Integral motivic sheaves and geometric representation theory. (English) Zbl 1502.14017
Sheaves on manifolds are ubiquitous in geometric representation theory, i.e., highest weight representations of a complex reductive Lie algebra can be realized in terms of perverse sheaves on a flag manifold and representations of a reductive algebraic group correspond to equivariant perverse sheaves on an affine Grassmannian. More refined formalisms of sheaves carry an additional notion of weights and a Tate twist functor that provide an additional grading on the category, e.g., mixed Hodge modules have a notion of weights through Hodge structures and mixed \(\ell\)-adic sheaves have a notion of eigenvalues of the Frobenius. But some drawbacks include characteristic zero coefficients and are hence not applicable in modular representation theory. Secondly, there are unwanted extensions between Tate objects which bear no representation-theoretic significance and thus yield technical problems. Third, the choice of a fixed cohomology theory leaves open the question to which extent the resulting categories are independent of the coefficients or base field.
The goal of the authors is to introduce a formalism of mixed sheaves that overcomes these problems. That is, their formalism works with (almost) arbitrary coefficients, carries a six functor formalism and has no extension of Tate objects. Moreover, under appropriate assumptions, it is independent of the base and specializes to the existing approaches to categories of mixed sheaves in the literature.
In summary, the authors construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. They show that reduced stratified Tate motives satisfy favorable properties including weight and \(t\)-structures. The authos also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory.
The goal of the authors is to introduce a formalism of mixed sheaves that overcomes these problems. That is, their formalism works with (almost) arbitrary coefficients, carries a six functor formalism and has no extension of Tate objects. Moreover, under appropriate assumptions, it is independent of the base and specializes to the existing approaches to categories of mixed sheaves in the literature.
In summary, the authors construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. They show that reduced stratified Tate motives satisfy favorable properties including weight and \(t\)-structures. The authos also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory.
Reviewer: Mee Seong Im (Annapolis)
MSC:
| 14C15 | (Equivariant) Chow groups and rings; motives |
| 22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |
| 32S60 | Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) |
Keywords:
geometric representation theory; motivic homotopy theory; mixed Tate motives; parity sheavesReferences:
| [1] | Arkhipov, S.; Bezrukavnikov, R.; Ginzburg, V., Quantum groups, the loop Grassmannian, and the Springer resolution, J. Am. Math. Soc., 17, 3, 595-678 (2004) · Zbl 1061.17013 |
| [2] | Arinkin, D.; Gaitsgory, D., The category of singularities as a crystal and global Springer fibers, J. Am. Math. Soc., 31, 1, 135-214 (2018) · Zbl 1388.14042 |
| [3] | Achar, P. N.; Hardesty, W., Co-t-Structures on derived categories of coherent sheaves and the cohomology of tilting modules (2020) |
| [4] | Achar, P. N.; Riche, S., Koszul duality and semisimplicity of Frobenius, Ann. Inst. Fourier, 63, 4, 1511-1612 (2013) · Zbl 1348.14041 |
| [5] | Achar, P. N.; Rider, L., Parity sheaves on the affine Grassmannian and the Mirković-Vilonen conjecture, Acta Math., 215, 2, 183-216 (January 2015) · Zbl 1344.14016 |
| [6] | Achar, P. N.; Riche, S., Modular perverse sheaves on flag varieties I: tilting and parity sheaves, Ann. Sci. Éc. Norm. Supér. (4), 49, 2, 325-370 (2016), With a joint appendix with Geordie Williamson · Zbl 1386.14178 |
| [7] | Achar, P. N.; Riche, S., Modular perverse sheaves on flag varieties, II: Koszul duality and formality, Duke Math. J., 165, 1, 161-215 (January 2016) · Zbl 1375.14162 |
| [8] | Achar, P. N.; Riche, S., Reductive groups, the loop Grassmannian, and the Springer resolution, Invent. Math., 214, 1, 289-436 (October 2018) · Zbl 1454.20095 |
| [9] | Ayoub, J., Les Six Opérations de Grothendieck et le Formalisme des Cycles Évanescents dans le Monde Motivique. I, Astérisque, vol. 314 (2008), x+466 pp., 2007 · Zbl 1146.14001 |
| [10] | Ayoub, J., Les Six Opérations de Grothendieck et le Formalisme des Cycles Évanescents dans le Monde Motivique. II, Astérisque, vol. 315 (2008), vi+364 pp., 2007 · Zbl 1153.14001 |
| [11] | Beĭlinson, A. A.; Bernstein, J.; Deligne, P., Faisceaux pervers, (Analysis and Topology on Singular Spaces, I. Analysis and Topology on Singular Spaces, I, Luminy, 1981. Analysis and Topology on Singular Spaces, I. Analysis and Topology on Singular Spaces, I, Luminy, 1981, Astérisque, vol. 100 (1982), Soc. Math. France: Soc. Math. France Paris), 5-171 · Zbl 1390.14055 |
| [12] | Beilinson, A.; Bezrukavnikov, R.; Mirkovic, I., Tilting exercises, Mosc. Math. J., 4, 3, 547-557 (2004) · Zbl 1075.14015 |
| [13] | Bunke, U.; Cisinski, D.-C.; Kasprowski, D.; Winges, C., Controlled objects in left-exact ∞-categories and the Novikov conjecture (2019) |
| [14] | Beilinson, A.; Ginzburg, V.; Soergel, W., Koszul duality patterns in representation theory, J. Am. Math. Soc., 9, 2, 473-527 (1996) · Zbl 0864.17006 |
| [15] | Bondarko, M. V., Weight structures vs. t-structures; weight filtrations, spectral sequences, and complexes (for motives and in general), J. K-Theory, 6, 3, 387-504 (2010) · Zbl 1303.18019 |
| [16] | Bondarko, M. V., On morphisms killing weights and Hurewicz-type theorems (2019) |
| [17] | Ben-Zvi, D.; Francis, J.; Nadler, D., Integral transforms and Drinfeld centers in derived algebraic geometry, J. Am. Math. Soc., 23, 4, 909-966 (2010) · Zbl 1202.14015 |
| [18] | Ben-Zvi, D.; Nadler, D., The character theory of a complex group (2009) |
| [19] | Cisinski, D.-C.; Déglise, F., Triangulated Categories of Mixed Motives, Springer Monogr. Math. (2019), Springer: Springer Cham · Zbl 07138952 |
| [20] | Cautis, S.; Kamnitzer, J., Quantum K-theoretic geometric Satake, Compos. Math., 154 (August 2015) |
| [21] | Cass, R.; van den Hove, T.; Scholbach, J., An integral Satake equivalence for motives (2022) |
| [22] | Deligne, P., Voevodsky’s lectures on cross functors, (Fall 2001 (2001)), 1-41, Unpublished |
| [23] | Drew, B., Réalisations tannakiennes des motifs triangulés (2015) |
| [24] | Eberhardt, J. N., K-Motives and Koszul duality, Bull. Lond. Math. Soc. (2022), in press · Zbl 1530.17007 |
| [25] | Eberhardt, J., Springer motives, Proc. Am. Math. Soc., 149, 5, 1845-1856 (May 2021) · Zbl 1505.14052 |
| [26] | Eberhardt, J. N., K-theory Soergel bimodules (August 2022) |
| [27] | Eberhardt, J. N.; Kelly, S., Mixed motives and geometric representation theory in equal characteristic, Sel. Math. New Ser., 25, 2 (2019), Art. 30, 54 · Zbl 1444.14046 |
| [28] | Eberhardt, J. N.; Stroppel, C., Motivic Springer theory, Indag. Math., 33, 1, 190-217 (January 2022) · Zbl 1482.14003 |
| [29] | Gaitsgory, D.; Rozenblyum, N., A Study in Derived Algebraic Geometry. Vol. I. Correspondences and Duality, Mathematical Surveys and Monographs, vol. 221 (2017), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1408.14001 |
| [30] | Haines, T., A proof of the Kazhdan-Lusztig purity theorem via the decomposition theorem of BBD |
| [31] | Ho, Q. P.; Li, P., Revisiting mixed geometry (2022) |
| [32] | Hoyois, M., The six operations in equivariant motivic homotopy theory, Adv. Math., 305, 197-279 (2017) · Zbl 1400.14065 |
| [33] | Hemo, T.; Richarz, T.; Scholbach, J., Constructible sheaves on schemes and a categorical Künneth formula (2021) |
| [34] | Juteau, D.; Mautner, C.; Williamson, G., Parity sheaves, J. Am. Math. Soc., 27, 4, 1169-1212 (May 2014) · Zbl 1344.14017 |
| [35] | Khan, A., Motivic homotopy theory in derived algebraic geometry (2016), Universität Duisburg-Essen, PhD thesis |
| [36] | Khan, A. A.; Ravi, C., Generalized cohomology theories for algebraic stacks (2021) |
| [37] | Levine, M., Tate motives and the vanishing conjectures for algebraic K-theory, (Algebraic K-Theory and Algebraic Topology. Algebraic K-Theory and Algebraic Topology, Lake Louise, AB, 1991. Algebraic K-Theory and Algebraic Topology. Algebraic K-Theory and Algebraic Topology, Lake Louise, AB, 1991, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 407 (1993), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 167-188 · Zbl 0885.19001 |
| [38] | Lurie, J., Higher Topos Theory, Annals of Mathematics Studies, vol. 170 (2009), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1175.18001 |
| [39] | Lurie, J., Higher algebra (2017) |
| [40] | Liu, Y.; Zheng, W., Enhanced six operations and base change theorem for higher Artin stacks (2011) |
| [41] | Mautner, C.; Riche, S., Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković-Vilonen conjecture, J. Eur. Math. Soc., 20, 9, 2259-2332 (July 2018) · Zbl 1401.14097 |
| [42] | Morel, F.; Voevodsky, V., \( \mathbf{A}^1\)-homotopy theory of schemes, Publ. Math. IHÉS, 90, 45-143 (1999) · Zbl 0983.14007 |
| [43] | Pauksztello, D., Compact corigid objects in triangulated categories and co-t-structures, Open Math., 6, 1, 25-42 (March 2008) · Zbl 1152.18009 |
| [44] | Ringel, C. M., The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z., 208, 2, 209-224 (1991) · Zbl 0725.16011 |
| [45] | Richarz, T.; Scholbach, J., The intersection motive of the moduli stack of shtukas, Forum Math. Sigma, 8, e8 (2020) · Zbl 1506.14045 |
| [46] | Richarz, T.; Scholbach, J., The motivic Satake equivalence, Math. Ann., 380, 3-4, 1595-1653 (2021) · Zbl 1491.14069 |
| [47] | Riehl, E.; Verity, D., Elements of ∞-Category Theory, Cambridge Studies in Advanced Mathematics (2022), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1492.18001 |
| [48] | Soergel, W., Kategorie \(\mathcal{O} \), perverse Garben und Moduln über den koinvarianten zur Weylgruppe, J. Am. Math. Soc., 3, 2, 421-445 (1990) · Zbl 0747.17008 |
| [49] | Soergel, W., On the relation between intersection cohomology and representation theory in positive characteristic, Commutative Algebra, Homological Algebra and Representation Theory. Commutative Algebra, Homological Algebra and Representation Theory, Catania/Genoa/Rome, 1998. Commutative Algebra, Homological Algebra and Representation Theory. Commutative Algebra, Homological Algebra and Representation Theory, Catania/Genoa/Rome, 1998, J. Pure Appl. Algebra, 152, 1-3, 311-335 (2000) · Zbl 1101.14302 |
| [50] | Sosnilo, V., Theorem of the heart in negative K-theory for weight structures, Doc. Math., 24, 2137-2158 (2019) · Zbl 1441.18008 |
| [51] | Spitzweck, M., Notes for a mini-course on “Mixed Tate Motives and Fundamental Groups” given in Bonn (2016) |
| [52] | Spitzweck, M., A commutative \(\mathbb{P}^1\)-spectrum representing motivic cohomology over Dedekind domains, Mém. Soc. Math. Fr., 157, 110 (2018) · Zbl 1408.14081 |
| [53] | Soergel, W.; Virk, R.; Wendt, M., Equivariant motives and geometric representation theory (2018), (with an appendix by F. Hörmann and M. Wendt) |
| [54] | Soergel, W.; Wendt, M., Perverse motives and graded derived category \(\mathcal{O} \), J. Inst. Math. Jussieu, 17, 2, 347-395 (2018) · Zbl 1436.14015 |
| [55] | Williamson, G., Algebraic representations and constructible sheaves, Jpn. J. Math., 12, 2, 211-259 (September 2017) · Zbl 1425.17013 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
