Nagoya Mathematical Journal

Gluing silting objects

Qunhua Liu, Jorge Vitória, and Dong Yang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Recent results by Keller and Nicolás and by Koenig and Yang have shown bijective correspondences between suitable classes of t-structures and co-t-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.

Article information

Nagoya Math. J., Volume 216 (2014), 117-151.

First available in Project Euclid: 9 January 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16E35: Derived categories
Secondary: 18E30: Derived categories, triangulated categories


Liu, Qunhua; Vitória, Jorge; Yang, Dong. Gluing silting objects. Nagoya Math. J. 216 (2014), 117--151. doi:10.1215/00277630-2847151.

Export citation


  • [1] T. Aihara and O. Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), 633–668.
  • [2] L. Angeleri Hügel, S. Koenig, and Q. Liu, Recollements and tilting objects, J. Pure Appl. Algebra 215 (2011), 420–438.
  • [3] Lidia Angeleri Hügel, Steffen Koenig, and Qunhua Liu, Jordan-Hölder theorems for derived module categories of piecewise hereditary algebras, J. Algebra 352 (2012), 361–381.
  • [4] Lidia Angeleri Hügel, Steffen Koenig, and Qunhua Liu, On the uniqueness of stratifications of derived module categories, J. Algebra 359 (2012), 120–137.
  • [5] A. A. Beilinson, J. Bernstein, and P. Deligne, “Faisceaux pervers” in Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque 100, Soc. Math. France, Paris, 1982, 5–171.
  • [6] A. Beligiannis and I. Reiten, Homological and Homotopical Aspects of Torsion Theories, Mem. Amer. Math. Soc. 188, 2007.
  • [7] A. I. Bondal and M. M. Kapranov, 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.
  • [8] M. V. Bondarko, Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6 (2010), 387–504.
  • [9] T. Bridgeland, t-structures on some local Calabi-Yau varieties, J. Algebra 289 (2005), 453–483.
  • [10] A. B. Buan, I. Reiten, and H. Thomas, Three kinds of mutation, J. Algebra 339 (2011), 97–113.
  • [11] E. Cline, B. Parshall, and L. L. Scott, Algebraic stratification in representation categories, J. Algebra 117 (1988), 504–521.
  • [12] E. Cline, B. Parshall, and L. L. Scott, Stratifying Endomorphism Algebras, Mem. Amer. Math. Soc. 124 (1996), no. 591.
  • [13] V. Franjou and T. Pirashvili, Comparison of abelian categories recollements, Doc. Math. 9 (2004), 41–56.
  • [14] D. Happel, I. Reiten, and S. O. Smalø, Tilting in Abelian Categories and Quasitilted Algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575.
  • [15] P. Jørgensen, Reflecting recollements, Osaka J. Math. 47 (2010), 209–213.
  • [16] B. Keller and P. Nicolás, Cluster hearts and cluster tilting objects, in preparation.
  • [17] B. Keller and D. Vossieck, “Aisles in derived categories” in Deuxième Contact Franco-Belge en Algèbre (Faulx-les-Tombes, 1987), Bull. Soc. Math. Belg. Sér. A 40 (1988), 239–253.
  • [18] S. Koenig and D. Yang, Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438.
  • [19] S. Ladkani, Derived equivalences of triangular matrix rings arising from extensions of tilting modules, Algebr. Represent. Theory 14 (2011), 57–74.
  • [20] Q. Liu and J. Vitória, t-structures via recollements for piecewise hereditary algebras, J. Pure Appl. Algebra 216 (2012), 837–849.
  • [21] O. Mendoza Hernández, E. C. Sáenz Valadez, V. Santiago Vargas, and M. J. Souto Salorio, Auslander-Buchweitz context and co-$t$-structures, Appl. Categ. Structures 21 (2013), 417–440.
  • [22] D. Pauksztello, Compact corigid objects in triangulated categories and co-$t$-structures, Cent. Eur. J. Math. 6 (2008), 25–42.
  • [23] I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295–366.
  • [24] J. Rickard, Morita theory for derived categories, J. Lond. Math. Soc. (2) 39 (1989), 436–456.
  • [25] J. Rickard and A. Schofield, Cocovers and tilting modules, Math. Proc. Cambridge Philos. Soc. 106 (1989), 1–5.
  • [26] M. J. Souto Salorio and S. Trepode, T-structures on the bounded derived category of the Kronecker algebra, Appl. Categ. Structures 20 (2012), 513–529.
  • [27] L. Alonso Tarrío, A. Jeremías López, and M. J. Souto Salorio, Construction of $t$-structures and equivalences of derived categories, Trans. Amer. Math. Soc. 355, no. 6 (2003), 2523–2543.
  • [28] Z. Wang, Nondegeneration and boundedness of $t$-structure induced by recollement, Xiamen Daxue Xuebao Ziran Kexue Ban 45 (2006), 10–13.
  • [29] J. Woolf, Stability conditions, torsion theories and tilting, J. Lond. Math. Soc. (2) 82 (2010), 663–682.