A Nullstellensatz for triangulated categories

TL;DR

This paper establishes a Nullstellensatz-like result for triangulated categories, characterizing when a subset of objects can be zeros of a cohomological functor, with implications for motives and weight structures.

Contribution

It proves a Nullstellensatz for cohomological functors in triangulated categories, linking object subsets to functor zeros and developing new methods relating categories to subcategories.

Findings

Existence of cohomological functors with prescribed zeros characterized by closure properties.

Equivalence of conditions for $R$-linear categories involving $R$-linear functors.

Application to objects belonging to envelopes via localizations at maximal ideals.

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.