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Recollements from generalized tilting


Author: Dong Yang
Journal: Proc. Amer. Math. Soc. 140 (2012), 83-91
MSC (2010): Primary 18E30, 16E45
Published electronically: May 19, 2011
MathSciNet review: 2833519
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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.


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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

DOI: https://doi.org/10.1090/S0002-9939-2011-10898-0
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.