Flat covers and cotorsion envelopes of sheaves (Q2781285)

After some rather technical assertions it is shown that if \(\mathcal O\) is a sheaf of rings on the topological space \(X\), then every \(\mathcal O\)-premodule has a flat cover (Theorem 2.6). The main result of the paper states that if \(\mathcal O\) is a sheaf of rings on a topological space \(X\), then every \(\mathcal O\)-module has a flat cover in the category of \(\mathcal O\)-modules. In fact, if \(F\to G\) is a flat cover of the \(\mathcal O\)-module \(G\) in the category of \(\mathcal O\)-premodules, then \(F\) is an \(\mathcal O\)-module and \(F\to G\) is a flat cover in the category of \(\mathcal O\)-modules (Theorem 2.7). Finally, if \(\mathcal O\) is a sheaf of rings on a topological space \(X\), then every \(\mathcal O\)-module has a cotorsion envelope in the category of \(\mathcal O\)-modules (Theorem 3.2).