When does the rational torsion split off for finitely generated modules.
Let \(C\) be a coalgebra with counit over a field \(k\). Let \(\text{Rat}(M)\) denote the rational part of a \(C^*\)-module, where \(C^*\) denotes the convolution algebra associated to \(C\). The splitting problem for commutative and noncommutative rings is a classical question. \textit{C. Năstăsescu} and the reviewer [J. Algebra 281, No. 1, 144-149 (2004; Zbl 1078.16047)] have shown that if all \(C^*\)-modules splits, i.e. \(\text{Rat}(M)\) is a direct summand of \(M\), then the coalgebra \(C\) is finite dimensional. In this paper the author considers the more general problem of characterizing coalgebras with the splitting property only for finitely generated modules. It is shown that if \(C\) is a co-local (infinite dimensional) coalgebra, then the coalgebra \(C\) has the left finitely generated Rat-splitting property if and only if \(C\) is chain coalgebra. Moreover, if the base field \(K\) is algebraically closed or the coalgebra \(C\) is pointed, this is equivalent to \(C\) being isomorphic to the divided power coalgebra. Some classes of examples of coalgebras with the splitting property are constructed.
scientific article; zbMATH DE number 5600713
scientific article