. This theory extends over suitable bases.

ABSTRACT In this article, we prove that if ( A , B , C ) is a recollement of abelian categories, then Wakamatsu tilting (resp. weak support τ -tilting) subcategories in A and C can induce Wakamatsu tilting (resp. weak support τ -tilting) subcategories in B , and the converses hold under natural assumptions. As an application, we mainly consider the relationship of τ -cotorsion torsion triples in ( A , B , C ).

In the forty years of existence of derived category, it was first thought as a tool in algebraic geometry, especially in the development of duality theories that were done by Hartshorne and others. After these first moments, the theory provided a powerful homological tool for the study of linear differential equations. The basic example in the literature that can be found about this is the Riemann-Hilbert problem of associating suitable regular systems of differential equations to constructible sheaves. This studies can be found in the work of Kashiwara and Schapira. See M. Kashiwara, P. Schapira “Sheaves on manifolds" ([15]). To understand the structure of the derived category is necessary to study the axioms of triangulated categories that were introduced in the mid 1960’s by J.L.Verdier in his thesis “Des catégories dérivées des catégories abéliennes" ([21]). The role of the triangles in the derived category is a similar role of the exact sequence in the abelian category. But it is important to remember that these axioms had their origins in algebraic geometry and algebraic topology. Nowadays there are important applications of triangulated categories in areas like algebraic geometry, algebraic topology (stable homotopy theory), commutative algebra, differential geometry and representation theory of artin algebras. See, for instance, the book of D. Happel- “Triangulated categories and the representation theory of finite dimensional algebras" ([11]). The objective of this notes is to present an introdutory material to the undergraduate and graduate students that would like to know some ideas about the derived category. These are the notes a one week series of introductory lectures which I gave in the XXIII-Escola de Algebra, in Maringá, Paraná, Brazil. Firstly we introduced the concepts of additive and abelian category to show the axioms of triangulated category that are our main objective. The triangulated category obey four axioms. We first introduced the first three axioms and their consequences on chapter one and then the octahedral axioms in various equivalent forms in a separate section of the first chapter. The objective of this section is to give a model capable of making this axiom more palatable since, in general, the form that it is presented in the literature does not remind the reader of any similar structure in other fields of mathematics. So, we make the necessary efforts here to present another form of this axiom that is similar to other tools that could be seen in the abelian categories. We present in chapter one the main example of triangulated category, the homotopy category of complexes. Secondly, to understand the morphisms in the derived category I introduced the concept of localization in chapter two. To those that are starting to study localization, we present the necessary background to understand the localization of non commutative ring. We believe that with this model in mind the student will profit more from the study of localization of categories. On chapter two, the student will find the necessary information and exercises to begin to manipulate morphisms in the derived category. So, on chapter three we introduce the definition of derived category of an abelian category and we explain how one sees the original abelian category as a subcategory of its derived category. After having done all this work, it is natural to have many questions about the behavior of the derived category or its applications. Therefore, we present here a bibliography in portuguese and in english that will help the students to make further investigations. The reader that whishes to know the history and the motivation of the begining of the derived category with many details, should read the introduction of the book "Sheaves on Manifolds - M. Kashiwara and M. Schapira ([15]). Acknowledgements: I am particularly grateful to Sônia Maria Fernandes-DMAUFV, Tanise Carnieri Pierin -DMAT-UFPR and Eduardo Nascimento Marcos IMEUSP, who carefully worked through the text and sent me detailed lists of corrections, questions and remarks. These notes were writen for the first time in 2014 and were used in a minicourse which I tough in the XXIII-Escola de Algebra in Maringá, Paraná, Brazil. The last version was written during my visit to IME-USP in 2018, where I got finantial help of Fapesp, process 2018/08104 - 3.

The aim of this paper is to introduce a tensor structure for the Serre quotient category of an abelian monoidal category with biexact tensor product to make the canonical functor a monoidal functor. In this tensor product, the Serre quotient category of a multiring category (resp. a multitensor category) by a two-sided Serre tensor-ideal is still a multiring category (resp. a multitensor category). Besides, a two-sided Serre tensor-ideal of a tensor category is always trivial. This result can be generalized to any tensor product. If the canonical functor is a monoidal functor, then the corresponding Serre subcategory of the tensor category is trivial.

Why is the Category of Near-Vector Spaces Abelian?

We study the transfer of (dual) (strongly) relative F-Baer objects with respect to a fully invariant short exact sequence via functors between abelian categories. Mainly fully faithful functors and adjoint pairs of functors are considered. Various results are obtained for (co)reflective subcategories, adjoint triples of functors and endomorphism rings of modules. In this vein, some applications to Grothendieck categories, (graded) module categories and comodule categories are deduced.

This article presents several results concerning the exactness of covariant and contravariant Hom functors and their derived functors in a balanced abelian category \( \mathscr{A} \). In particular: (i) The functors \( \mathrm{Hom}_{\mathscr{A}}(X,-) \) and \( \mathrm{Hom}_{\mathscr{A}}(-,X) \) are left exact, and become exact if and only if \( X \) is projective (resp. injective). (ii) The functors \( \mathrm{Hom}_{\mathrm{Comp}(\mathscr{A})}(X,-) \) and \( \mathrm{Hom}_{\mathrm{Comp}(\mathscr{A})}(-,X) \) on the category of complexes \( \mathrm{Comp}(\mathscr{A}) \) preserve this behavior. (iii) The homological functors \( \tilde{H}_n(X,-) \) and \( \tilde{H}_n(-,X) \) are constructed for all \( n \in \mathbb{Z} \). (iv) For projective \( X \), the connecting morphism \( \lambda_n: \tilde{H}_n(X,-)((T,\gamma)) \to \tilde{H}_{n+1}(X,-)((Y,\alpha)) \) allows \( \tilde{H}_n(X,-) \) to send short exact sequences in \( \mathrm{Comp}(\mathscr{A}) \) into long exact sequences in \( \mathrm{Ab} \). (v) Similarly, for injective \( X \), the morphism \( \delta_n: \tilde{H}_n(-,X)((Y,\alpha)) \to \tilde{H}_{n+1}(-,X)((T,\gamma)) \) shows that \( \tilde{H}_n(-,X) \) also preserves long exact sequences.

An Abelian category of cofinite modules

We develop a functorial approach to the study of n-abelian categories by reformulating their axioms in terms of their categories of finitely presented functors. Such an approach allows the use of classical homological algebra and representation theory techniques to understand higher homological algebra. As an application, we present two possible generalizations of the axioms “every monomorphism is a kernel” and “every epimorphism is a cokernel” of an abelian category to n-abelian categories. We also specialize our results to modules over rings, thereby describing when the category of finitely generated projective modules over a ring is n-abelian. Moreover, we establish a correspondence for n-abelian categories with additive generators, which extends the higher Auslander correspondence.

Hopf algebras, most generally in a semisimple abelian symmetric monoidal category, are here supposed to be commutative but not to be of finite-type, and their (equivariant) smoothness are discussed. Given a Hopf algebra H in a category such as above, it is proved that the following are equivalent: (i) H is smooth as an algebra; (ii) H is smooth as an H-comodule algebra; (iii) the product morphism S H 2 ( H + ) → H + defined on the 2nd symmetric power is monic. Working over a field k of characteristic zero, we prove: (a) every ordinary Hopf algebra, i.e., such in the category Vec of vector spaces, satisfies the equivalent conditions (i)–(iii) and some others; (b) every Hopf algebra in the category sVec of super-vector spaces has a certain property that is stronger than (i). In the case where char k = p > 0 , there are shown weaker properties of ordinary Hopf algebras and of Hopf algebras in sVec or in the ind-completion Ve r p ind of the Verlinde category.