Note on the Serre-Swan Theorem

TL;DR

This paper identifies specific ringed spaces where locally free sheaves correspond to finitely generated projective modules, extending the Serre-Swan theorem to new contexts.

Contribution

It generalizes the Serre-Swan theorem by characterizing classes of ringed spaces with this categorical equivalence.

Findings

Established a class of ringed spaces satisfying the theorem

Derived Serre-Swan theorems for affine schemes and differentiable manifolds

Extended the theorem to Stein spaces

Abstract

We determine a class of ringed space X, for which the category of locally free sheaves of bounded rank is equivalent to the category of finitely generated projective A(X)-modules, where A(X) denote the ring of global sections of X. The well-known Serre-Swan theorems for affine schemes, differentiable manifolds, Stein spaces, etc., are then derived.