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

MathSciNet review:
3589221

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Abstract |
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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
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References
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*Homological and homotopical aspects of torsion theories*, Mem. Amer. Math. Soc. **188** (2007), no. 883. MR **2327478**
- M. V. Bondarko,
*$\mathbb {Z}[\frac {1}{p}]$-motivic resolution of singularities*, Compos. Math. **147** (2011), no. 5, 1434–1446. MR **2834727**
- —,
*Gersten weight structures for motivic homotopy categories; direct summands of cohomology of function fields and coniveau spectral sequences*, Preprint, 2013, arXiv:1312.7493.
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*Étale motives*, Comp. Math. **152** (2016), no. 3, 556–666. MR **3477640**
- S. Kelly,
*Triangulated categories of motives in positive characteristic*, Dissertation, 2012, arXiv: 1305.5349.
- H. 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**
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*The additivity of traces in triangulated categories*, Adv. Math. **163** (2001), no. 1, 34–73. MR **1867203**
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*A note on compactly generated co-t-structures*, Comm. Algebra **40** (2012), no. 2, 386–394. MR **2889469**
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*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**
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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

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