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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Derived equivalence induced by infinitely generated $ n$-tilting modules


Authors: Silvana Bazzoni, Francesca Mantese and Alberto Tonolo
Journal: Proc. Amer. Math. Soc. 139 (2011), 4225-4234
MSC (2010): Primary 16D90, 18E30, 18E35
Published electronically: April 28, 2011
MathSciNet review: 2823068
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Abstract: Let $ T_R$ be a right $ n$-tilting module over an arbitrary associative ring $ R$. In this paper we prove that there exists an $ n$-tilting module $ T'_R$ equivalent to $ T_R$ which induces a derived equivalence between the unbounded derived category $ \mathcal{D}(R)$ and a triangulated subcategory $ \mathcal E_{\perp}$ of $ \mathcal{D}(\operatorname{End}(T'))$ equivalent to the quotient category of $ \mathcal{D}(\operatorname{End}(T'))$ modulo the kernel of the total left derived functor $ -\otimes^{\mathbb{L}}_{S'}T'$. If $ T_R$ is a classical $ n$-tilting module, we have that $ T=T'$ and the subcategory $ \mathcal E_{\perp}$ coincides with $ \mathcal{D}(\operatorname{End}\vert(T))$, recovering the classical case.


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Additional Information

Silvana Bazzoni
Affiliation: Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, I-35121 Padova, Italy
Email: bazzoni@math.unipd.it

Francesca Mantese
Affiliation: Dipartimento di Informatica, Università degli Studi di Verona, strada Le Grazie 15, I-37134 Verona, Italy
Email: francesca.mantese@univr.it

Alberto Tonolo
Affiliation: Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, I-35121 Padova, Italy
Email: tonolo@math.unipd.it

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10900-6
PII: S 0002-9939(2011)10900-6
Received by editor(s): February 18, 2010
Received by editor(s) in revised form: September 9, 2010, and October 21, 2010
Published electronically: April 28, 2011
Additional Notes: This research was supported by grant CPDA071244/07 of Padova University and MIUR PRIN 2007
Communicated by: Harm Derksen
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.