Cotorsion pairs and adjoint functors in the homotopy category of N-complexes

In this paper, we first construct some complete cotorsion pairs on the category [Formula: see text] of unbounded [Formula: see text]-complexes of Grothendieck category [Formula: see text], from two given cotorsion pairs in [Formula: see text]. Next, as an application, we focus on particular homotopy categories and the existence of adjoint functors between them. These are an [Formula: see text]-complex version of the results that were shown by Neeman in the category of ordinary complexes.

Local cohomology in Grothendieck categories

Let [Formula: see text] be a locally noetherian Grothendieck category. In this paper, we define and study the section functor on [Formula: see text] with respect to an open subset of [Formula: see text]. Next, we define and study local cohomology theory in [Formula: see text] in terms of the section functors. Finally, we study abstract local cohomology functor on the derived category [Formula: see text].

Pure exact structures and the pure derived category of a scheme

Let$\mathcal{C}$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.

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 OSOFSKY–SMITH THEOREM

We recall a version of the Osofsky–Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which characterizes semi-simple rings as those rings whose every cyclic module is injective.

Cohomology for Bicomodules. Separable and Maschke functors

We introduce the category of bicomodules for a comonad on a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups using cointegrations into bicomodules. We present two applications: the characterization of coseparable corings stated in [14], and the characterization of coseparable coalgebra coextensions stated in [19].