Local units versus local projectivity. Dualisations: Corings with local structure maps

The structure of corings: Induction functors, Maschke-type theorem, and Frobenius and Galois-type properties

Given a ring $A$ and an $A$-coring $\cC$ we study when the forgetful functor from the category of right $\cC$-comodules to the category of right $A$-modules and its right adjoint $-\otimes_A\cC$ are separable. We then proceed to study when the induction functor $-\otimes_A\cC$ is also the left adjoint of the forgetful functor. This question is closely related to the problem when $A\to {}_A{\rm Hom}(\cC,A)$ is a Frobenius extension. We introduce the notion of a Galois coring and analyse when the tensor functor over the subring of $A$ fixed under the coaction of $\cC$ is an equivalence. We also …

The structure of corings with a grouplike element

Characteristic properties of corings with a grouplike element are analysed. Associated differential graded rings are studied. A correspondence between categories of comodules and flat connections is established. A generalisation of the Cuntz-Quillen theorem relating existence of connections in a module to projectivity of this module is proven.

Bimodule herds

The notion of a bimodule herd is introduced and studied. A bimodule herd consists of a $B$-$A$ bimodule, its formal dual, called a pen, and a map, called a shepherd, which satisfies untiality and coassociativity conditions. It is shown that every bimodule herd gives rise to a pair of corings and coactions. If, in addition, a bimodule herd is tame i.e. it is faithfully flat and a progenerator, then these corings are associated to entwining structures; the bimodule herd is a Galois comodule of these corings. The notion of a bicomodule coherd is introduced as a formal dualisation of the definitio…

有限生成$(H, {\rm C})$-余模的自同态环上的投射性与平坦性

Projectivity and flatness over the endomorphism ring of a finitely generated $(H, {\mathcal C})$-comodule

设k为交换环，H为k上双代数，C为(H,A)-余环，若C作为左A-模平坦，有限生成的右(H,C)-余模Λ对应的自同态环上的模具有投射性与平坦性的充要条件被给出；当C含固定H-群样元时，可将Λ替换为A。

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…