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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

From triangulated categories to module categories via localisation


Authors: Aslak Bakke Buan and Robert J. Marsh
Journal: Trans. Amer. Math. Soc. 365 (2013), 2845-2861
MSC (2010): Primary 18E30, 18E35, 16G20; Secondary 13F60
Published electronically: November 7, 2012
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Abstract: We show that the category of finite dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class of maps. This generalises the $ 2$-Calabi-Yau tilting theorem of Keller-Reiten, in which the module category is obtained as a factor category, to the rigid case.


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Additional Information

Aslak Bakke Buan
Affiliation: Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway
Email: aslakb@math.ntnu.no

Robert J. Marsh
Affiliation: School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom
Email: marsh@maths.leeds.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05631-5
PII: S 0002-9947(2012)05631-5
Keywords: Triangulated categories, $2$-Calabi-Yau categories, localisation, module categories, rigid objects, cluster-tilting objects, approximations
Received by editor(s): November 17, 2010
Received by editor(s) in revised form: March 1, 2011
Published electronically: November 7, 2012
Additional Notes: This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/G007497/1] and by the NFR [FRINAT grant number 196600].
Article copyright: © Copyright 2012 Aslak Bakke Buan and Robert J. Marsh