Hochschild (co-)homology of schemes with tilting object

Abstract

Given akk–schemeXXthat admits a tilting objectTT, we prove that the Hochschild (co-)homology ofXXis isomorphic to that ofA=EndX⁡(T)A=\operatorname {End}_{X}(T). We treat more generally the relative case whenXXis flat over an affine schemeY=Spec⁡RY=\operatorname {Spec} R, and the tilting object satisfies an appropriate Tor-independence condition overRR. Among applications, Hochschild homology ofXXoverYYis seen to vanish in negative degrees, smoothness ofXXoverYYis shown to be equivalent to that ofAAoverRR, and forXXa smooth projective scheme we obtain that Hochschild homology is concentrated in degree zero. Using the Hodge decomposition of Hochschild homology in characteristic zero, forXXsmooth overYYthe Hodge groupsHq(X,ΩX/Yp)H^{q}(X,\Omega _{X/Y}^{p})vanish forp>qp > q, while in the absolute case they even vanish forp≠qp\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.