Well-closed subschemes of noncommutative schemes

TL;DR

This paper explores the concept of well-closed subcategories in noncommutative geometry, providing characterizations and applications to points in quasi-schemes, with the aim of defining noncommutative blowups.

Contribution

It characterizes well-closed subcategories via projective effacements and describes the associated functor explicitly, advancing the theory of noncommutative blowups.

Findings

Well-closedness characterized by projective effacements

Explicit description of the functor F in terms of effacements

Closed points are well-closed in general quasi-schemes

Abstract

Van den Bergh has defined the blowup of a noncommutative surface at a point lying on a commutative divisor. We study one aspect of the construction, with an eventual aim of defining more general kinds of noncommutative blowups. Our basic object of study is a quasi-scheme X (a Grothendieck category). Given a closed subcategory Z, in order to define a blowup of X along Z one first needs to have a functor F which is an analog of tensoring with the defining ideal of Z. Following Van den Bergh, a closed subcategory Z which has such a functor is called well-closed. We show that well-closedness can be characterized by the existence of certain projective effacements for each object of X, and that the needed functor F has an explicit description in terms of such effacements. As an application, we prove that closed points are well-closed in quite general quasi-schemes.