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Gluing Silting Objects

Published online by Cambridge University Press:  11 January 2016

Qunhua Liu ,
Jorge Vitória  and
Dong Yang
Qunhua Liu
Affiliation:
Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, People’s Republic of China, [email protected]
Jorge Vitória
Affiliation:
Department of Computer Science—Sector of Mathematics, University of Verona, I-37134 Verona, Italy, [email protected]
Dong Yang
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China, [email protected]
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Abstract

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Recent results by Keller and Nicolás and by Koenig and Yang have shown bijective correspondences between suitable classes of t-structures and cot-structures with certain objects of the derived category: silting objects. On the other hand, the techniques of gluing (co-)t-structures along a recollement play an important role in the understanding of derived module categories. Using the above correspondence with silting objects, we present explicit constructions of gluing of silting objects, and, furthermore, we answer the question of when the glued silting is tilting.

Keywords

16E3518E30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 216 , 2014 , pp. 117 - 151
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2014

References

[1] Aihara, T. and Iyama, O., Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), 633–668. MR 2927802. DOI 10.1112/jlms/jdr055.Google Scholar
[2] Angeleri, L., Hügel Koenig, S., and Liu, Q., Recollements and tilting objects, J. Pure Appl. Algebra 215 (2011), 420–438. MR 2738361. DOI 10.1016/j.jpaa.2010.04.027.Google Scholar
[3] Angeleri, L. Hügel Koenig, S., and Liu, Q., Jordan-Hölder theorems for derived module categories of piece-wise hereditary algebras, J. Algebra 352 (2012), 361–381. MR 2862193. DOI 10.1016/j.jalgebra.2011.09.041.Google Scholar
[4] Angeleri, L. Hügel Koenig, S., and Liu, Q., On the uniqueness of stratifications of derived module categories, J. Algebra 359 (2012), 120–137. MR 2914629. DOI 10.1016/j.jalgebra.2012.02.022.CrossRefGoogle Scholar
[5] Beilinson, A. A., Bernstein, J., and Deligne, P., “Faisceaux pervers” in Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque 100, Soc. Math. France, Paris, 1982, 5–171. MR 0751966.Google Scholar
[6] Beligiannis, A. and Reiten, I., Homological and Homotopical Aspects of Torsion Theories, Mem. Amer. Math. Soc. 188, 2007. MR 2327478. DOI 10.1090/memo/0883.Google Scholar
[7] Bondal, A. I. and Kapranov, M. M., Representable functors, Serre functors, and reconstructions (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 53, no. 6 (1989), 1183–1205, 1337; English translation in Math. USSR-Izv. 35, no. 3 (1990), 519–541. MR 1039961.Google Scholar
[8] Bondarko, M. V., Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6 (2010), 387–504. MR 2746283. DOI 10.1017/is010012005jkt083.Google Scholar
[9] Bridgeland, T., t-structures on some local Calabi-Yau varieties, J. Algebra 289 (2005), 453–483. MR 2142382. DOI 10.1016/j.jalgebra.2005.03.016.Google Scholar
[10] Buan, A. B., Reiten, I., and Thomas, H., Three kinds of mutation, J. Algebra 339 (2011), 97–113. MR 2811314. DOI 10.1016/j.jalgebra.2011.04.030.Google Scholar
[11] Cline, E., Parshall, B., and Scott, L. L., Algebraic stratification in representation categories, J. Algebra 117 (1988), 504–521. MR 0957457. DOI 10.1016/0021-8693(88)90123-8.CrossRefGoogle Scholar
[12] Cline, E., Parshall, B., and Scott, L. L., Stratifying Endomorphism Algebras, Mem. Amer. Math. Soc. 124 (1996), no. 591. MR 1350891. DOI 10.1090/memo/0591.Google Scholar
[13] Franjou, V. and Pirashvili, T., Comparison of abelian categories recollements, Doc. Math. 9 (2004), 41–56. MR 2054979.CrossRefGoogle Scholar
[14] Happel, D., Reiten, I., and Small, S. O., Tilting in Abelian Categories and Quasitilted Algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575. MR 1327209. DOI 10.1090/memo/0575.Google Scholar
[15] Jörgensen, P., Reflecting recollements, Osaka J. Math. 47 (2010), 209–213. MR 2666132.Google Scholar
[16] Keller, B. and Nicolás, P., Cluster hearts and cluster tilting objects, in preparation.Google Scholar
[17] Keller, B. and Vossieck, D., “Aisles in derived categories” in Deuxiàme Contact Franco-Belge en Algàbre (Faulx-les-Tombes, 1987), Bull. Soc. Math. Belg. Sér. A 40 (1988), 239–253. MR 0976638.Google Scholar
[18] Koenig, S. and Yang, D., Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438. MR 3178243.Google Scholar
[19] Ladkani, S., Derived equivalences of triangular matrix rings arising from extensions of tilting modules, Algebr. Represent. Theory 14 (2011), 57–74. MR 2763291. DOI 10.1007/s10468-009-9175-0.CrossRefGoogle Scholar
[20] Liu, Q. and Vitória, J., t-structures via recollements for piecewise hereditary algebras, J. Pure Appl. Algebra 216 (2012), 837–849. MR 2864858. DOI 10.1016/j.jpaa.2011.08.010.Google Scholar
[21] Mendoza Hernández, O., Sáenz Valadez, E. C., Santiago Vargas, V., and Souto Salorio, M. J., Auslander-Buchweitz context and co-t-structures, Appl. Categ. Structures 21 (2013), 417–440. MR 3097052. DOI 10.1007/s10485-011-9271-2.Google Scholar
[22] Pauksztello, D., Compact corigid objects in triangulated categories and co-t-structures, Cent. Eur. J. Math. 6 (2008), 25–42. MR 2379950. DOI 10.2478/s11533-008-0003-2.Google Scholar
[23] Reiten, I. and Van den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295–366. MR 1887637. DOI 10.1090/S0894-0347-02-00387-9.Google Scholar
[24] Rickard, J., Morita theory for derived categories, J. Lond. Math. Soc. (2) 39 (1989), 436–456. MR 1002456. DOI 10.1112/jlms/s2-39.3.436.Google Scholar
[25] Rickard, J. and Schofield, A., Cocovers and tilting modules, Math. Proc. Cambridge Philos. Soc. 106 (1989), 1–5. MR 0994075. DOI 10.1017/S0305004100067931.CrossRefGoogle Scholar
[26] Souto Salorio, M. J. and Trepode, S., T-structures on the bounded derived category of the Kronecker algebra, Appl. Categ. Structures 20 (2012), 513–529. MR 2957314. DOI 10.1007/s10485-011-9248-1.Google Scholar
[27] Alonso Tarr´ıo, L., Jerem´ıas López, A., and Souto Salorio, M. J., Construction of t-structures and equivalences of derived categories, Trans. Amer. Math. Soc. 355, no. 6 (2003), 2523–2543. MR 1974001. DOI 10.1090/S0002-9947-03-03261-6.CrossRefGoogle Scholar
[28] Wang, Z., Nondegeneration and boundedness of t-structure induced by recollement, Xiamen Daxue Xuebao Ziran Kexue Ban 45 (2006), 10–13. MR 2193734.Google Scholar
[29] Woolf, J., Stability conditions, torsion theories and tilting, J. Lond. Math. Soc. (2) 82 (2010), 663–682. MR 2739061. DOI 10.1112/jlms/jdq035.Google Scholar