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A Nullstellensatz for triangulated categories


Authors: M. V. Bondarko and V. A. Sosnilo
Translated by: M. V. Bondarko
Original publication: Algebra i Analiz, tom 27 (2015), nomer 6.
Journal: St. Petersburg Math. J. 27 (2016), 889-898
MSC (2010): Primary 18E30
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 $ \ushort {C}$ and $ E\subset \mathrm {Obj} \ushort {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 $ \ushort {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: \ushort {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} \ushort {C}$ whenever the same is true for the images of $ Y$ and $ D$ in all the categories $ \ushort {C}_p$ obtained from $ \ushort {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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Additional Information

M. V. Bondarko
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia
Email: mbondarko@gmail.com

V. A. Sosnilo
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia
Email: vsosnilo@gmail.com

DOI: https://doi.org/10.1090/spmj/1425
Keywords: Triangulated categories, cohomological functors, separating functors, envelopes, localization of the coefficients
Received by editor(s): August 19, 2015
Published electronically: September 30, 2016
Additional Notes: The first author was supported by RFBR (grant no. 14-01-00393-a), by Dmitry Zimin’s Foundation “Dynasty”, and by the Scientific schools grant no. 3856.2014.1.
The second author was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under the RF Government grant 11.G34.31.0026, and also by the JSC “Gazprom Neft”. Both authors were supported by the RFBR grant no. 15-01-03034-a.
Dedicated: To Sergei Vladimirovich Vostokov with our best wishes
Article copyright: © Copyright 2016 American Mathematical Society