Verdier quotients of homotopy categories of rings and Gorenstein-projective precovers

2023/09/20 by Cortés-Izurdiaga, Manuel
#16E05 #16E65 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2309.11209

Abstract

Let R be a ring, \textrmProj be the class of all projective right R-modules, \mathcal K be the full subcategory of the homotopy category \mathbf K(\textrmProj) whose class of objects consists of all totally acyclic complexes, and \textrmMor_{\mathcal K} be the class of all morphisms in \mathbf K(\textrmProj) whose cones belong to \mathcal K. We prove that if \mathbf K(\textrmProj) has enough \textrmMor_{\mathcal K}-injective objects, then the Verdier quotient \mathbf K(\textrmProj)/\mathcal K has small Hom-sets, and this last condition implies the existence of Gorenstein-projective precovers in \textrmMod-R and of totally acyclic precovers in \mathbf C(\textrmMod-R).

Related

• Verdier quotients of homotopy categories
• Homotopy category of projective complexes and complexes of Gorenstein projective modules
• Acyclicity versus total acyclicity for complexes over noetherian rings
• The Gorenstein defect category
• Maximum deconstructibility in module categories