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Six operations and Lefschetz-Verdier formula for Deligne-Mumford stacks. (English) Zbl 1353.14024

Summary: Y. Laszlo and M. Olsson [Publ. Math., Inst. Hautes Étud. Sci. 107, 109–168 (2008; Zbl 1191.14002); Publ. Math., Inst. Hautes Étud. Sci. 107, 169–210 (2008; Zbl 1191.14003)] constructed Grothendieck’s six operations for constructible complexes on Artin stacks in étale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. We give a more direct construction of the six operations for complexes on Deligne-Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed by W. Zheng [“Gluing pseudo functors via \(n\)-fold categories”, Preprint, arXiv:1211.1877]. As an application, we prove a Lefschetz-Verdier formula for Deligne-Mumford stacks. We include both torsion and \(\ell\)-adic coefficients.

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
14A20 Generalizations (algebraic spaces, stacks)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
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[1] Artin, M.; Grothendieck, A.; Verdier, J-L; etal., Théorie des topos et cohomologie étale des schémas, No. 269 (1972), Berlin · Zbl 0234.00007
[2] Behrend, K. A., Derived l-adic categories for algebraic stacks (2003) · Zbl 1051.14023
[3] Beĭlinson, A. A.; Bernstein, J.; Deligne, P., Faisceaux pervers, No. 100, 5-171 (1982), Paris · Zbl 0536.14011
[4] Conrad B. The Keel-Mori theorem via stacks. Preprint, 2005
[5] Conrad B, Lieblich M, Olsson M. Nagata compactification for algebraic spaces. J Inst Math Jussieu, 2012, 11: 747-814 · Zbl 1255.14003 · doi:10.1017/S1474748011000223
[6] Deligne P. La conjecture de Weil, II. Inst Hautes études Sci Publ Math, 1980, 52: 137-252 · Zbl 0456.14014 · doi:10.1007/BF02684780
[7] Deligne P, avec la collaboration de Boutot J-F, Grothendieck A, et al. Cohomologie étale. Lecture Notes in Mathematics, vol. 569. Berlin: Springer-Verlag, 1977 · Zbl 0345.00010 · doi:10.1007/BFb0091516
[8] Ekedahl, T., On the adic formalism, No. 87, 197-218 (1990), Boston, MA · Zbl 0821.14010
[9] Gabriel, P.; Zisman, M., Calculus of Fractions and Homotopy Theory (1967), New York · Zbl 0186.56802
[10] Grothendieck A. Sur quelques points d’algèbre homologique. Tôhoku Math J, 1957, 9: 119-221 · Zbl 0118.26104
[11] Grothendieck, A.; etal., Cohomologie ℓ-adique et fonctions L, No. 589 (1977), Berlin
[12] Illusie, L.; Laszlo, Y.; Orgogozo, F., Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents, 363-364 (2014), Paris
[13] Illusie, L., Cohomological dimension: First results, 455-459 (2014), Paris · Zbl 1320.14026
[14] Illusie L, Zheng W. Odds and ends on finite group actions and traces. Int Math Res Not, 2013, 2013: 1-62 · Zbl 1360.14120
[15] Illusie L, Zheng W. Quotient stacks and equivariant étale cohomology algebras: Quillen’s theory revisited. ArXiv:1305.0365, 2013 · Zbl 1358.14009
[16] Kashiwara M, Schapira P. Categories and Sheaves. Grundlehren der Mathematischen Wissenschaften, vol. 332. Berlin: Springer, 2006 · Zbl 1118.18001
[17] Keel S, Mori S. Quotients by groupoids. Ann of Math, 1997, 145: 193-213 · Zbl 0881.14018 · doi:10.2307/2951828
[18] Knutson D. Algebraic Spaces. Lecture Notes in Mathematics, vol. 203. Berlin: Springer-Verlag, 1971 · Zbl 0221.14001
[19] Laszlo Y, Olsson M. The six operations for sheaves on Artin stacks, I: Finite coefficients. Publ Math Inst Hautes études Sci, 2008, 107: 109-168 · Zbl 1191.14002 · doi:10.1007/s10240-008-0011-6
[20] Laszlo Y, Olsson M. The six operations for sheaves on Artin stacks, II: Adic coefficients. Publ Math Inst Hautes études Sci, 2008, 107: 169-210 · Zbl 1191.14003 · doi:10.1007/s10240-008-0012-5
[21] Laumon G. Comparaison de caractéristiques d’Euler-Poincaré en cohomologie l-adique. C R Acad Sci Paris Sér I Math, 1981, 292: 209-212 · Zbl 0468.14005
[22] Laumon G, Moret-Bailly L. Champs Algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 39. Berlin: Springer-Verlag, 2000 · Zbl 0945.14005
[23] Liu Y, Zheng W. Gluing restricted nerves of ∞-categories. ArXiv:1211.5294, 2012
[24] Liu Y, Zheng W. Enhanced six operations and base change theorem for Artin stacks. ArXiv:1211.5948, 2012
[25] Liu Y, Zheng W. Enhanced adic formalism and perverse t-structures for higher Artin stacks. ArXiv:1404.1128, 2014
[26] Olsson, M., Fujiwara’s theorem for equivariant correspondences (2014) · Zbl 1348.14058
[27] Orgogozo, F., Le théorème de finitude, 261-275 (2014), Paris · Zbl 1320.14006
[28] Rickard J. Derived categories and stable equivalence. J Pure Appl Algebra, 1989, 61: 303-317 · Zbl 0685.16016 · doi:10.1016/0022-4049(89)90081-9
[29] Riou, J., Dualité, 351-453 (2014), Paris · Zbl 1320.14032
[30] Rydh D. Compactification of tame Deligne-Mumford stacks. Preprint, 2011
[31] Rydh D. Existence and properties of geometric quotients. J Algebraic Geom, 2013, 22: 626-669 · Zbl 1278.14003 · doi:10.1090/S1056-3911-2013-00615-3
[32] Serre J-P. Représentations linéaires des groupes finis. 5th ed. Paris: Hermann, 1998 · Zbl 0926.20003
[33] The Stacks Project Authors. Stacks Project, http://math.columbia.edu/algebraic_geometry/stacks-git
[34] Varshavsky Y. Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara. Geom Funct Anal, 2007, 17: 271-319 · Zbl 1131.14019 · doi:10.1007/s00039-007-0596-9
[35] Zheng W. Sur l’indépendance de l en cohomologie l-adique sur les corps locaux. Ann Sci éc Norm Supér, 2009, 42: 291-334 · Zbl 1203.14023
[36] Zheng W. Gluing pseudofunctors via n-fold categories. ArXiv:1211.1877v3, 2014
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