Category-Theoretic Reconstruction of Schemes from Categories of Reduced 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.