Salce's problem on cotorsion pairs is undecidable. Issue 4 (18th April 2022)

• Record Type: Journal Article
• Authors: Cox, Sean
• Is Part Of: Bulletin of the London Mathematical Society. Volume 54:Issue 4(2022)
• Page Start: 1363
• Page End: 1374
• Publication Date: 2022-04-18
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12634
• Languages: English
• ISSNs: 0024-6093

Abstract

Abstract: In the 1970's, Salce introduced the notion of a cotorsion pair of classes of abelian groups, and asked whether every such pair is complete (that is, has enough injectives and projectives); we refer to this as Salce's Problem (for Ab ). We prove that it is consistent, relative to the consistency of Vopěnka's Principle (VP), that the answer is affirmative. Combined with a previous result of Eklof and Shelah , this shows that Salce's Problem for Ab, and in fact for R‐Mod when R $R$ is hereditary, is independent of the ZFC axioms (modulo the consistency of VP).