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

 

Hochschild (co-)homology of schemes with tilting object


Authors: Ragnar-Olaf Buchweitz and Lutz Hille
Journal: Trans. Amer. Math. Soc. 365 (2013), 2823-2844
MSC (2010): Primary 14F05, 16S38, 16E40, 18E30
Published electronically: December 11, 2012
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Abstract: Given a $ k$-scheme $ X$ that admits a tilting object $ T$, we prove that the Hochschild (co-)homology of $ X$ is isomorphic to that of $ A=\operatorname {End}_{X}(T)$. We treat more generally the relative case when $ X$ is flat over an affine scheme $ Y=\operatorname {Spec} R$, and the tilting object satisfies an appropriate Tor-independence condition over $ R$. Among applications, Hochschild homology of $ X$ over $ Y$ is seen to vanish in negative degrees, smoothness of $ X$ over $ Y$ is shown to be equivalent to that of $ A$ over $ R$, and for $ X$ a smooth projective scheme we obtain that Hochschild homology is concentrated in degree zero. Using the Hodge decomposition of Hochschild homology in characteristic zero, for $ X$ smooth over $ Y$ the Hodge groups $ H^{q}(X,\Omega _{X/Y}^{p})$ vanish for $ p < q$, while in the absolute case they even vanish for $ p\neq q$.

We illustrate the results for crepant resolutions of quotient singularities, in particular for the total space of the canonical bundle on projective space.


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Additional Information

Ragnar-Olaf Buchweitz
Affiliation: Department of Computer and Mathematical Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada M1C 1A4
Email: ragnar@utsc.utoronto.ca

Lutz Hille
Affiliation: Mathematisches Institut der Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
Email: lutzhille@uni-muenster.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05577-2
PII: S 0002-9947(2012)05577-2
Received by editor(s): September 13, 2010
Received by editor(s) in revised form: February 25, 2011
Published electronically: December 11, 2012
Additional Notes: The first author gratefully acknowledges partial support through NSERC grant 3-642-114-80, while the second author thanks SFB 478 “Geometrische Strukturen in der Mathematik” for its support.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.