J. R. García Rozas

Flat Covers in the Category of Quasi-coherent Sheaves Over the Projective Line

Abstract In this paper we prove the existence of a flat cover and a cotorsion envelope for any quasi-coherent sheaf over the projective line , where R is any commutative ring. We first prove a general result that guarantees the existence of ℱ-covers and ℱ⊥-envelopes in the general setting of a Grothendieck category (not necessarily with enough projectives) provided that the class ℱ satisfies some “standard” conditions. This will generalize some results of the earlier work. [Aldrich, S. T., Enochs, E., García Rozas, J. R., Oyonarte, L. (2001). Covers and envelopes in Grothendieck categories. Flat covers of complexes with applications. J. Algebra 243:615–630].

Preserving and reflecting covers by functors. Applications to graded modules

Abstract We study C -covers in the context of Grothendieck categories. Namely, we analyse when a functor between two Grothendieck categories preserves or reflects C -covers. We apply our general study to the category of graded modules over a graded ring, by showing that relative injective covers with respect to a torsion theory are preserved and reflected, in some cases, among the categories R-gr, R1-Mod and R-Mod.

Exact and Semisimple Differential Graded Algebras

Abstract In this paper we provide a classification theorem and a structure theorem for exact differential graded algebras, and we use the classification theorem to show that a differential graded algebra A is semisimple (as a differential graded algebra) precisely when the graded algebra Z(A) is semisimple (as a graded algebra) and A is an exact complex. We also relate exact differential graded algebras with a graded version of Hochschild cohomology.

On the existence of covers by injective modules relative to a torsion theory

Let R be a ring with identity. In this note we study covers of left R-modules by r-injectives left R-modules, where r is a hereditary torsion theory defined in the category of all left R-modules and all R-morphisms. When R is an artinian commutative ring, a complete answer about the existence of such covers for every R-module is given. In case that T is a centrally splitting torsion theory, we can characterize those T for which every left R-module has a T-injective cover. Also we analyze R-modules such that the injective and the T-injective cover are the same. At the end of this note we relate the concepts of colocalization and cover

ON THE EXISTENCE OF FLAT COVERS IN R-gr

Recently, a proof of the existence of a flat cover of any module over an arbitrary associative ring with unit has been finally given (see 4-5). In this paper we prove the existence of flat covers in the category of graded modules over a graded ring. Some graded theoretical machinery is introduced to make the proof possible and new graded homological tools are developed.

Subgroups of the Brauer Group of a Cocommutative Coalgebra

We introduce Shur and projective Schur subgroup of the Brauer group of a cocommutative coalgebra by means of twisted cogroup coalgebras and we study their properties. In particular we show that these subgroups are always torsion (in contrast with the whole Brauer group). Moreover, when C is coreflexive and irreducible both subgroups coincide with the coradical ones. We illustrate the theory with several examples.

FINITELY GENERATED COTORSION MODULES

We describe the structure of nitely generated cotorsion modules over commutative noethe- rian rings. Also we characterize the so-called covering morphisms between nitely generated modules over these rings.

Divisorially graded coalgebras

Abstract In this paper, we classify torsion theories in the category of graded comodules over a graded coalgebra. Moreover, we give a structure theorem for divisorially graded coalgebras in terms of Picard groups.

Rugged modules: The opposite of flatness

ABSTRACT Relative notions of flatness are introduced as a mean to gauge the extent of the flatness of any given module. Every module is thus endowed with a flatness domain and, for every ring, the collection of flatness domains of all of its modules is a lattice with respect to class inclusion. This lattice, the flatness profile of the ring, allows us, in particular, to focus on modules which have a smallest flatness domain (namely, one consisting of all regular modules.) We establish that such modules exist over arbitrary rings and we call them Rugged Modules. Rings all of whose (cyclic) modules are rugged are shown to be precisely the von Neumann regular rings. We consider rings without a flatness middle class (i.e., rings for which modules must be either flat or rugged.) We obtain that, over a right Noetherian ring every left module is rugged or flat if and only if every right module is poor or injective if and only if R = S×T, where S is semisimple Artinian and T is either Morita equivalent to a right PCI-domain, or T is right Artinian whose Jacobson radical properly contains no nonzero ideals. Character modules serve to bridge results about flatness and injectivity profiles; in particular, connections between rugged and poor modules are explored. If R is a ring whose regular left modules are semisimple, then a right module M is rugged if and only if its character left module M+ is poor. Rugged Abelian groups are fully characterized and shown to coincide precisely with injectively poor and projectively poor Abelian groups. Also, in order to get a feel for the class of rugged modules over an arbitrary ring, we consider the homological ubiquity of rugged modules in the category of all modules in terms of the feasibility of rugged precovers and covers for arbitrary modules.

Relative Projective and Injective Dimensions

We study the concepts of the 𝒫C-projective and the ℐC-injective dimensions of a module in the noncommutative case, weakening the condition of C being semidualizing. We give the relations between these dimensions and the C-relative Gorenstein dimensions (GC-projective and GC-injective dimensions) of the module. Finally, we compare, in some circumstances, the global 𝒫C-projective dimension of a ring and the global dimension of the endomorphisms ring of C.