Hochschild (Co-)Homology of Schemes with Tilting Object

Details Hochschild (Co-)Homology of Schemes with Tilting Object Hochschild (Co-)Homology of Schemes with Tilting Object

2010-03-22

arxiv.org/abs/1003.4201v1

Mathematics

Algebraic Geometry

21 pages, no figures

Scientific paper

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 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 \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.

Affiliated with

Also associated with

No associations

Rating

Hochschild (Co-)Homology of Schemes with Tilting Object does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hochschild (Co-)Homology of Schemes with Tilting Object, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hochschild (Co-)Homology of Schemes with Tilting Object will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-206681