An Alternative Model for Coherent Sheaves over Noetherian Schemes

TL;DR

This paper presents a new way to describe the category of coherent sheaves over noetherian schemes using the scheme's underlying poset, enabling a better understanding of their structure through order-theoretic methods.

Contribution

The paper introduces an equivalent description of the category of coherent sheaves over noetherian schemes via poset-based presheaves, extending the topological reconstruction to sheaf categories.

Findings

Established an equivalence between coherent sheaves and presheaves over the poset

Constructed an analogous structure sheaf in the poset framework

Demonstrated the functor's full faithfulness and characterized its essential image

Abstract

The category of coherent sheaves over a noetherian scheme is very important for studying the properties of a given scheme. For noetherian schemes it is a well-known fact that the topology can be fully recovered from the corresponding partially ordered set of its points together with the specializaion partial order. Here we show that the same thing holds for the category of coherent sheaves over a noetherian scheme, i.e. there exists an equivalent description of this category in the "language" of this poset. First we build the equivalent notion of the structure sheaf, then we introduce the desired functor from the category of coherent sheaves to a certain category of presheaves over it, show this functor is fully faithful, describe its essential image and hence find an equivalent category for the category of coherent sheaves over a noetherian scheme. This paper is based on the author's master's thesis.