Well-closed subschemes of noncommutative schemes

D. Rogalski

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.

Keywords: Grothendieck category, noncommutative blowing up, adjoint functors, locally noetherian, closed subcategory

2010 MSC: 18E15, 18A40, 14A22

Theory and Applications of Categories, Vol. 34, 2019, No. 14, pp 375-404.

Published 2019-04-26.


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