The hearts of weight structures are the weakly idempotent complete categories
https://doi.org/10.22405/2226-8383-2020-21-3-29-38
Abstract
This paper proves that additive categories that occur as hearts of weight structures are
precisely the weakly idempotent complete categories, that is, the categories where all split
monomorphisms give direct sum decompositions. The work also gives several other conditions
equivalent to weak idempotent completeness (some of them are completely new) and discusses
weak idempotent completions of additive categories
Keywords
weakly idempotent complete category,
idempotent completion,
weak retractionclosure,
triangulated category,
weight structure,
heart
Supporting agencies: This work was funded by the Russian Science Foundation under grant no. 16-11-00200
About the Authors
Mikhail Vladimirovich Bondarko
Saint Petersburg State University
Russian Federation
Russian Federation
Doctor of Physical and Mathematical Sciences, RAS Professor
Sergei Vladimirovich Vostokov
Saint Petersburg State University
Russian Federation
Doctor of Physical and Mathematical Sciences, Professor
Russian Federation
Doctor of Physical and Mathematical Sciences, Professor
References
1. Balmer P., Schlichting M., 2001, “Idempotent completion of triangulated categories”, Journal of Algebra, 236, no. 2, 819–834.
2. Beilinson A., Bernstein J., Deligne P., 1982, “Faisceaux pervers” Asterisque, 100, 5–171.
3. Bondarko M. V., 2010, “Weight structures vs. t-structures; weight filtrations, spectral sequences and complexes (for motives and in general)”, J. of K-theory, v. 6(3), 387–504, see also http://arxiv.org/abs/0704.4003
4. Bondarko M. V., 2018, “On weight complexes, pure functors, and detecting weights”, preprint,
5. https://arxiv.org/abs/1812.11952
6. Bondarko M. V., Sosnilo V. A., 2018, “On constructing weight structures and extending them
7. to idempotent extensions”, Homology, Homotopy and Appl., vol. 20(1), 37–57.
8. B..uhler T., 2010, “Exact categories”, Expo. Math. 28, 1–69.
9. Freyd P., 1965, “Splitting homotopy idempotents”, Proceedings of the Conference on Categorical Algebra, La Jolla, CA, Springer, New York, 1966, 173–176.
10. Pauksztello D., 2008, “Compact cochain objects in triangulated categories and co-t-structures”, Central European Journal of Mathematics, vol. 6(1), 25–42.
11. Neeman A., 1990, “The derived category of an exact category”, J. Algebra 135 (2), 388–394.
12. Rose D., 2015, “A note on the Grothendieck group of an additive category”, Vestn. Chelyab.
13. Gos. Univ., 17(3), 135–139.
Review
For citations:
Bondarko M.V., Vostokov S.V. The hearts of weight structures are the weakly idempotent complete categories. Chebyshevskii Sbornik. 2020;21(3):29-38. (In Russ.) https://doi.org/10.22405/2226-8383-2020-21-3-29-38
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