Abstract
Let \({c : C \rightarrow X \times X}\) be a correspondence with C and X quasi-projective schemes over an algebraically closed field k. We show that if \({u_\ell : c_1^*\mathbb{Q}_\ell \rightarrow c_2^!\mathbb{Q}_\ell}\) is an action defined by the localized Chern classes of a c 2-perfect complex of vector bundles on C, where ℓ is a prime invertible in k, then the local terms of u ℓ are given by the class of an algebraic cycle independent of ℓ. We also prove some related results for quasi-finite correspondences. The proofs are based on the work of Cisinski and Deglise on triangulated categories of motives.
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References
Artin, M., Grothendieck, A., Verdier, J.-L.: Théorie des topos et cohomologie étale des schémas. Springer Lecture Notes in Mathematics, 269, 270, 305. Springer, Berlin (1972)
Balmer P., Schlichting M.: Idempotent completion of triangulated categories. J. Algebra 236, 819–834 (2001)
Bondarko, M.: Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky vs. Hanamura. arXiv:math/0601713 (2008)
Cisinski, D.-C., Déglise, F.: Triangulated categories of mixed motives (2013, preprint)
Cisinski, D.-C., Déglise, F.: Étale Motives. Compositio Math. (to appear)
Conrad B., Lieblich M., Olsson M.: Nagata compactification for algebraic spaces. J. Inst. Math. Jussieu 11, 747–814 (2012)
Déglise F.: Around the Gysin triangle II. Doc. Math. 13, 613–675 (2008)
de Jong A.J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996)
de Jong A.J.: Families of curves and alterations. Ann. Inst. Fourier 47, 599–621 (1997)
Fujiwara K.: Rigid geometry, Lefschetz–Verdier trace formula, and Deligne’s conjecture. Invent. Math. 127, 489–533 (1997)
Fulton W.: Intersection Theory, Second Edition, Ergebnisse der Mathematik. Springer, Berlin (1998)
Fulton W.: Rational equivalence on singular varieties. Publ. Math. IHES 45, 147–167 (1975)
Grothendieck, A.: Cohomologie l-adic et Fonctions L. Springer Lectures Notes in Math, vol. 589. Springer, Berlin (1977)
Illusie L.: Miscellany on traces in ℓ-adic cohomology: a survey. Jpn. J. Math 1, 107–136 (2006)
Iversen B.: Local Chern classes. Ann. Sci. École Norm. Supér. 9, 155–169 (1976)
Krause H.: On Neeman’s well generated triangulated categories. Doc. Math. 6, 121–126 (2001)
Laumon, G.: Homologie étale, Exposé VIII in Séminaire de géométrie analytique (École Norm. Sup., Paris, 1974-75), pp. 163–188. Asterisque 36–37, Soc. Math. France, Paris (1976)
Olsson, M.: Borel–Moore homology, intersection products, and local terms. Adv. Math. 273, 56–123 (2015)
Olsson, M.: Fujiwara’s theorem for equivariant correspondences. J. Algebr. Geom. (2014). doi:10.1090/S1056-3911-2014-00628-7
Riou, J.: Classes de Chern, morphismes de Gysin et pureté absolue, Exposé XVI in “Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents”, Séminaire à l’Ecole polytechnique 2006–2008, dirigé par L. Illusie, Y. Laszlo, F. Orgogozo, to appear in Astérisque
Soulé C.: Opérations en K-théorie algébrique. Can. J. Math. 37, 488–550 (1985)
Varshavsky Y.: Lefschetz–Verdier trace formula and a generalization of a theorem of Fujiwara. Geom. Funct. Anal. 17, 271–319 (2007)
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Olsson, M. Motivic cohomology, localized Chern classes, and local terms. manuscripta math. 149, 1–43 (2016). https://doi.org/10.1007/s00229-015-0765-3
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DOI: https://doi.org/10.1007/s00229-015-0765-3