Hochschild (co-)homology of schemes with tilting object

Buchweitz, Ragnar-Olaf; Hille, Lutz

doi:10.1090/S0002-9947-2012-05577-2

Hochschild (co-)homology of schemes with tilting object
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Trans. Amer. Math. Soc. 365 (2013), 2823-2844 Request permission

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 $A=\operatorname {End}_{X}(T)$. We treat more generally the relative case when $X$ is flat over an affine scheme $Y=\operatorname {Spec} R$, 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 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.

References

Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR 1476671
Dagmar Baer, Tilting sheaves in representation theory of algebras, Manuscripta Math. 60 (1988), no. 3, 323–347. MR 928291, DOI 10.1007/BF01169343
Arend Bayer and Yuri I. Manin, (Semi)simple exercises in quantum cohomology, The Fano Conference, Univ. Torino, Turin, 2004, pp. 143–173. MR 2112573
A. A. Beĭlinson, Coherent sheaves on $\textbf {P}^{n}$ and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69 (Russian). MR 509388
George M. Bergman and Warren Dicks, Universal derivations and universal ring constructions, Pacific J. Math. 79 (1978), no. 2, 293–337. MR 531320, DOI 10.2140/pjm.1978.79.293
Raf Bocklandt, Travis Schedler, and Michael Wemyss, Superpotentials and higher order derivations, J. Pure Appl. Algebra 214 (2010), no. 9, 1501–1522. MR 2593679, DOI 10.1016/j.jpaa.2009.07.013
Christian Böhning, Derived categories of coherent sheaves on rational homogeneous manifolds, Doc. Math. 11 (2006), 261–331. MR 2262935
A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25–44 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 23–42. MR 992977, DOI 10.1070/IM1990v034n01ABEH000583
A. I. Bondal, Helices, representations of quivers and Koszul algebras, Helices and vector bundles, London Math. Soc. Lecture Note Ser., vol. 148, Cambridge Univ. Press, Cambridge, 1990, pp. 75–95. MR 1074784, DOI 10.1017/CBO9780511721526.008
Alexei Bondal and Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Math. 125 (2001), no. 3, 327–344. MR 1818984, DOI 10.1023/A:1002470302976
A. Bondal and M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 (2003), no. 1, 1–36, 258 (English, with English and Russian summaries). MR 1996800, DOI 10.17323/1609-4514-2003-3-1-1-36
Tom Bridgeland, t-structures on some local Calabi-Yau varieties, J. Algebra 289 (2005), no. 2, 453–483. MR 2142382, DOI 10.1016/j.jalgebra.2005.03.016
Tom Bridgeland and David Stern, Helices on del Pezzo surfaces and tilting Calabi-Yau algebras, Adv. Math. 224 (2010), no. 4, 1672–1716. MR 2646308, DOI 10.1016/j.aim.2010.01.018
Brylinski, J.-L.: A correspondence dual to McKay’s, preprint 1996, 16 pp.; arXiv.org:alg-geom/9612003
Ragnar-Olaf Buchweitz, Finite representation type and periodic Hochschild (co-)homology, Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997) Contemp. Math., vol. 229, Amer. Math. Soc., Providence, RI, 1998, pp. 81–109. MR 1676212, DOI 10.1090/conm/229/03311
Ragnar-Olaf Buchweitz and Hubert Flenner, Global Hochschild (co-)homology of singular spaces, Adv. Math. 217 (2008), no. 1, 205–242. MR 2357326, DOI 10.1016/j.aim.2007.06.012
Ragnar-Olaf Buchweitz and Hubert Flenner, The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah-Chern character, Adv. Math. 217 (2008), no. 1, 243–281. MR 2357327, DOI 10.1016/j.aim.2007.06.013
Căldăraru, A.: The Mukai pairing, I: the Hochschild structure, preprint 2003, 32 pp., arXiv.org:math/0308079
Andrei Căldăraru, Anthony Giaquinto, and Sarah Witherspoon, Algebraic deformations arising from orbifolds with discrete torsion, J. Pure Appl. Algebra 187 (2004), no. 1-3, 51–70. MR 2027895, DOI 10.1016/j.jpaa.2003.10.004
J. Daniel Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998), no. 2, 284–339. MR 1626856, DOI 10.1006/aima.1998.1735
L. Costa and R. M. Miró-Roig, Derived categories of projective bundles, Proc. Amer. Math. Soc. 133 (2005), no. 9, 2533–2537. MR 2146195, DOI 10.1090/S0002-9939-05-07846-9
Boris Dubrovin, Geometry and analytic theory of Frobenius manifolds, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 315–326. MR 1648082
Marco Farinati, Hochschild duality, localization, and smash products, J. Algebra 284 (2005), no. 1, 415–434. MR 2115022, DOI 10.1016/j.jalgebra.2004.09.009
Victor Ginzburg and Dmitry Kaledin, Poisson deformations of symplectic quotient singularities, Adv. Math. 186 (2004), no. 1, 1–57. MR 2065506, DOI 10.1016/j.aim.2003.07.006
A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222 (French). MR 217084
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085
Yang Han, Hochschild (co)homology dimension, J. London Math. Soc. (2) 73 (2006), no. 3, 657–668. MR 2241972, DOI 10.1112/S002461070602299X
Dieter Happel, Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 108–126. MR 1035222, DOI 10.1007/BFb0084073
Lutz Hille and Markus Perling, Exceptional sequences of invertible sheaves on rational surfaces, Compos. Math. 147 (2011), no. 4, 1230–1280. MR 2822868, DOI 10.1112/S0010437X10005208
Lutz Hille and Michel Van den Bergh, Fourier-Mukai transforms, Handbook of tilting theory, London Math. Soc. Lecture Note Ser., vol. 332, Cambridge Univ. Press, Cambridge, 2007, pp. 147–177. MR 2384610, DOI 10.1017/CBO9780511735134.007
G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58–67. MR 11076, DOI 10.2307/1969145
Bernhard Keller, Derived categories and tilting, Handbook of tilting theory, London Math. Soc. Lecture Note Ser., vol. 332, Cambridge Univ. Press, Cambridge, 2007, pp. 49–104. MR 2384608, DOI 10.1017/CBO9780511735134.005
Bernhard Keller, Hochschild cohomology and derived Picard groups, J. Pure Appl. Algebra 190 (2004), no. 1-3, 177–196. MR 2043327, DOI 10.1016/j.jpaa.2003.10.030
Bernhard Keller, Invariance and localization for cyclic homology of DG algebras, J. Pure Appl. Algebra 123 (1998), no. 1-3, 223–273. MR 1492902, DOI 10.1016/S0022-4049(96)00085-0
Amnon Neeman, The connection between the $K$-theory localization theorem of Thomason, Trobaugh and Yao and the smashing subcategories of Bousfield and Ravenel, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 547–566. MR 1191736, DOI 10.24033/asens.1659
Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205–236. MR 1308405, DOI 10.1090/S0894-0347-96-00174-9
Daniel Quillen, On the (co-) homology of commutative rings, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 65–87. MR 0257068
Jeremy Rickard, Derived equivalences as derived functors, J. London Math. Soc. (2) 43 (1991), no. 1, 37–48. MR 1099084, DOI 10.1112/jlms/s2-43.1.37
Helices and vector bundles, London Mathematical Society Lecture Note Series, vol. 148, Cambridge University Press, Cambridge, 1990. Seminaire Rudakov; Translated from the Russian by A. D. King, P. Kobak and A. Maciocia. MR 1074776
Alexander Samokhin, Some remarks on the derived categories of coherent sheaves on homogeneous spaces, J. Lond. Math. Soc. (2) 76 (2007), no. 1, 122–134. MR 2351612, DOI 10.1112/jlms/jdm038
A. V. Samokhin, The derived category of coherent sheaves on $LG_3^\textbf {C}$, Uspekhi Mat. Nauk 56 (2001), no. 3(339), 177–178 (Russian); English transl., Russian Math. Surveys 56 (2001), no. 3, 592–594. MR 1859740, DOI 10.1070/RM2001v056n03ABEH000410
Anne V. Shepler and Sarah Witherspoon, Finite groups acting linearly: Hochschild cohomology and the cup product, Adv. Math. 226 (2011), no. 4, 2884–2910. MR 2764878, DOI 10.1016/j.aim.2010.09.022
Richard G. Swan, Hochschild cohomology of quasiprojective schemes, J. Pure Appl. Algebra 110 (1996), no. 1, 57–80. MR 1390671, DOI 10.1016/0022-4049(95)00091-7
Bertrand Toën, The homotopy theory of $dg$-categories and derived Morita theory, Invent. Math. 167 (2007), no. 3, 615–667. MR 2276263, DOI 10.1007/s00222-006-0025-y
Michel van den Bergh, Erratum to: “A relation between Hochschild homology and cohomology for Gorenstein rings” [Proc. Amer. Math. Soc. 126 (1998), no. 5, 1345–1348; MR1443171 (99m:16013)], Proc. Amer. Math. Soc. 130 (2002), no. 9, 2809–2810. MR 1900889, DOI 10.1090/S0002-9939-02-06684-4