A Nullstellensatz for triangulated categories
Bondarko, Mikhail V. · Sosnilo, Vladimir A.
Original · EN
The main goal of this paper is to prove the following: for a triangulated category C and E⊂ Obj C 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, we obtain a certain Nullstellensatz for functors of this type). Moreover, for C being an R-linear category (where R is a commutative ring) this is also equivalent to the existence of an R-linear F: Cop→ R-mod satisfying this property. As a corollary, we prove that an object Y belongs to the corresponding "envelope" of some D⊂ Obj C whenever the same is true for the images of Y and D in all the categories Cₚ obtained from C by means of "localizing the coefficients" at maximal ideals p R. Moreover, to prove our theorem we develop certain new methods for relating triangulated categories to their (non-full) countable triangulated subcategories. The results of this paper can be applied to the study of weight structures and of triangulated categories of motives.
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