A NULLSTELLENSATZ FOR TRIANGULATED CATEGORIES

作者：Bondarko M V^*; Sosnilo V A

来源：St Petersburg Mathematical Journal, 2016, 27(6): 889-898.

DOI：10.1090/spmj/1425

• Triangulated categories
• cohomological functors
• separating functors
• envelopes
• localization of the coefficients

The paper is aimed at proving the following: for a triangulated category (C) under bar and E subset of Obj (C) under bar, there exists a cohomological functor F (with values in some Abelian category) such that E is its set of zeros if (and only if) E is closed with respect to retracts and extensions (so, a certain Nullstellensatz is obtained for functors of this type). Moreover, if (C) under bar is an R-linear category (where R is a commutative ring), this is also equivalent to the existence of an R-linear functor F : (C) under bar (oop) -> R-mod with this property. As a corollary, it is proved that an object Y belongs to the corresponding "envelope" of some D subset of Obj (C) under bar whenever the same is true for the images of Y and D in all the categories (C) under barp obtained from (C) under bar via "localizing the coefficients" at maximal ideals p sic R. Moreover, certain new methods are developed for relating triangulated categories to their (nonfull) countable triangulated subcategories.
The results of this paper can be applied to weight structures and triangulated categories of motives.

• 出版日期2016-12