Category-Theoretic Reconstruction of Schemes from Categories of Reduced Schemes

TL;DR

This paper develops a category-theoretic method to reconstruct a locally Noetherian normal scheme from categories of its reduced schemes with certain properties, generalizing previous results.

Contribution

It introduces a functorial algorithm to recover the scheme from categories of reduced schemes satisfying specific properties, extending prior work by Mochizuki and de Bruyn.

Findings

Reconstruction of schemes from categories of reduced schemes.

Extension of Mochizuki's and de Bruyn's results.

Applicable to locally Noetherian normal schemes.

Abstract

Let $S$ be a locally Noetherian normal scheme and $\blacklozenge/S$ a set of properties of $S$-schemes. Then we shall write Sch$_{\blacklozenge/S}$ for the full subcategory of the category of $S$-schemes Sch$_{/S}$ determined by the objects $X\in {\rm Sch}_{\blacklozenge/S}$ that satisfy every property of $\blacklozenge/S$. In the present paper, we shall mainly be concerned with the properties "reduced", "quasi-compact over $S$", "quasi-separated over $S$", and "separated over $S$". We give a functorial category-theoretic algorithm for reconstructing $S$ from the intrinsic structure of the abstract category Sch$_{\blacklozenge/S}$. This result is analogous to a result of Mochizuki \cite{Mzk04} and may be regarded as a partial generalization of a result of de Bruyn \cite{deBr19} in the case where $S$ is a locally Noetherian normal scheme.