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

 

Tilting theory and the finitistic dimension conjectures


Authors: Lidia Angeleri-Hügel and Jan Trlifaj
Journal: Trans. Amer. Math. Soc. 354 (2002), 4345-4358
MSC (2000): Primary 16E10, 16E30, 16G10
Published electronically: June 24, 2002
MathSciNet review: 1926879
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Abstract: Let $R$ be a right noetherian ring and let $\mathcal{P}^{<\infty}$ be the class of all finitely presented modules of finite projective dimension. We prove that findim $R = n < \infty$ iff there is an (infinitely generated) tilting module $T$ such that pd$T = n$ and $T ^\perp = (\mathcal P^{<\infty})^\perp$. If $R$ is an artin algebra, then $T$ can be taken to be finitely generated iff $\mathcal P^{<\infty}$ is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.


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Additional Information

Lidia Angeleri-Hügel
Affiliation: Mathematisches Institut der Universität, Theresienstrasse 39, D-80333 München, Germany
Address at time of publication: Universitat Autònoma de Barcelona, Departament de Matemàtiques, E-08193 Bellaterra (Barcelona), Spain
Email: angeleri@rz.mathematik.uni-muenchen.de

Jan Trlifaj
Affiliation: Katedra algebry MFF UK, Sokolovsk\a’a 83, 186 75 Prague 8, Czech Republic
Email: trlifaj@karlin.mff.cuni.cz

DOI: http://dx.doi.org/10.1090/S0002-9947-02-03066-0
PII: S 0002-9947(02)03066-0
Received by editor(s): July 13, 2001
Received by editor(s) in revised form: February 7, 2002
Published electronically: June 24, 2002
Additional Notes: Research supported by an HWP-grant of LMU Munich and by grants GAČR 201/00/0766 and MSM 113200007
Dedicated: Dedicated to Idun Reiten on the occasion of her sixtieth birthday
Article copyright: © Copyright 2002 American Mathematical Society