Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules
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- by Pramod N. Achar and William Hardesty;
- Represent. Theory 28 (2024), 49-89
- DOI: https://doi.org/10.1090/ert/655
- Published electronically: February 2, 2024
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Abstract:
We construct a co-$t$-structure on the derived category of coherent sheaves on the nilpotent cone $\mathcal {N}$ of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the “exotic parity objects” (considered by Achar, Hardesty, and Riche [Transform. Groups 24 (2019), pp. 597–657]), along the (classical) Springer resolution, give indecomposable objects inside the coheart of the co-$t$-structure on $\mathcal {N}$. We also demonstrate how the various parabolic co-$t$-structures can be related by introducing an analogue to the usual translation functors. As an application, we give a proof of a scheme-theoretic formulation of the relative Humphreys conjecture on support varieties of tilting modules in type $A$ for $p>h$.References
- Pramod N. Achar, Perverse coherent sheaves on the nilpotent cone in good characteristic, Recent developments in Lie algebras, groups and representation theory, Proc. Sympos. Pure Math., vol. 86, Amer. Math. Soc., Providence, RI, 2012, pp. 1–23. MR 2976995, DOI 10.1090/pspum/086/1409
- Pramod N. Achar, On exotic and perverse-coherent sheaves, Representations of reductive groups, Progr. Math., vol. 312, Birkhäuser/Springer, Cham, 2015, pp. 11–49. MR 3495792, DOI 10.1007/978-3-319-23443-4_{2}
- Pramod N. Achar, Nicholas Cooney, and Simon Riche, The parabolic exotic $t$-structure, Épijournal Géom. Algébrique 2 (2018), Art. 8, 31. MR 3894857, DOI 10.46298/epiga.2018.volume2.4520
- Pramod N. Achar and William D. Hardesty, Calculations with graded perverse-coherent sheaves, Q. J. Math. 70 (2019), no. 4, 1327–1352. MR 4045103, DOI 10.1093/qmath/haz016
- P. Achar and W. Hardesty, Silting complexes of coherent sheaves and the Humphreys conjecture, arXiv:2106.04268, 2021.
- Pramod N. Achar, William Hardesty, and Simon Riche, On the Humphreys conjecture on support varieties of tilting modules, Transform. Groups 24 (2019), no. 3, 597–657. MR 3989684, DOI 10.1007/s00031-019-09513-y
- Pramod N. Achar, William Hardesty, and Simon Riche, Integral exotic sheaves and the modular Lusztig-Vogan bijection, J. Lond. Math. Soc. (2) 106 (2022), no. 3, 2403–2458. MR 4498557, DOI 10.1112/jlms.12638
- P. Achar, W. Hardesty, and S. Riche, Conjectures on tilting modules and antispherical $p$-cells, arXiv:1812.09960, 2018.
- Pramod N. Achar and Simon Riche, Reductive groups, the loop Grassmannian, and the Springer resolution, Invent. Math. 214 (2018), no. 1, 289–436. MR 3858401, DOI 10.1007/s00222-018-0805-1
- Pramod N. Achar and Laura Rider, The affine Grassmannian and the Springer resolution in positive characteristic, Compos. Math. 152 (2016), no. 12, 2627–2677. MR 3594290, DOI 10.1112/S0010437X16007661
- Henning Haahr Andersen and Jens Carsten Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann. 269 (1984), no. 4, 487–525. MR 766011, DOI 10.1007/BF01450762
- Dmitry Arinkin and Roman Bezrukavnikov, Perverse coherent sheaves, Mosc. Math. J. 10 (2010), no. 1, 3–29, 271 (English, with English and Russian summaries). MR 2668828, DOI 10.17323/1609-4514-2010-10-1-3-29
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- Roman Bezrukavnikov, Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone, Represent. Theory 7 (2003), 1–18. MR 1973365, DOI 10.1090/S1088-4165-03-00158-4
- Roman Bezrukavnikov, Cohomology of tilting modules over quantum groups and $t$-structures on derived categories of coherent sheaves, Invent. Math. 166 (2006), no. 2, 327–357. MR 2249802, DOI 10.1007/s00222-006-0514-z
- Roman Bezrukavnikov, Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group, Israel J. Math. 170 (2009), 185–206. MR 2506323, DOI 10.1007/s11856-009-0025-x
- M. V. Bondarko, Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory 6 (2010), no. 3, 387–504. MR 2746283, DOI 10.1017/is010012005jkt083
- Xiao-Wu Chen, Yu Ye, and Pu Zhang, Algebras of derived dimension zero, Comm. Algebra 36 (2008), no. 1, 1–10. MR 2378361, DOI 10.1080/00927870701649184
- Eric M. Friedlander and Brian J. Parshall, Cohomology of Lie algebras and algebraic groups, Amer. J. Math. 108 (1986), no. 1, 235–253 (1986). MR 821318, DOI 10.2307/2374473
- Robert Gordon and Edward L. Green, Graded Artin algebras, J. Algebra 76 (1982), no. 1, 111–137. MR 659212, DOI 10.1016/0021-8693(82)90240-X
- Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. MR 1489894, DOI 10.1007/s002220050197
- William D. Hardesty, On support varieties and the Humphreys conjecture in type $A$, Adv. Math. 329 (2018), 392–421. MR 3783418, DOI 10.1016/j.aim.2018.01.023
- Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 222093, DOI 10.1007/BFb0080482
- Sebastian Herpel, On the smoothness of centralizers in reductive groups, Trans. Amer. Math. Soc. 365 (2013), no. 7, 3753–3774. MR 3042602, DOI 10.1090/S0002-9947-2012-05745-X
- J. E. Humphreys, Comparing modular representations of semisimple groups and their Lie algebras, Modular interfaces (Riverside, CA, 1995) AMS/IP Stud. Adv. Math., vol. 4, Amer. Math. Soc., Providence, RI, 1997, pp. 69–80. MR 1483904, DOI 10.1090/amsip/004/05
- Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
- Jens Carsten Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211. MR 2042689
- Peter Jørgensen, Co-t-structures: the first decade, Surveys in representation theory of algebras, Contemp. Math., vol. 716, Amer. Math. Soc., [Providence], RI, [2018] ©2018, pp. 25–36. MR 3852398, DOI 10.1090/conm/716/14425
- Steffen Koenig and Dong Yang, Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403–438. MR 3178243, DOI 10.4171/dm/451
- George Lusztig, Cells in affine Weyl groups. IV, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), no. 2, 297–328. MR 1015001
- George Lusztig and Nan Hua Xi, Canonical left cells in affine Weyl groups, Adv. in Math. 72 (1988), no. 2, 284–288. MR 972764, DOI 10.1016/0001-8708(88)90031-X
- Carl Mautner and Simon Riche, On the exotic t-structure in positive characteristic, Int. Math. Res. Not. IMRN 18 (2016), 5727–5774. MR 3567258, DOI 10.1093/imrn/rnv330
- George J. McNinch and Donna M. Testerman, Completely reducible $\rm SL(2)$-homomorphisms, Trans. Amer. Math. Soc. 359 (2007), no. 9, 4489–4510. MR 2309195, DOI 10.1090/S0002-9947-07-04289-4
- Octavio Mendoza Hernández, Edith Corina Sáenz Valadez, Valente Santiago Vargas, and María José Souto Salorio, Auslander-Buchweitz context and co-$t$-structures, Appl. Categ. Structures 21 (2013), no. 5, 417–440. MR 3097052, DOI 10.1007/s10485-011-9271-2
Bibliographic Information
- Pramod N. Achar
- Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
- MR Author ID: 701892
- ORCID: 0000-0002-3447-5165
- Email: pramod.achar@math.lsu.edu
- William Hardesty
- Affiliation: School of Mathematics and Statistics, University of Sydney, Camperdown, New South Wales 2006, Australia
- MR Author ID: 1143324
- Email: hardes11@gmail.com
- Received by editor(s): February 14, 2021
- Received by editor(s) in revised form: March 22, 2022, and April 25, 2023
- Published electronically: February 2, 2024
- Additional Notes: The first author was supported by NSF Grant No. DMS-1802241. The second author was supported by the ARC Discovery Grant No. DP170104318.
- © Copyright 2024 American Mathematical Society
- Journal: Represent. Theory 28 (2024), 49-89
- MSC (2020): Primary 20G05, 18G80
- DOI: https://doi.org/10.1090/ert/655
- MathSciNet review: 4700051
Dedicated: Dedicated to the memory of Jim Humphreys