Prescribing endomorphism algebras. The cotorsion-free case (Q1120647)

This paper is one of a series of articles on the representation of endomorphism algebras, work which originated with \textit{A. L. S. Corner}'s celebrated paper [Proc. Lond. Math. Soc., III. Ser. 13, 687-710 (1963; Zbl 0116.024)]. Here, R is a commutative ring with identity, and S is a fixed multiplicatively closed countable subset of R containing 1 but no zerodivisors. If H is an R-module, topological notions like cotorsion- free, pure and dense are defined with respect to the S-topology on H which has the set \(\{\) sH\(|\) \(s\in S\}\) as a basis of neighborhoods. Given an R-algebra A, a faithful A-module H and an infinite cardinal \(\lambda\geq | H|\), the main result is the equivalence of the following statements: (a) H is a cotorsion-free R-module and A is a pure subalgebra of End H which is closed in the \(\aleph_ 1\)-topology of End H; and (b) there exists a cotorsion-free R-module G of cardinality \(2^{\aleph_ 0}\) such that End G is topologically isomorphic to A (under suitable topologies), and G contains a dense and pure A-submodule which is isomorphic to the direct sum of \(\lambda\) copies of H. Consequences include results contained in joint work of \textit{A. L. S. Corner} and \textit{R. Göbel} [ibid. 50, 447-479 (1985; Zbl 0562.20030)] and \textit{R. Göbel} and \textit{S. Shelah} [J. Algebra 93, 136-150 (1985; Zbl 0554.20018)]. The methods of proof use many combinatorical ideas dating back to work by Shelah, Corner and Göbel, and Göbel and Shelah including an extended ``Black Box'' Principle due to \textit{S. Shelah} [CISM Courses Lect. 287, 37-86 (1984; Zbl 0581.20052)].