A local finiteness theorem for corings over semihereditary rings (Q292077)

Let \(R\) be an algebra over the commutative ring \(k\). The main result of this paper is the following: If \(C\) is an \(R\)-coring, \(R\) is right or left semihereditary, and \(C\) is projective as a left and right \(R\)-module, then \(C\) is locally finite (i.e., any finite subset of \(C\) is contained in a subcoring which is finitely generated as an \(R-R\)-bimodule). This generalizes several results: the fundamental theorem of coalgebras (a coalgebra over a field is locally finite), a result of \textit{M. Hazewinkel} [J. Pure Appl. Algebra 183, No. 1--3, 61--103 (2003; Zbl 1048.16022)] saying that a coalgebra over a principal ideal domain whose underlying module is free is locally finite, and an (unpublished) result of Bergman, mentioned on page 173 of [\textit{G. M. Bergman} and \textit{A. O. Hausknecht}, Cogroups and co-rings in categories of associative rings. Providence, RI: AMS, American Mathematical Society (1996; Zbl 0857.16001)], which says that corings over a semisimple Artinian ring are locally finite.