从约化概形范畴出发的概形的范畴论重构
Category-Theoretic Reconstruction of Schemes from Categories of Reduced Schemes
AI总结：
本文研究由满足约化及相对拟紧、拟分离、分离等性质的 $S$-概形所成抽象范畴，提出函子性范畴论算法以从其内在结构重构局部 Noether 正规概形 $S$。
摘要：
设 $S$ 为局部 Noether 正规概形，且 $\blacklozenge/S$ 为 $S$-概形的一组性质。于是我们将 Sch$_{\blacklozenge/S}$ 记为 $S$-概形范畴 Sch$_{/S}$ 的全子范畴，它由满足 $\blacklozenge/S$ 中每一项性质的对象 $X\in{\rm Sch}_{\blacklozenge/S}$ 所确定。在本文中，我们主要关注“约化”“在 $S$ 上拟紧”“在 $S$ 上拟分离”以及“在 $S$ 上分离”这些性质。我们给出了一种函子性的范畴论算法，用于从抽象范畴 Sch$_{\blacklozenge/S}$ 的内在结构重构 $S$。该结果与 Mochizuki 的结果类似，并且可被视为 de Bruyn 的结果在 $S$ 为局部 Noether 正规概形情形下的部分推广。
英文摘要：
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.
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