Notes on formal smoothness

The definition of an S-category is proposed by weakening the axioms of a Q-category introduced by Kontsevich and Rosenberg. Examples of Q- and S-categories and (co)smooth objects in such categories are given.Comment: 11 page

Towers of corings

The notion of a Frobenius coring is introduced, and it is shown that any such coring produces a tower of Frobenius corings and Frobenius extensions. This establishes a one-to-one correspondence between Frobenius corings and extensions.Comment: 8 pages, LaTeX; uses xypic; Prop 2.9 replaced by a remar

On modules associated to coalgebra Galois extensions

For a given entwining structure $(A,C)_\psi$ involving an algebra $A$, a coalgebra $C$, and an entwining map $\psi: C\otimes A\to A\otimes C$, a category \M_A^C(\psi) of right $(A,C)_\psi$-modules is defined and its structure analysed. In particular, the notion of a measuring of $(A,C)_\psi$ to (\tA,\tC)_\tpsi is introduced, and certain functors between \M_A^C(\psi) and \M_\tA^\tC(\tpsi) induced by such a measuring are defined. It is shown that these functors are inverse equivalences iff they are exact (or one of them faithfully exact) and the measuring satisfies a certain Galois-type condition. Next, left modules $E$ and right modules $\bar{E}$ associated to a $C$-Galois extension $A$ of $B$ are defined. These can be thought of as objects dual to fibre bundles with coalgebra $C$ in the place of a structure group, and a fibre $V$. Cross-sections of such associated modules are defined as module maps $E\to B$ or $\bar{E}\to B$. It is shown that they can be identified with suitably equivariant maps from the fibre to $A$. Also, it is shown that a $C$-Galois extension is cleft if and only if A=B\tens C as left $B$-modules and right $C$-comodules. The relationship between the modules $E$ and $\bar{E}$ is studied in the case when $V$ is finite-dimensional and in the case when the canonical entwining map is bijective.Comment: 31 pages, LaTeX, uses amscd and amssymb. Some changes in Section 3. Version to appear in J. Algebr

Deformation of Algebra Factorisations

A general deformation theory of algebras which factorise into two subalgebras is studied. It is shown that the classification of deformations is related to the cohomology of a certain double complex reminiscent of the Gerstenhaber-Schack complex of a bialgebra.Comment: 12 pages, LaTeX, uses amscd; proof of Theorem 3.1 corrected, new examples and references added; final version to appear in Commun. Algebr