A Nullstellensatz for triangulated categories

Abstract

The main goal of this paper is to prove the following: for a triangulated category $ \underline{C}$ and $E\subset \operatorname{Obj} \underline{C}$ 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 $ \underline{C}$ being an $R$-linear category (where $R$ is a commutative ring) this is also equivalent to the existence of an $R$-linear $F: \underline{C}^{op}\to R-\operatorname{mod}$ satisfying this property. As a corollary, we prove that an object $Y$ belongs to the corresponding "envelope" of some $D\subset \operatorname{Obj} \underline{C}$ whenever the same is true for the images of $Y$ and $D$ in all the categories $ \underline{C}_p$ obtained from $ \underline{C}$ by means of "localizing the coefficients" at maximal ideals $p\triangleleft R$. 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.