Curved differential graded algebras and corings

A relationship between curved differential algebras and corings is established and explored. In particular it is shown that the category of semi-free curved differential graded algebras is equivalent to the category of corings with surjective counits. Under this equivalence, comodules over a coring correspond to integrable connections or quasi-cohesive curved modules, while contramodules over a coring correspond to a specific class of curved modules introduced and termed Z-divergences in here.Comment: 25 page

Morita theory of comodules over corings

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule $\Sigma$ of an $A$-coring \cC. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring \cC or the comodule $\Sigma$ is finitely generated and projective as an $A$-module. That is, we obtain relations between the category of \cC-comodules and the category of firm modules for a firm ring $R$, which is an ideal of the endomorphism algebra ^\cC(\Sigma). For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.Comment: LaTeX, 35 pages. v2: Minor changes including the title, examples added in Section