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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_Z 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_Z 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.
@article{TAC_2019_34_a13,
author = {D. Rogalski},
title = {Well-closed subschemes of noncommutative schemes},
journal = {Theory and applications of categories},
pages = {375--404},
publisher = {mathdoc},
volume = {34},
year = {2019},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2019_34_a13/}
}
D. Rogalski. Well-closed subschemes of noncommutative schemes. Theory and applications of categories, Tome 34 (2019), pp. 375-404. http://geodesic.mathdoc.fr/item/TAC_2019_34_a13/