The Heart of Twin Left n-cotorsion Pairs in Triangulated Categories

Abstract

Inspired by Nakaoka’s series of investigations on the construction of abelian structure from a triangulated category with the help of cotorsion pairs, we want to know whether the “high dimension” ones, i.e., (left) n-cotorsion pairs, also have the same “function”. Through some special constructions, we obtain that the heart of a twin left n-cotorsion pair in a triangulated category is a preabelian category. Moreover, if we only consider a single left n-cotorsion pair, then its heart becomes an abelian category. At last, we give an application of some results of a twin left n-cotorsion pair on the colocalization sequence.

Keywords

Triangulated category / n-cotorsion pair / heart / abelian category / 18E10 / 18E35 / 18G80

References

[1] Beligiannis A., Reiten I., Homological and homotopical aspects of torsion theories. Mem. Amer. Math. Soc., 2007, 188(883): viii+207 pp.

[2] Borceux F. Handbook of Categorical Algebra 1—Basic Category Theory, 1994, Cambridge, Cambridge University Press 50

[3] Buan AB, Marsh R, Reineke M, Reiten I, Todorov G. Tilting theory and cluster combinatorics. Adv. Math., 2006, 204(2): 572-618

[4] Bühler T. Exact categories. Expo. Math., 2010, 28(1): 1-69

[5] Caldero P, Keller B. From triangulated categories to cluster algebras, II. Ann. Sci. École Norm. Sup. (4), 2006, 39(6): 983-1009

[6] Caldero P, Keller B. From triangulated categories to cluster algebras. Invent. Math., 2008, 172(1): 169-211

[7] Demonet L, Liu Y. Quotients of exact categories by cluster tilting subcategories as module categories. J. Pure Appl. Algebra, 2013, 217(12): 2282-2297

[8] Gillespie J. Cotorsion pairs and degreewise homological model structures. Homology Homotopy Appl., 2008, 10(1): 283-304

[9] Gillespie J. Gorenstein complexes and recollements from cotorsion pairs. Adv. Math., 2016, 291: 859-911

[10] Happel D. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, 1988, Cambridge, Cambridge University Press 119

[11] He J., Zhou P., Abelian quotients of extriangulated categories. Proc. Indian Acad. Sci. Math. Sci., 2019, 129 (4): Paper No. 61, 11 pp.

[12] He J., Zhou P., On the relation between n-cotorsion pairs and (n + 1)-cluster tilting subcategories. J. Algebra Appl., 2022, 21 (1): Paper No. 2250011, 12 pp.

[13] Hovey M. Cotorsion pairs, model category structures, and representation theory. Math. Z., 2022, 241(3): 553-592

[14] Huang Z, Iyama O. Auslander-type conditions and cotorsion pairs. J. Algebra, 2007, 318(1): 93-100

[15] Huerta M., Mendoza O., Pérez M., n-Cotorsion pairs. J. Pure Appl. Algebra, 2021, 225 (5): Paper No. 106556, 34 pp.

[16] Iyama O. Cluster tilting for higher Auslander algebra. Adv. Math., 2011, 226(1): 1-61

[17] Iyama O, Kato K, Miyachi J. Recollement of homotopy categories and Cohen–Macaulay modules. J. K-Theory, 2011, 8(3): 507-542

[18] Keller B. Chain complexes and stable categories. Manuscripta Math., 1990, 67(4): 379417

[19] Keller B, Reiten I. Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math., 2007, 211(1123-151

[20] Koenig S, Zhu B. From triangulated categories to abelian categories: cluster tilting in a general framework. Math. Z., 2008, 258(1): 143-160

[21] Krause H. Homological Theory of Representations, 2022, Cambridge, Cambridge University Press195

[22] Liu Y. Hearts of twin cotorsion pairs on exact categories. J. Algebra, 2013, 394: 245-284

[23] Liu Y, Nakaoka H. Hearts of twin cotorsion pairs on extriangulated categories. J. Algebra, 2019, 528: 96-149

[24] Nakaoka H. General heart construction on a triangulated category (I): unifying t-structures and cluster tilting subcategories. Appl. Categ. Structures, 2011, 19(6): 879899

[25] Nakaoka H. General heart construction for twin torsion pairs on triangulated categories. J. Algebra, 2013, 374: 195-215

[26] Nakaoka H. Equivalence of hearts of twin cotorsion pairs on triangulated categories. Comm. Algebra, 2016, 44(104302-4326

[27] Nakaoka H, Palu Y. Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah. Topol. Géom. Différ. Catég., 2019, 60(2): 117-193

[28] Neeman A. Trangulated Categories, 2001, Princeton, NJ, Princeton University Press 148

[29] Salce L. Cotorsion Theories for Abelian Groups, 1979, London–New York, Academic Press

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Peking University