Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 055, 43 pages      arXiv:2009.12785
Contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday

Equivariant Tilting Modules, Pfaffian Varieties and Noncommutative Matrix Factorizations

Yuki Hirano
Department of Mathematics, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan

Received September 29, 2020, in final form May 28, 2021; Published online June 02, 2021

We show that equivariant tilting modules over equivariant algebras induce equivalences of derived factorization categories. As an application, we show that the derived category of a noncommutative resolution of a linear section of a Pfaffian variety is equivalent to the derived factorization category of a noncommutative gauged Landau-Ginzburg model $(\Lambda,\chi, w)^{\mathbb{G}_m}$, where $\Lambda$ is a noncommutative resolution of the quotient singularity $W/\operatorname{GSp}(Q)$ arising from a certain representation $W$ of the symplectic similitude group $\operatorname{GSp}(Q)$ of a symplectic vector space $Q$.

Key words: equivariant tilting module; Pfaffian variety; matrix factorization.

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  1. Alonso Tarrío L., Jeremías López A., Pérez Rodríguez M., Vale Gonsalves M.J., A functorial formalism for quasi-coherent sheaves on a geometric stack, Expo. Math. 33 (2015), 452-501, arXiv:1304.2520.
  2. Ballard M., Deliu D., Favero D., Isik M.U., Katzarkov L., Resolutions in factorization categories, Adv. Math. 295 (2016), 195-249, arXiv:1212.3264.
  3. Ballard M., Favero D., Katzarkov L., A category of kernels for equivariant factorizations and its implications for Hodge theory, Publ. Math. Inst. Hautes Études Sci. 120 (2014), 1-111, arXiv:1105.3177.
  4. Ballard M., Favero D., Katzarkov L., A category of kernels for equivariant factorizations, II: further implications, J. Math. Pures Appl. 102 (2014), 702-757, arXiv:1310.2656.
  5. Halpern-Leistner D., Sam S.V., Combinatorial constructions of derived equivalences, J. Amer. Math. Soc. 33 (2020), 735-773, arXiv:1601.02030.
  6. Happel D., Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, Vol. 119, Cambridge University Press, Cambridge, 1988.
  7. Hirano Y., Derived Knörrer periodicity and Orlov's theorem for gauged Landau-Ginzburg models, Compos. Math. 153 (2017), 973-1007, arXiv:1602.04769.
  8. Hirano Y., Equivalences of derived factorization categories of gauged Landau-Ginzburg models, Adv. Math. 306 (2017), 200-278, arXiv:1506.00177.
  9. Kashiwara M., Schapira P., Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften, Vol. 292, Springer-Verlag, Berlin, 1994.
  10. Kashiwara M., Schapira P., Categories and sheaves, Grundlehren der Mathematischen Wissenschaften, Vol. 332, Springer-Verlag, Berlin, 2006.
  11. Lipman J., Notes on derived functors and Grothendieck duality, in Foundations of Grothendieck duality for diagrams of schemes, Lecture Notes in Math., Vol. 1960, Springer, Berlin, 2009, 1-259.
  12. Lunts V.A., Schnürer O.M., Matrix factorizations and semi-orthogonal decompositions for blowing-ups, J. Noncommut. Geom. 10 (2016), 907-979, arXiv:1212.2670.
  13. Okonek C., Teleman A., Graded tilting for gauged Landau-Ginzburg models and geometric applications, Pure Appl. Math. Q. 17 (2021), 185-235, arXiv:1907.10099.
  14. Olsson M., Algebraic spaces and stacks, American Mathematical Society Colloquium Publications, Vol. 62, Amer. Math. Soc., Providence, RI, 2016.
  15. Positselski L., Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence, Mem. Amer. Math. Soc. 212 (2011), vi+133 pages, arXiv:0905.2621.
  16. Rennemo J.V., Segal E., Hori-mological projective duality, Duke Math. J. 168 (2019), 2127-2205, arXiv:1609.04045.
  17. Rennemo J.V., Segal E., Van den Bergh M., A non-commutative Bertini theorem, J. Noncommut. Geom. 13 (2019), 609-616, arXiv:1705.01366.
  18. Špenko Š., Van den Bergh M., Non-commutative resolutions of quotient singularities for reductive groups, Invent. Math. 210 (2017), 3-67, arXiv:1502.05240.
  19. The Stacks Project Authors, Stacks Project, available at
  20. Thomason R.W., Equivariant resolution, linearization, and Hilbert's fourteenth problem over arbitrary base schemes, Adv. Math. 65 (1987), 16-34.

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