WELL-CLOSED SUBSCHEMES OF NONCOMMUTATIVE SCHEMES.

Van den Bergh has defined the blowup of a noncommutative surface at a point lying on a commutative divisor [VdB]. 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 FZ 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 FZ 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. [ABSTRACT FROM AUTHOR]