## A Nullstellensatz for triangulated categories

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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.

## References

- Apostolos Beligiannis and Idun Reiten,
*Homological and homotopical aspects of torsion theories*, Mem. Amer. Math. Soc.**188**(2007), no. 883, viii+207. MR**2327478**, DOI 10.1090/memo/0883 - M. V. Bondarko,
*$\Bbb Z[1/p]$-motivic resolution of singularities*, Compos. Math.**147**(2011), no. 5, 1434–1446. MR**2834727**, DOI 10.1112/S0010437X11005410 - —,
*Gersten weight structures for motivic homotopy categories; direct summands of cohomology of function fields and coniveau spectral sequences*, Preprint, 2013, arXiv:1312.7493. - Denis-Charles Cisinski and Frédéric Déglise,
*Étale motives*, Compos. Math.**152**(2016), no. 3, 556–666. MR**3477640**, DOI 10.1112/S0010437X15007459 - S. Kelly,
*Triangulated categories of motives in positive characteristic*, Dissertation, 2012, arXiv: 1305.5349. - Henning Krause,
*Localization theory for triangulated categories*, Triangulated categories, London Math. Soc. Lecture Note Ser., vol. 375, Cambridge Univ. Press, Cambridge, 2010, pp. 161–235. MR**2681709**, DOI 10.1017/CBO9781139107075.005 - J. P. May,
*The additivity of traces in triangulated categories*, Adv. Math.**163**(2001), no. 1, 34–73. MR**1867203**, DOI 10.1006/aima.2001.1995 - David Pauksztello,
*A note on compactly generated co-$t$-structures*, Comm. Algebra**40**(2012), no. 2, 386–394. MR**2889469**, DOI 10.1080/00927872.2010.528714 - Jan Šťovíček and David Pospíšil,
*On compactly generated torsion pairs and the classification of co-$t$-structures for commutative noetherian rings*, Trans. Amer. Math. Soc.**368**(2016), no. 9, 6325–6361. MR**3461036**, DOI 10.1090/tran/6561 - Leovigildo Alonso Tarrío, Ana Jeremías López, and María José Souto Salorio,
*Construction of $t$-structures and equivalences of derived categories*, Trans. Amer. Math. Soc.**355**(2003), no. 6, 2523–2543. MR**1974001**, DOI 10.1090/S0002-9947-03-03261-6

## Bibliographic 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
- 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. - © Copyright 2016 American Mathematical Society
- Journal: St. Petersburg Math. J.
**27**(2016), 889-898 - MSC (2010): Primary 18E30
- DOI: https://doi.org/10.1090/spmj/1425
- MathSciNet review: 3589221

Dedicated: To Sergei Vladimirovich Vostokov with our best wishes