Projective modules over rings of global sections of flasque sheaves

We consider a quasi-compact ringed space (X,) for which the sheaf is flasque and locally Serre. We prove that for such a space the functor of global sections Γ defines an equivalence of the category of locally free left -modules of finite rank and the category of projective finitely generated Γ()-modules. Then we give a condition on an open covering U of X sufficient for every -module which is free on U to be free. If U satisfies this condition and besides is flaque and Serre on U then it follows that Γ() is also Serre. As an application we obtain a proof of Vorst's theorem that the face ring of an arbitrary finite complex is Serre.