On matrix type corings, algebra coverings and Cech cohomology

The Brauer group of Azumaya corings and the second cohomology group

Let $R$ be a commutative ring. An Azumaya coring consists of a couple $(S,\Cc)$, with $S$ a faithfully flat commutative $R$-algebra, and an $S$-coring $\Cc$ satisfying certain properties. If $S$ is faithfully projective, then the dual of $\Cc$ is an Azumaya algebra. Equivalence classes of Azumaya corings form an abelian group, called the Brauer group of Azumaya corings. This group is canonically isomorphic to the second flat cohomology group. We also give algebraic interpretations of the second Amitsur cohomology group and the first Villamayor-Zelinsky cohomology group in terms of corings.

Descent cohomology and corings

A coring approach to non-Abelian descent cohomology of [P Nuss and M Wambst, Non-Abelian Hopf cohomology, Preprint arXiv:math.KT/0511712, (2005)] is described and a definition of a Galois cohomology for partial group actions is proposed.

Corings、它们的对偶环与相对（上）Hochschild上同调

Corings, their dual rings and relative (co)Hochschild cohomology

对于作为左模有限生成投射的coring，本文证明其Cartier上同调同构于右代数的相对Hochschild上同调，该同构可提升至$B_{\infty}$-代数层面，并应用于entwining structures的等变上同调描述。

Characterization of Contextuality with Semi-Module Čech Cohomology and its Relation with Cohomology of Effect Algebras

I present a generalized notion of obstruction in Čech cohomology on semi-modules, which allows one to characterize non-disturbing contextual behaviors with any semi-field. This framework generalizes the usual Čech cohomology used in the sheaf approach to contextuality, avoiding false positives. I revise a similar work done in the framework of effect algebras with cyclic and order cohomologies and explore the relationship with the one presented here.

The relative Picard group of a comodule algebra and Harrison cohomology

Let $A$ be a commutative comodule algebra over a commutative bialgebra $H$. The group of invertible relative Hopf modules maps to the Picard group of $A$, and the kernel is described as a quotient group of the group of invertible grouplike elements of the coring $A\ot H$, or as a Harrison cohomology group. Our methods are based on elementary $K$-theory. The Hilbert 90 Theorem follows as a corollary. The part of the Picard group of the coinvariants that becomes trivial after base extension embeds in the Harrison cohomology group, and the image is contained in a well-defined subgroup $E$. It equ…

Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures

We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (an…