Relative homological algebra in the category of quasi-coherent sheaves

Abstract

In this paper, we prove the existence of a flat cover and of a cotorsion envelope for any quasi-coherent sheaf over a scheme (X, OX). Indeed we prove something more general. We define what it is understood by the category of quasi-coherent R-modules, where R is a representation by rings of a quiver Q, and we prove the existence of a flat cover and a cotorsion envelope for quasi-coherent R-modules. Then we use the fact that the category of quasi-coherent sheaves on (X, OX is equivalent to the category of quasi-coherent R-modules for some Q and R to get our result. © 2004 Elsevier Inc. All rights reserved.