Purity and chains of modules

Abstract

In this thesis we relate the notions of purity and pure-injectivity in the category Mod RA[infinity, infinity] to the corresponding notions in Mod R. Where R is a von Neumann regular ring, we give a complete characterisation of pure monomorphisms of Mod RA[infinity, infinity] in terms of embeddings in Mod R, as well as a description of the indecomposable pure-injectives of Mod RA[infinity, infinity] in terms of those of Mod R. We also relate the Cantor-Bendixson rank of the Ziegler spectrum of Mod RA[infinity, infinity] to that of the Ziegler spectrum of Mod R, showing that if CB(Zg( R )) = a then CB(Zg( Mod RA[infinity, infinity] )) = a + 2. We generalise these results by defining, for any ring R, a definable subcategory FR(infinity) of Mod RA[infinity, infinity] that coincides with Mod RA[infinity, infinity] precisely when R is von Neumann regular, giving a complete characterisation of pure monomorphisms and indecomposable pure-injectives of FR(infinity) in terms of those of Mod R, and showing that if CB(Zg( R )) = a, then CB(Zg( FR(infinity) )) = a + 2. On the way to the above, we consider questions of purity, pure-injectivity, and the Ziegler spectrum in the context of the quasi-abelian (definable) subcategories Mono R and Epi R of Mod RA[infinity, infinity], as well as their (respective) definable subcategories MR and ER that coincide with Mono R, respectively Epi R, when R is von Neumann regular. This leads to descriptions of the absolutely pure and flat objects of these categories, as well as what we call the `strictly pure' and `strictly flat' objects.

Date of Award	1 Aug 2024
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Michael Prest (Supervisor) & Marcus Tressl (Supervisor)