Category-theoretic reconstruction of schemes from categories of reduced schemes

Abstract

Let S be a locally Noetherian normal scheme and ⧫/S a set of properties of S-schemes. Then we shall write Sch_⧫/S for the full subcategory of the category of S-schemes Sch_/S determined by the objects X ∈ Sch_⧫/S that satisfy every property of ⧫/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_⧫/S. This result is analogous to a result of Mochizuki [5] and may be regarded as a partial generalization of a result of de Bruyn [6] in the case where S is a locally Noetherian normal scheme.