A Nullstellensatz for triangulated categories

Bondarko, M.; Sosnilo, V.

doi:10.1090/spmj/1425

A Nullstellensatz for triangulated categories
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by M. V. Bondarko and V. A. Sosnilo
Translated by: M. V. Bondarko

St. Petersburg Math. J. 27 (2016), 889-898

DOI: https://doi.org/10.1090/spmj/1425

Published electronically: September 30, 2016

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Abstract:

The paper is aimed at proving the following: for a triangulated category $\underline {C}$ and $E\subset \mathrm {Obj} \underline {C}$, 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, a certain Nullstellensatz is obtained for functors of this type). Moreover, if $\underline {C}$ is an $R$-linear category (where $R$ is a commutative ring), this is also equivalent to the existence of an $R$-linear functor $F: \underline {C}^{\mathrm {oop}}\to R-\bmod$ with this property. As a corollary, it is proved that an object $Y$ belongs to the corresponding “envelope” of some $D\subset \mathrm {Obj} \underline {C}$ whenever the same is true for the images of $Y$ and $D$ in all the categories $\underline {C}_p$ obtained from $\underline {C}$ via “localizing the coefficients” at maximal ideals $p \triangleleft R$. 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.

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