Relatively big projective modules, direct sum decompositions of modules, and representations of finite groups

Abstract

In this dissertation we study relatively big projective modules and their applications to direct sum decompositions. The first part addresses the class of direct sums of finitely generated torsion-free modules over a commutative domain, focusing on its closure under direct summands. For local domains of Krull dimension 1, we characterize this closure in terms of endomorphism rings of indecomposable modules, and extend to h-local domains by establishing necessary and sufficient conditions involving two-generated ideals and integral closure. In the second part we show that in broad classes of rings ---including right noetherian PI-rings, semiperfect rings, and suitable locally semiperfect torsion-free algebras over h-local domains--- every countably generated projective module is relatively big. Going back to applications we include structural results for endomorphism rings of torsion-free modules and conditions ensuring decompositions into finitely generated summands. Finally, we extend the analysis to locally semiperfect algebras with semisimple artinian rings of quotients. This leads to precise criteria for the existence and classification of projective modules via dimension sequences and decomposition matrices, obtaining representation-theoretic descriptions of projectives. The main examples of this part are group algebras of finite groups over a suitable Dedekind domain.

Date of Award	9 Feb 2026
Original language	English
Awarding Institution	Universitat Autònoma de Barcelona (UAB)
Supervisor	Dolors Herbera Espinal (Director) & Pavel Prihoda (Director)