Recollements from generalized tilting
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Abstract:
Let $\mathcal {A}$ be a small dg category over a field $k$ and let $\mathcal {U}$ be a small full subcategory of the derived category $\mathcal {D}\mathcal {A}$ which generates all free dg $\mathcal {A}$-modules. Let $(\mathcal {B},X)$ be a standard lift of $\mathcal {U}$. We show that there is a recollement such that its middle term is $\mathcal {D}\mathcal {B}$, its right term is $\mathcal {D}\mathcal {A}$, and the three functors on its right side are constructed from $X$. This applies to the pair $(A,T)$, where $A$ is a $k$-algebra and $T$ is a good $n$-tilting module, and we obtain a result of Bazzoni–Mantese–Tonolo. This also applies to the pair $(\mathcal {A}, \mathcal {U})$, where $\mathcal {A}$ is an augmented dg category and $\mathcal {U}$ is the category of ‘simple’ modules; e.g., $\mathcal {A}$ is a finite-dimensional algebra or the Kontsevich–Soibelman $A_\infty$-category associated to a quiver with potential.References
- Lidia Angeleri Hügel, Steffen Koenig and Qunhua Liu, Recollements and tilting objects, J. Pure Appl. Algebra 215 (2011), no. 4, 420–438.
- Silvana Bazzoni, Equivalences induced by infinitely generated tilting modules, Proc. Amer. Math. Soc. 138 (2010), no. 2, 533–544. MR 2557170, DOI 10.1090/S0002-9939-09-10120-X
- Silvana Bazzoni, Francesca Mantese and Alberto Tonolo, Derived equivalence induced by $n$-tilting modules, arXiv:0905.3696, to appear in Proc. Amer. Math. Soc.
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- A. I. Bondal and M. M. Kapranov, Framed triangulated categories, Mat. Sb. 181 (1990), no. 5, 669–683 (Russian); English transl., Math. USSR-Sb. 70 (1991), no. 1, 93–107. MR 1055981, DOI 10.1070/SM1991v070n01ABEH001253
- Hongxing Chen and Changchang Xi, Good tilting modules and recollements of derived module categories, arXiv:1012.2176.
- Victor Ginzburg, Calabi-Yau algebras, arXiv:math/0612139v3 [math.AG].
- Dieter Happel, Partial tilting modules and recollement, Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 345–361. MR 1175843, DOI 10.1016/0022-4049(92)90093-u
- Peter Jørgensen, Recollement for differential graded algebras, J. Algebra 299 (2006), no. 2, 589–601. MR 2228328, DOI 10.1016/j.jalgebra.2005.07.027
- Bernhard Keller, Deriving DG categories, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 1, 63–102. MR 1258406, DOI 10.24033/asens.1689
- Bernhard Keller and Dong Yang, Derived equivalences from mutations of quivers with potential, Adv. Math. 226 (2011), no. 3, 2118–2168.
- Steffen König, Tilting complexes, perpendicular categories and recollements of derived module categories of rings, J. Pure Appl. Algebra 73 (1991), no. 3, 211–232. MR 1124785, DOI 10.1016/0022-4049(91)90029-2
- Maxim Kontsevich and Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
- Henning Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162. MR 2157133, DOI 10.1112/S0010437X05001375
- Jun-Ichi Miyachi, Recollement and tilting complexes, J. Pure Appl. Algebra 183 (2003), no. 1-3, 245–273. MR 1992048, DOI 10.1016/S0022-4049(03)00072-0
- Amnon Neeman, The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 547–566. MR 1191736, DOI 10.24033/asens.1659
- Pedro Nicolás and Manuel Saorín, Parametrizing recollement data for triangulated categories, J. Algebra 322 (2009), no. 4, 1220–1250. MR 2537682, DOI 10.1016/j.jalgebra.2009.04.035
- Jeremy Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436–456. MR 1002456, DOI 10.1112/jlms/s2-39.3.436
- Jeremy Rickard, Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), no. 3, 303–317. MR 1027750, DOI 10.1016/0022-4049(89)90081-9
Additional Information
- Dong Yang
- Affiliation: Max-Planck-Institut für Mathematik in Bonn, Vivatsgasse 7, 53111 Bonn, Germany
- Address at time of publication: HIM, Hausdorff Research Institute for Mathematics, Poppelsdorff Allee 82, D-53115, Bonn, Germany
- Email: yangdong98@mails.thu.edu.cn
- Received by editor(s): June 21, 2010
- Received by editor(s) in revised form: October 11, 2010, and November 8, 2010
- Published electronically: May 19, 2011
- Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 83-91
- MSC (2010): Primary 18E30, 16E45
- DOI: https://doi.org/10.1090/S0002-9939-2011-10898-0
- MathSciNet review: 2833519