Abstract
In the work of Hoshino, Kato and Miyachi, [11], the authors look at t-structures induced by a compact object, \( C \), of a triangulated category, \( \mathcal{T} \), which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a non-degenerate t-structure on \( \mathcal{T} \) whose heart is equivalent to Mod(End(\( C \))op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, \( \mathcal{S} \), of a triangulated category, \( \mathcal{T} \), induces a structure similar to a t-structure which we shall call a co-t-structure. We also show that the coheart of this non-degenerate co-t-structure is equivalent to Mod(End(\( \mathcal{S} \))op), and hence an abelian subcategory of \( \mathcal{T} \).
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References
Aldrich S.T., García Rozas J.R., Exact and semisimple differential graded algebras, Comm. Algebra, 2002, 30, 1053–1075
Avramov L., Halperin S., Through the looking glass: a dictionary between rational homotopy theory and local algebra, Algebra algebraic topology and their interactions (Stockholm, 1983), 1–27, Lecture Notes in Math. 1183, Springer, Berlin, 1986
Beilinson A.A., Bernstein J., Deligne P., Faisceaux pervers, Astérique, 100, Soc. Math. France, Paris, 1982 (in French)
Beligiannis A., Reiten I., Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc., 2007, 188, no. 883
Van den Bergh M., A remark on a theorem by Deligne, Proc. Amer. Math. Soc., 2004, 132, 2857–2858
Bernstein J., Lunts V, Eguivariant sheaves and functors, Lecture Notes in Math. 1578, Springer-Verlag, Berlin Heidelberg, 1994
Bondarko M.V., Weight structures for triangulated categories: weight filtrations, weight spectral sequences and weight complexes, applications to motives and to the stable homotopy category, preprint available at http://arxiv.org/abs/0704.4003v1
Enochs E., Injective and flat covers, envelopes and resolvents, Israel J. Math., 1981, 39, 189–209
Félix Y., Halperin S., Thomas J-C., Rational homotopy theory, Graduate Texts in Mathematics 205, Springer, New York, 2001
Frankild A., Jørgensen P., Homological identities for differential graded algebras, J. Algebra, 2003, 265, 114–135
Hoshino M., Kato Y., Miyachi J., On t-structures and torsion theories induced by compact objects, J. Pure Appl. Algebra, 2002, 167, 15–35
Iyama O., Yoshino Y., Mutations in triangulated categories and rigid Cohen-Macaulay modules, preprint available at http://arxiv.org/abs/math/0607736
Kashiwara M., Schapira P., Sheaves on manifolds, A Series of Comprehensive Studies in Mathematics 292, Springer-Verlag, Berlin Heidelberg, 1990
MacLane S., Categories for the working mathematician, Springer, New York, 1971
Neeman A., The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc., 1996, 9, 205–236
Neeman A., Triangulated categories, Annals of Mathematics Studies, Princeton University Press, Princeton and Oxford, 2001
Rotman J.J., An introduction to algebraic topology, Graduate Texts in Mathematics 119, Springer-Verlag, New York, 1988
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Pauksztello, D. Compact corigid objects in triangulated categories and co-t-structures. centr.eur.j.math. 6, 25–42 (2008). https://doi.org/10.2478/s11533-008-0003-2
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DOI: https://doi.org/10.2478/s11533-008-0003-2