Projective modules over rings of global sections of flasque sheaves
Let (X,\({\mathcal O})\) be a quasicompact ringed space for which the sheaf \({\mathcal O}\) is flasque and locally Serre. (A ring is Serre if all projective modules of finite type are free.) The present paper proves (corollary 1.3) that the global sections functor \(\Gamma\) gives an equivalence of categories between the category of locally free (left) \({\mathcal O}\)-modules of finite rank and the category of projective finitely generated (left) \(\Gamma\) (\({\mathcal O})\)-modules. A sufficient condition is given (corollary 2.2) for \(\Gamma\) (\({\mathcal O})\) to be Serre. From this the author concludes (theorem 3.1) that the face ring of a finite simplicial complex over an arbitrary field is Serre, thus reproving a result of \textit{T. Vorst} [Math. Z. 184, 425-433 (1983; Zbl 0504.13008)].
scientific article; zbMATH DE number 4163036