2010/03/22 by Ragnar-Olaf Buchweitz, Buchweitz, Ragnar-Olaf, Lutz Hille +1 · 1 citation
Mathematics · #14F05 #16E40 #16S38 #18E30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #math.AG #msc:14F05 #msc:16E40 #msc:16S38 #msc:18E30
paper · pdf · doi:10.48550/arxiv.1003.4201
21 pages, no figures
arxiv created 2010/03/22 · openalex publication_date 2010/03/22 · arxiv updated 2010/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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= EndX(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 \citeBFl2 of Hochschild homology in characteristic zero, for X smooth over Y the Hodge groups Hq(X,ΩX/Yp) vanish for p < q, while in the absolute case they even vanish for p≠ q. We illustrate the results for crepant resolutions of quotient singularities, in particular for the total space of the canonical bundle on projective space.