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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Hochschild (co-)homology of schemes with tilting object
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by Ragnar-Olaf Buchweitz and Lutz Hille PDF
Trans. Amer. Math. Soc. 365 (2013), 2823-2844 Request permission


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
  • MR Author ID: 42840
  • Email:
  • Lutz Hille
  • Affiliation: Mathematisches Institut der Universität Münster, Einsteinstraße 62, 48149 Münster, Germany
  • Email:
  • 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.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 2823-2844
  • MSC (2010): Primary 14F05, 16S38, 16E40, 18E30
  • DOI:
  • MathSciNet review: 3034449