#
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.

http://www.tac.mta.ca/tac/volumes/34/14/34-14.pdf

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