Abstract
We study triangulated categories and torsion theories in them, and compare two definitions of torsion theories in this work. The most important types of torsion theories—weight structures and t-structures (and admissible triangulated subcategories) are also been considered. One of the aims of this paper is to show that a number of basic definitions and properties of weight and t-structures naturally extend to arbitrary torsion theories (in particular, we define smashing and cosmashing torsion theories). This can optimize some of the proofs. Similarly, the definitions of orthogonal and adjacent weight and t-structures are generalized. We relate the adjacency of torsion theories with Brown-Comenetz duality and Serre functors. These results may be applied to the study of t-structures in compactly generated triangulated categories and in derived categories of coherent sheaves. The relationship between torsion theories and projective classes is described.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
A. Beilinson, J. Bernstein, and P. Deligne, “Faisceaux pervers,” Asterisque 100, 5–171 (1982).
M. Bondarko, “Weight structures vs. t-structures; Weight filtrations, spectral sequences, and complexes (for motives and in general),” J. K-Theory 6, 387–504 (2010). http://arxiv.org/abs/0704.4003. Accessed October 27, 2018.
D. Pauksztello, “Compact cochain objects in triangulated categories and co-t-structures,” Cent. Eur. J. Math. 6, 25–42 (2008).
M. V. Bondarko and V. A. Sosnilo, “On constructing weight structures and extending them to idempotent extensions,” Homol., Homotopy Appl. 20, 37–57 (2018).
O. Iyama and Y. Yoshino, “Mutation in triangulated categories and rigid Cohen-Macaulay modules,” Invent. Math. 172, 117–168 (2008).
M. V. Bondarko, “On morphisms killing weights, weight complexes, and Eilenberg—Maclane (co)homology of spectra,” Preprint (2015). http://arxiv.org/abs/1509.08453. Accessed October 27, 2018.
A. I. Bondal and M. M. Kapranov, “Representable functors, Serre functors, and mutations,” Math. USSR-Izv. 35, 519–541 (1990).
J. Christensen, “Ideals in triangulated categories: Phantoms, ghosts and skeleta,” Adv. Math. 136, 284–339 (1998).
A. Neeman, Triangulated Categories (Princeton Univ. Press, Princeton, NJ, 2001), in Ser: Annals of Mathematics Studies, Vol. 148.
A. Beligiannis, “Relative homology, higher cluster-tilting theory and categorified Auslander-Iyama correspondence,” J. Algebra 444, 367–503 (2015).
M. V. Bondarko, “Motivically functorial coniveau spectral sequences; direct summands of cohomology of function fields,” Doc. Math., extra volume: Andrei Suslin’s Sixtieth Birthday, 33–117 (2010). http://arxiv.org/abs/0812.2672. Accessed October 27, 2018.
M. V. Bondarko, “Gersten weight structures for motivic homotopy categories; Retracts of cohomology of function fields, motivic dimensions, and coniveau spectral sequences,” Preprint (2018). https://arxiv.org/abs/1803.01432. Accessed October 27, 2018.
D. Pospíšil and J. Št’ovíček, “On compactly generated torsion pairs and the classification of co-t-structures for commutative Noetherian rings,” Trans. Am. Math. Soc. 368, 6325–6361 (2016).
M. Saorín and J. Št’ovícek, “On exact categories and applications to triangulated adjoints and model structures,” Adv. Math. 228, 968–1007 (2011).
M. V. Bondarko and V. A. Sosnilo, “On purely generated α-smashing weight structures and weightexact localizations,” Preprint (2017). http://arxiv.org/abs/1712.00850. Accessed October 27, 2018.
M. V. Bondarko and V. A. Sosnilo, “On the weight lifting property for localizations of triangulated categories,” Lobachevskii J. Math. 39, 970–984 (2018).
H. Krause, “A Brown representability theorem via coherent functors,” Topology 41, 853–861 (2002).
B. Keller, “A remark on the generalized smashing conjecture,” Manuscripta Math. 84, 193–198 (1994).
Author information
Authors and Affiliations
Corresponding authors
About this article
Cite this article
Bondarko, M.V., Vostokov, S.V. On Torsion Theories, Weight and t-Structures in Triangulated Categories. Vestnik St.Petersb. Univ.Math. 52, 19–29 (2019). https://doi.org/10.3103/S1063454119010047
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1063454119010047