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Derived equivalences induced by nonclassical tilting objects


Authors: Luisa Fiorot, Francesco Mattiello and Manuel Saorín
Journal: Proc. Amer. Math. Soc. 145 (2017), 1505-1514
MSC (2010): Primary 18E30; Secondary 18E10, 18G55
DOI: https://doi.org/10.1090/proc/13368
Published electronically: November 21, 2016
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Abstract: Suppose that $ \mathcal {A}$ is an abelian category whose derived category $ \mathcal {D}(\mathcal {A})$ has $ Hom$ sets and arbitrary (small) coproducts, let $ T$ be a (not necessarily classical) ($ n$-)tilting object of $ \mathcal {A}$ and let $ \mathcal {H}$ be the heart of the associated t-structure on $ \mathcal {D}(\mathcal {A})$. We show that there is a triangulated equivalence of unbounded derived categories $ \mathcal {D}(\mathcal {H})\stackrel {\cong }{\longrightarrow }\mathcal {D}(\mathcal {A})$ which is compatible with the inclusion functor $ \mathcal {H}\hookrightarrow \mathcal {D}(\mathcal {A})$. The result admits a straightforward dualization to cotilting objects in abelian categories whose derived category has $ Hom$ sets and arbitrary products.


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Additional Information

Luisa Fiorot
Affiliation: Dipartimento di Matematica, Università degli studi di Padova, Via Trieste 63, 35121 Padova, Italy
Email: fiorot@math.unipd.it

Francesco Mattiello
Affiliation: Dipartimento di Matematica, Università degli studi di Padova, Via Trieste 63, 35121 Padova, Italy
Email: mattiell@math.unipd.it

Manuel Saorín
Affiliation: Departamento de Matemáticas, Universidad de Murcia, Aptdo. 4021, 30100 Espinardo, Murcia, Spain
Email: msaorinc@um.es

DOI: https://doi.org/10.1090/proc/13368
Received by editor(s): November 21, 2015
Received by editor(s) in revised form: December 8, 2015, May 2, 2016, and June 27, 2016
Published electronically: November 21, 2016
Additional Notes: The second author was supported by Assegno di ricerca “Tilting theory in triagulated categories” del Dipartimento di Matematica dell’Università degli Studi di Padova and by Progetto di Eccellenza della fondazione Cariparo.
The third author was supported by research projects from the Spanish Ministerio de Economía y Competitividad (MTM2013-46837-P) and from the Fundación ‘Séneca’ of Murcia (19880/GERM/15), with a part of FEDER funds.
The authors thank their institutions for their help. They also thank Jorge Vitória and Alexandra Zvonareva for pointing out some omissions in a first draft of the paper.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2016 American Mathematical Society