Koszul duality and semisimplicity of Frobenius

[Dualité de Koszul et semi-simplicité du Frobenius]

Achar, Pramod N. ; Riche, Simon

Annales de l'Institut Fourier, Tome 63 (2013), p. 1511-1612 / Harvested from Numdam

Résumé

D’après un résultat fondamental de Beĭlinson–Ginzburg–Soergel, sur les variétés de drapeaux et certains autres espaces, une version modifiée de la catégorie des faisceaux pervers $\ell $-adiques possède des propriétés liées à la dualité de Koszul. Cette catégorie modifiée est obtenue en éliminant les objets où l’action du Frobenius sur les fibres n’est pas semi-simple. Dans cet article, nous démontrons que de nombreuses opérations faisceautiques s’étendent à cette catégorie modifiée et sa version triangulée. En particulier, ces foncteurs préservent la semi-simplicité de l’action du Frobenius.

Abstract

A fundamental result of Beĭlinson–Ginzburg–Soergel states that on flag varieties and related spaces, a certain modified version of the category of $\ell $-adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue. In particular, we show that these functors preserve semisimplicity of the Frobenius action.

Détail

Publié le : 2013-01-01

MR 3137361 | Zbl 06359595

DOI : https://doi.org/10.5802/aif.2809
Classification:  16S37,  14F05,  14M15
Mots clés: Dualité de Koszul ; faisceaux pervers ; variété de drapeaux

Bibliographie

[1] Achar, P.; Riche, S. Constructible sheaves on affine Grassmannians and geometry of the dual nilpotent cone (preprint)

[2] Achar, P.; Treumann, D. Baric structures on triangulated categories and coherent sheaves, Int. Math. Res. Not. IMRN, Tome 2010 (2000) (doi:10.1093/imrn/rnq226, 56 pages) | MR 2824842 | Zbl 1239.18009

[3] Andersen, H. H.; Jantzen, J. C. Cohomology of induced representations for algebraic groups, Math. Ann., Tome 269 (1984), p. 487-252 | Article | MR 766011 | Zbl 0529.20027

[4] Arkhipov, S.; Bezrukavnikov, R.; Ginzburg, V. Quantum groups, the loop Grassmannian, and the Springer resolution, J. Amer. Math. Soc., Tome 17 (2004), pp. 595-678 | Article | MR 2053952 | Zbl 1061.17013

[5] Auslander, M.; Reiten, I.; Smalø, S. Representation theory of Artin algebras, Cambridge University Press, Cambridge Studies in Advanced Mathematics (1995) no. 36 | MR 1314422 | Zbl 0834.16001

[6] Beĭlinson, A. On the derived category of perverse sheaves, $K$-theory, arithmetic and geometry (Moscow, 1984–1986), Springer-Verlag, Berlin (Lecture Notes in Mathematics) Tome 1289 (1987), pp. 27-41 | MR 923133 | Zbl 0652.14008

[7] Beĭlinson, A.; Bernstein, J.; Deligne, P. Faisceaux pervers, Analyse et topologie sur les espaces singuliers, I (Luminy, 1981), Soc. Math. France, Paris (Astérisque) Tome 100 (1982), pp. 5-171 | MR 751966

[8] Beĭlinson, A.; Bezrukavnikov, R.; Mirković, I. Tilting exercises, Mosc. Math. J., Tome 4 (2004), pp. 547-557 | MR 2119139 | Zbl 1075.14015

[9] Beĭlinson, A.; Ginzburg, V.; Soergel, W. Koszul duality patterns in representation theory, J. Amer. Math. Soc., Tome 9 (1996), pp. 473-527 | Article | MR 1322847 | Zbl 0864.17006

[10] Bernstein, J.; Gel’Fand, I.; Gel’Fand, S. Schubert cells and cohomology of the spaces $G/P$, Upsehi Mat. Nauk, Tome 28 (1973), pp. 3-26 | Zbl 0289.57024

[11] Bernstein, J.; Lunts, V. Equivariant sheaves and functors, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1578 (1994) | MR 1299527 | Zbl 0808.14038

[12] Bezrukavnikov, R. Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone, Represent. Theory, Tome 7 (2003), pp. 1-18 | Article | MR 1973365 | Zbl 1065.20055

[13] Bezrukavnikov, R. Cohomology of tilting modules over quantum groups and $t$-structures on derived categories of coherent sheaves, Invent. Math., Tome 166 (2006), pp. 327-357 | Article | MR 2249802 | Zbl 1123.17002

[14] Bezrukavnikov, R.; Yun, Z. On Koszul duality for Kac–Moody groups (http://arxiv.org/abs/1101.1253)

[15] Broer, A. Line bundles on the cotangent bundle of the flag variety, Invent. Math., Tome 113 (1993), pp. 1-20 | Article | MR 1223221 | Zbl 0807.14043

[16] Deligne, P. La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. (1974) no. 43, pp. 273-307 | Article | Numdam | MR 340258 | Zbl 0287.14001

[17] Deligne, P. La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980) no. 52, pp. 137-252 | Article | Numdam | MR 601520 | Zbl 0456.14014

[18] Goresky, M.; Macpherson, R. Intersection homology. II, Invent. Math. (1983) no. 72, pp. 77-129 | Article | MR 696691 | Zbl 0529.55007

[19] Kazhdan, D.; Lusztig, G. Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math., XXXVI) (1980), pp. 185-203 | MR 573434 | Zbl 0461.14015

[20] Milne, J. S. Étale cohomology, Princeton University Press, Princeton, Princeton Mathematical Series (1980) no. 33 | MR 559531 | Zbl 0433.14012

[21] Milne, J. S. Motives over finite fields, Motives (Seattle, WA, 1991), Part 1, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 55 (1994) | MR 1265538 | Zbl 0811.14018

[22] Morel, S. Complexes d’intersection des compactifications de Baily–Borel : Le cas des groupes unitaires sur $\mathbb{Q}$, Université Paris 11 Orsay (2005) (Ph. D. Thesis)

[23] Orlov, D. Equivalences of derived categories and K3 surfaces, J. Math. Sci. (1997) no. 84, pp. 1361-1381 | Article | MR 1465519 | Zbl 0938.14019

[24] Ostrik, V. On the equivariant $K$-theory of the nilpotent cone, Represent. Theory (2000) no. 4, pp. 296-305 | Article | MR 1773863 | Zbl 0986.20045

[25] Tate, J. Conjectures on algebraic cycles in $\ell $-adic cohomology, Motives (Seattle, WA, 1991), Part 1, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 55 (1994) | MR 1265523 | Zbl 0814.14009

[26] Yun, Z. Weights of mixed tilting sheaves and geometric Ringel duality, Sel. Math., New. ser. (2009) no. 14, pp. 299-320 | Article | MR 2480718 | Zbl 1197.14016