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 [Mzk04] and may be regarded as a partial generalization of a result of de Bruyn [vDdB19] in the case where S is a locally Noetherian normal scheme.