Cotorsion theories and splitters

Abstract

Abstract. Let R be a subring of the rationals. We want to investigate self splitting R-modules G (that is Ext R (G, G) = 0). Following Schultz, we call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples of splitters. Are there others? Answering an open problem posed by Schultz, we will show that there are more splitters, in fact we are able to prescribe their endomorphism R-algebras with a free R-module structure. As a by-product we are able to solve a problem of Salce, showing that all rational cotorsion theories have enough injectives and enough projectives. This is also basic for answering the flat-cover-conjecture.

Complete Cotorsion Pairs in Exact Categories

“…One fundamental result on completeness of cotorsion pairs is due to Eklof and Trlifaj, they proved that any cotorsion pair cogenerated by a set of modules is complete [5], which is a generalization of the corresponding result of Göbel and Shelah on abelian groups in [7]. Hovey extended this result to Grothendieck categories [12], and later, Saorín and Šť ovíček generalized it further to their efficient exact categories [16].…”

Commuting Properties of Ext

“…We introduced these groups in [20], pointing out the obvious facts that free groups and cotorsion groups are splitters, and posed the question whether these classes are the only examples. This question was answered immediately in the negative by Göbel and Shelah [13], who by an elaborate construction using generators and relations found large classes of counterexamples.…”

Philip Hall's problem on non-Abelian splitters

“…An arbitrary group L is cotorsion-free if all its abelian subgroups are cotorsion-free. Other splitters have only recently been constructed (see [10,11]). They were also fundamental for solving the flat cover conjecture for modules.…”