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Formality of $\mathbb{P}$-objects

Part of: Abelian categories Homological methods (Co)homology theory

Published online by Cambridge University Press:  03 May 2019

Andreas Hochenegger  and
Andreas Krug
Andreas Hochenegger
Affiliation:
Dipartimento di Matematica ‘Federigo Enriques’, Università degli Studi di Milano, via Cesare Saldini 50, 20133 Milano, Italy email [email protected]
Andreas Krug
Affiliation:
FB 12 Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße 6, 35032 Marburg, Germany email [email protected]
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Abstract

We show that a $\mathbb{P}$-object and simple configurations of $\mathbb{P}$-objects have a formal derived endomorphism algebra. Hence the triangulated category (classically) generated by such objects is independent of the ambient triangulated category. We also observe that the category generated by the structure sheaf of a smooth projective variety over the complex numbers only depends on its graded cohomology algebra.

Keywords

ℙ-objectformality of endomorphism algebrastriangulated category

MSC classification

Primary: 18E30: Derived categories, triangulated categories 14F05: Sheaves, derived categories of sheaves and related constructions 16E40: (Co)homology of rings and algebras (e.g. Hochschild, cyclic, dihedral, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 5 , May 2019 , pp. 973 - 994
Copyright
© The Authors 2019 

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References

Abuaf, R., Homological units , Int. Math. Res. Not. IMRN 2017 (2017), 6943–6960.Google Scholar
Addington, N., New derived symmetries of some hyperkähler varieties , Algebr. Geom. 3 (2016), 223–260.Google Scholar
Anno, R. and Logvinenko, T., Spherical DG-functors , J. Eur. Math. Soc. (JEMS) 19 (2017), 2577–2656.Google Scholar
Anno, R. and Logvinenko, T., On uniqueness of $\mathsf{P}\!$ -twists, Preprint (2017), arXiv:1711.06649.Google Scholar
Anthes, B., Cattaneo, A., Rollenske, S. and Tomassini, A., ∂ ∂ -complex symplectic and Calabi–Yau manifolds: Albanese map, deformations and period maps , Ann. Global Anal. Geom. 54 (2018), 377–398.Google Scholar
Angella, D., Cohomological aspects in complex non-Kähler geometry, Lecture Notes in Mathematics, vol. 2095 (Springer, Cham, 2014).Google Scholar
Balmer, P. and Schlichting, M., Idempotent completion of triangulated categories , J. Algebra 236 (2001), 819–834.Google Scholar
Barth, W., Hulek, K., Peters, C. and van de Ven, A., Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 4, second edition (Springer, Berlin, 2004).Google Scholar
Bernstein, J. and Lunts, V. A., Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578 (Springer, Berlin, 1994).Google Scholar
Block, J., Duality and equivalence of module categories in noncommutative geometry , in A celebration of the mathematical legacy of Raoul Bott, CRM Proceedings Lecture Notes, vol. 50, ed. Kotiuga, P. R. (American Mathematical Society, Providence, RI, 2010), 311–339.10.1090/crmp/050/24Google Scholar
Bridgeland, T., King, A. and Reid, M., The McKay correspondence as an equivalence of derived categories , J. Amer. Math. Soc. 14 (2001), 535–554.Google Scholar
Bongartz, K., Algebras and quadratic forms , J. Lond. Math. Soc. (2) 28 (1983), 462–469.Google Scholar
Butler, M. C. R. and King, A. D., Minimal resolutions of algebras , J. Algebra 212 (1999), 323–362.Google Scholar
Căldăraru, A., Katz, S. and Sharpe, E., D-branes, B fields, and Ext groups , Adv. Theor. Math. Phys. 7 (2003), 381–404.Google Scholar
Căldăraru, A., Derived categories of twisted sheaves on Calabi–Yau manifolds, PhD thesis, Cornell University (2000).Google Scholar
Collins, D. J., Relations among the squares of the generators of the braid group , Invent. Math. 117 (1994), 525–529.10.1007/BF01232254Google Scholar
Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D., Real homotopy theory of Kähler manifolds , Invent. Math. 29 (1975), 245–274.Google Scholar
Eilenberg, S., Homological dimension and syzygies , Ann. of Math. (2) 64 (1956), 328–336.Google Scholar
Ganter, N. and Kapranov, M., Symmetric and exterior powers of categories , Transform. Groups 19 (2014), 57–103.10.1007/s00031-014-9255-zGoogle Scholar
Haiman, M., Hilbert schemes, polygraphs and the Macdonald positivity conjecture , J. Amer. Math. Soc. 14 (2001), 941–1006.Google Scholar
Hochenegger, A., Kalck, M. and Ploog, D., Spherical subcategories in algebraic geometry , Math. Nachr. 289 (2016), 1450–1465.Google Scholar
Hochenegger, A., Kalck, M. and Ploog, D., Spherical subcategories in representation theory , Math. Z. 291 (2019), 113–147.10.1007/s00209-018-2075-4Google Scholar
Hochenegger, A. and Ploog, D., Rigid divisors on surfaces , Izv. Math. 83 (2019), doi:10.1070/IM8721.Google Scholar
Huybrechts, D., Fourier–Mukai transforms in algebraic geometry (Clarendon Press, Oxford, 2006).Google Scholar
Huybrechts, D. and Thomas, R. P., ℙ-objects and autoequivalences of derived categories , Math. Res. Lett. 13 (2006), 87–98.10.4310/MRL.2006.v13.n1.a7Google Scholar
Kadeishvili, T., Structure of A (∞)-algebra and Hochschild and Harrison cohomology , Proc. A. Razmadze Math. Inst. 91 (1988), 20–27 (in Russian); also Preprint (2002), arXiv:math/0210331 (in English).Google Scholar
Keller, B., Deriving DG categories , Ann. Sci. Éc. Norm. Supér (4) 27 (1994), 63–102.Google Scholar
Keller, B., On differential graded categories , in Proceedings of the International Congress of Mathematicians, Madrid 2006, vol. 2, eds Sanz-Solé, M., Soria, J., Varona, J. L. and Verdera, J. (European Mathematical Society, Zurich, 2006), 151–190.Google Scholar
Keller, B., A-infinity algebras, modules and functor categories , in Trends in representation theory of algebras and related topics, Contemporary Mathematics, vol. 406, eds de la Peña, J. and Bautista, R. (American Mathematical Society, Providence, RI, 2006), 67–93.Google Scholar
Keller, B., Derived categories and tilting , in Handbook of tilting theory, London Mathematical Society Lecture Note Series, vol. 332, eds Angeleri Hügel, L., Happel, D. and Krause, H. (Cambridge University Press, Cambridge, 2007), 49–104.Google Scholar
Keller, B., Yang, D. and Zhou, G., The Hall algebra of a spherical object , J. Lond. Math. Soc. (2) 80 (2009), 771–784.Google Scholar
Krug, A., On derived autoequivalences of Hilbert schemes and generalised Kummer varieties , Int. Math. Res. Not. IMRN 2015 (2015), 10680–10701.Google Scholar
Krug, A., Varieties with ℙ-units , Trans. Amer. Math. Soc. 370 (2018), 7959–7983.Google Scholar
Krug, A., Ploog, D. and Sosna, P., Derived categories of resolutions of cyclic quotient singularities , Quart. J. Math. 69 (2018), 509–548.Google Scholar
Kuznetsov, A. and Lunts, V., Categorical resolutions of irrational singularities , Int. Math. Res. Not. IMRN 2015 (2015), 4526–4625.Google Scholar
Lehn, M., Chern classes of tautological sheaves on Hilbert schemes of points on surfaces , Invent. Math. 136 (1999), 157–207.Google Scholar
Lunts, V. and Orlov, D., Uniqueness of enhancement for triangulated categories , J. Amer. Math. Soc. 23 (2010), 853–908.Google Scholar
Lunts, V. and Schnürer, O., New enhancements of derived categories of coherent sheaves and applications , J. Algebra 446 (2016), 203–274.10.1016/j.jalgebra.2015.09.017Google Scholar
Mak, C. Y. and Wu, W., Dehn twists exact sequences through Lagrangian cobordism , Compos. Math. 154 (2018), 2485–2533.Google Scholar
Neeman, A., Triangulated categories, Annals of Mathematics Studies, vol. 148 (Princeton University Press, Princeton, NJ, 2001).10.1515/9781400837212Google Scholar
Neisendorfer, J. and Taylor, L., Dolbeault homotopy theory , Trans. Amer. Math. Soc. 245 (1978), 183–210.Google Scholar
Ploog, D. and Sosna, P., On autoequivalences of some Calabi–Yau and hyperkähler varieties , Int. Math. Res. Not. IMRN 2014 (2014), 6094–6110.Google Scholar
Roitzheim, C. and Whitehouse, S., Uniqueness of A ∞ -structures and Hochschild cohomology , Algebr. Geom. Topol. 11 (2011), 107–143.Google Scholar
Scala, L., Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles , Duke Math. J. 150 (2009), 211–267.Google Scholar
Seidel, P. and Thomas, R. P., Braid group actions on derived categories of coherent sheaves , Duke Math. J. 108 (2001), 37–108.Google Scholar
Segal, E., All autoequivalences are spherical , Int. Math. Res. Not. IMRN 2018 (2018), 3137–3154.Google Scholar
Toën, B., Lectures on DG-categories , in Topics in algebraic and topological K-theory, Lecture Notes in Mathematics, vol. 2008, ed. Cortiñas, G. (Springer, Heidelberg, 2011), 243–302.Google Scholar
Voisin, C., Hodge theory and complex algebraic geometry I, Cambridge Studies in Advanced Mathematics, vol. 76 (Cambridge University Press, Cambridge, 2002).Google Scholar
Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38 (Cambridge University Press, Cambridge, 1994).Google Scholar