Koszul duality and semisimplicity of Frobenius

Achar, Pramod N.; Riche, Simon

Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1511-1612.

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.

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.

[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, Volume 2010 (2000) (doi:10.1093/imrn/rnq226, 56 pages) | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[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., Volume 113 (1993), pp. 1-20 | DOI | MR | Zbl

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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

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

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