Cointegrations, relative cohomology for comodules, and coseparable corings (Q1825261)

Let \(S\) and \(R\) be rings, \(K\) an \(S\)-coring, \(L\) an \(R\)-coring, and \(M\) and \(N\) K-L bicomodules. A (left) cointegration from \(N\) to \(M\) is an \(S\)-\(L\) map from \( N\) to \(K\otimes_ S M\) satisfying a condition extending that of a coderivation from \(N\) to \(K\) (when \(N\) is a \(K\)-\(K\) bicomodule). Using the existence of couniversal cointegrations and coderivations, the author shows that in the category of comodules over a coring, the relative functor Ext is the relative derived functor of cointegrations, i.e., \(H^ n(N,M)\cong \text{Ext}^{n+1}_{\text{( K-L,S- L)}}(N,M)\) for \(n\geq 1\). As a corollary, the relative cohomology of a coring is the derived functor of coderivations. The main result is dual to the result of \textit{M. Kleiner} [J. Pure Appl. Algebra 38, 71- 86 (1985; Zbl 0569.16027)] for integrations in the category of modules over a ring. As an application, the author obtains ten conditions, each of which is equivalent to \(K\) being coseparable (i.e., there is a \(K\)-\(K\) map from \(K\otimes K\) to \(K\) splitting the comultiplication \(\Delta\) of \(K\)). Three of them involve certain cointegrations or coderivations being inner, and one is that \(\text{Ext} ^ n_{\text{(K-L,S-L)}}=0\) for all \(L\) and all \(n\geq1\), which is dual to a characterization of finite-dimensional separable algebras over a field.