Flat covers and cotorsion envelopes of sheaves

Abstract

In this paper we prove that any sheaf of modules over any topological space (in fact, any O \mathcal {O} -module where O \mathcal {O} is a sheaf of rings on the topological space) has a flat cover and a cotorsion envelope. This result is very useful, as we shall explain later in the introduction, in order to compute cohomology, due to the fact that the category of sheaves ( O \mathcal {O} -modules) does not have in general enough projectives.