Hochschild (co-)homology of schemes with tilting object (Q2838106)

If a variety \(X\) possesses a tilting object, then this places strong constraints on its Hochschild (co-)homology, and thence (when working in characteristic zero) on its Hodge groups. The authors explain these consequences, and how they arise from invariance of Hochschild cohomology under derived equivalence, giving vanishing of the Hodge groups \(H^p(X, \Omega^q_X)\) for \(p\neq q\). They also consider the relative situation, where \(X\) possesses a tilting object relative to an affine \(Y\). In this case, for \(X\) smooth over \(Y\), they show that the groups \(H^p(X, \Omega^q_{X/Y})\) vanish for \(p<q\). Some explicit calculations of Hochschild (co-)homology are supplied in examples arising from resolutions of quotient singularities, along with a detailed review of tilting theory.NEWLINENEWLINEThe key tool here is the invariance of Hochschild cohomology under the derived equivalence (provided by the tilting object \(T\)) between the category \(\text{QCoh}(X)\) of quasicoherent sheaves on \(X\), and the category of \(A\)-modules for the tilting algebra \(A=\text{End}_X(T)\). A similar result is shown in the relative case, under a condition of flatness, or more generally \(\text{Tor}\)-independence. The above vanishing results then follow from vanishing of Hochschild cohomology in negative degrees. A crucial technical step is to show that \(T\) induces a tilting object on the product \(X\times X\), with tilting algebra given by the enveloping algebra \(A^{\mathrm{e}}\).NEWLINENEWLINEAs a corollary of their analysis, the authors show that smoothness of the variety \(X\) is equivalent to smoothness of the tilting algebra \(A\), in the sense of \textit{M. van den Bergh} (see erratum to [Proc. Am. Math. Soc. 126, No. 5, 1345--1348 (1998); erratum ibid. 130, No. 9, 2809--2810 (2002; Zbl 0894.16005)]). The authors also note that the results obtained should follow in the broader context provided by \textit{B. Toën} [Invent. Math. 167, No. 3, 615--667 (2007; Zbl 1118.18010)], where algebras are replaced by DG-algebras, and flatness conditions may be relaxed.