Algebraic Geometry over Non-Algebraically Closed Fields -- A-Coherent Sheaves over a Ringed Space

TL;DR

This paper explores the properties and categorical equivalences of $A$-coherent sheaves over ringed spaces in algebraic geometry over non-algebraically closed fields, linking sheaves to modules over global sections.

Contribution

It establishes conditions under which $A$-coherent sheaves are equivalent to finitely presented modules over the global section ring, extending classical results to non-closed fields.

Findings

Proves categorical equivalence between $A$-coherent sheaves and finitely presented modules.

Demonstrates the faithful flatness of homomorphisms from Nash to analytic functions.

Utilizes Cartan's Theorem B to show vanishing higher cohomology.

Abstract

In this paper, we investigate the properties of $A$-coherent and $A$-quasi-coherent sheaves within the framework of algebraic geometry over non-algebraically closed fields. We define an $\mathcal{O}_X$-module to be $A$-coherent (resp. $A$-quasi-coherent) if it admits a global presentation by free modules of finite rank (resp. arbitrary rank) over a ringed space $X$. We establish a fundamental correspondence between these sheaves and modules over the ring of global sections $A = \Gamma(X,\mathcal{O}_X)$. Specifically, we prove that under conditions of flatness for the canonical morphism and the exactness of the global section functor, there exists an equivalence of categories between $A$-coherent $\mathcal{O}_X$-modules and finitely presented modules over $A$. We further demonstrate the utility of these results by proving the faithful flatness of the canonical homomorphisms from rings of Nash functions to rings of analytic functions, utilizing the vanishing of higher cohomology groups as guaranteed by Cartan's Theorem~B.