Hochschild (Co)Homology of Schemes with Tilting Object
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 $A= End_{X}(T)$. We treat more generally the relative case when $X$ is flat over an affine scheme $Y=\Spec R$ and the tilting object satisfies an appropriate Torindependence 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 \cite{BFl2} 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2010
 arXiv:
 arXiv:1003.4201
 Bibcode:
 2010arXiv1003.4201B
 Keywords:

 Mathematics  Algebraic Geometry;
 14F05;
 16S38;
 16E40;
 18E30
 EPrint:
 21 pages, no figures