Subdirect Products of Preadditive Categories and Weak Equivalences

We study the notions of coproduct and subdirect product of preadditive categories and prove a Birkhoff type theorem showing that every skeletally small preadditive category is a subdirect product of subdirectly irreducible, skeletally small, preadditive categories. Moreover, we show that every direct-sum decomposition of the monoid of the isomorphism classes of objects of is weakly induced by a coproduct decomposition of the preadditive category . Partially supported by Gruppo Nazionale Strutture Algebriche e Geometriche e loro Applicazioni of Istituto Nazionale di Alta Matematica, Italy. This paper was written during a visit of the second author at the Dipartimento di Matematica Pura e Applicata (Università di Padova, Italy). He acknowledges the kind hospitality received.     

Rigid complexes via DG algebras

Let be a commutative ring, a commutative -algebra and a complex of -modules. We begin by constructing the square , which is also a complex of -modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism , then the pair is called a rigid complex over relative to (there are some finiteness conditions). There is an obvious notion of rigid morphism between rigid complexes.

We establish several properties of rigid complexes, including their uniqueness, existence (under some extra hypothesis), and formation of pullbacks (resp. ) along a finite (resp. essentially smooth) ring homomorphism .

In the subsequent paper, Rigid Dualizing Complexes over Commutative Rings, we consider rigid dualizing complexes over commutative rings, building on the results of the present paper. The project culminates in our paper Rigid Dualizing Complexes and Perverse Sheaves on Schemes, where we give a comprehensive version of Grothendieck duality for schemes.

The idea of rigid complexes originates in noncommutative algebraic geometry, and is due to Van den Bergh (1997).

     

Wedge Products and Cotensor Coalgebras in Monoidal Categories

The construction of the cotensor coalgebra for an “abelian monoidal” category which is also cocomplete, complete and AB5, was performed in Ardizzoni et al. (Comm Algebra 35(1):25–70, 2007). It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra E in is filled by considering a direct limit of a filtration consisting of wedge products of a subcoalgebra D of E. The main aim of this paper is to characterize hereditary coalgebras , where D is a coseparable coalgebra in , by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra if and only if it is a cotensor coalgebra , where N is a certain D-bicomodule in . Because of our choice, even when we apply our results in the category of vector spaces, new results are obtained. This paper was written while A. Ardizzoni was member of G.N.S.A.G.A. with partial financial support from Mi.U.R.     

Cohomology theories in triangulated categories

Let C be a triangulated category with a proper class E of triangles.We prove that there exists an Avramov–Martsinkovsky type exact sequence in C,which connects E-cohomology,E-Tate cohomology and E-Gorenstein cohomology.     

Hereditary triangular matrix comonads

We recognize Harada’s generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler proofs, some of them under more general hypothesis, besides with a characterization of the normal triangular matrix comonads that are hereditary, that is, of homological dimension less than or equal to 1. Our methods rest on a matrix representation of additive functors and natural transformations, which allows us to adapt typical algebraic manipulations from Linear Algebra to the additive categorical setting.     

A monoidal approach to splitting morphisms of bialgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra such that its Jacobson radical is a nilpotent Hopf ideal and is a semisimple algebra. We prove that the canonical projection of on has a section which is an -colinear algebra map. Furthermore, if is cosemisimple too, then we can choose this section to be an -bicolinear algebra morphism. This fact allows us to describe as a `generalized bosonization' of a certain algebra in the category of Yetter-Drinfeld modules over . As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of into which is an -linear coalgebra morphism. Furthermore, if is semisimple too, then we can choose this retraction to be an -bilinear coalgebra morphism. Then, also in this case, we can describe as a `generalized bosonization' of a certain coalgebra in the category of Yetter-Drinfeld modules over .

     

A recollement construction of Gorenstein derived categories

We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause’s work [Math. Ann., 2012, 353: 765–781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.     

On Some Categories and Functors in the Theory of Quaternionic Bergman Spaces

Basic facts are presented about the theory of quaternionic Bergman spaces with special emphasis on what is happening with them under conformal transformations of the domains. Constructing a series of categories of quaternion-valued functions as well as functors acting between them we show that the arising spaces and operators have conformally covariant or invariant characters in terms of the theory of categories. The second-named and the third-named authors were partially supported by CONACYT projects as well as by IPN in the framework of COFAA and SIP programs.     

The small quantum group as a quantum double

We prove that the quantum double of the quasi-Hopf algebra of dimension attached in [P. Etingof, S. Gelaki, On radically graded finite-dimensional quasi-Hopf algebras, Mosc. Math. J. 5 (2) (2005) 371–378] to a simple complex Lie algebra and a primitive root of unity q of order n² is equivalent to Lusztig's small quantum group (under some conditions on n). We also give a conceptual construction of using the notion of de-equivariantization of tensor categories.