Viewing Quasi-Coherent Sheaves of Ideals as Ideals of a Ring

TL;DR

This paper develops a method to interpret quasi-coherent sheaves of ideals on blowups as regular ideals within a constructed ring, linking geometric models with algebraic ideals through scheme morphisms.

Contribution

It introduces a construction of a ring D* from Nagata function rings that captures the ideal structure of projective models, bridging geometric and algebraic perspectives.

Findings

Constructed a ring D* containing the projective model

Characterized relevant ideals as regular ideals of D*

Established a faithfully flat scheme morphism

Abstract

This paper presents a technique for viewing quasi-coherent sheaves of ideals of a given blowup as regular ideals of a ring. In the paper, we first describe (Zariski) models as integral schemes that are separated and of finite type over an integral domain $D$. We then construct a ring $D^*$ for a given projective model (e.g. blowup of $D$ over a finitely generated ideal) by intersecting Nagata function rings. The spectrum of $D^*$ contains the projective model, but similar to the Proj-construction, it includes additional prime ideals. We characterize the relevant ideals and construct a faithfully flat morphism of schemes from the spectrum of $D^*$ to the model. Finally, using Abhyankar's definition of ideals on models, we identify the relevant ideals of $D^*$ with the quasi-coherent sheaves of ideals of the corresponding projective model.