Abstract
In this paper, we prove the ℓ-adic abelian class field theory for henselian regular local rings of equi-characteristic assuming the surjectivity of Galois symbol maps, which is an ℓ-adic variant of a result of Matsumi (Class field theory for \(\mathbb{F}_{q}[\hspace{-1.4pt}[X_{1},\dotsc,X_{n}]\hspace{-1.4pt}]\), preprint, 2002).
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References
Bloch S., Kato K.: p-adic étale cohomology. Inst. Hautes Études Sci. Publ. Math. 63, 107–152 (1986)
Bloch S., Ogus A.: Gersten’s conjecture and the homology of schemes. Ann. Sci. École Norm. Sup. (4) 7, 181–202 (1974)
Colliot-Thélène, J.-L., Hoobler, R. T., Kahn, B.: The Bloch–Ogus–Gabber theorem. In: Algebraic K-theory. Toronto, 1996, (Fields Inst. Commun. 16), pp. 31–94. American Mathematical Society, Providence (1997)
Fujiwara K.: Theory of tubular neibourhood in etale topology. Duke Math. J. 80, 15–56 (1995)
Grothendieck A., Artin M., Verdier J.-L., with Deligne P., Saint-Donat B.: Théorie des Topos et Cohomologie Étale des Schémas, Tome 3. Lecture Notes in Mathematics, vol. 305. Springer, Berlin (1973)
Grothendieck A., with Bucur I., Houzel C., Illusie L., Jouanolou J.-P., Serre J.-P.: Cohomologie ℓ-adique et Fonctions L. Lecture Notes in Mathematics, vol. 589. Springer, Berlin (1977)
Illusie, L.: Groupe de travail sur les travaux de Gabber en cohomologie étale. http://www.math.polytechnique.fr/~laszlo/gdtgabber/gabberluc.pdf
Kato K.: A generalization of local class field theory using K-groups. II. J. Fac. Sci. Univ. of Tokyo, Sec. IA 27, 602–683 (1980)
Kato, K., Saito, S.: Global class field theory of arithmetic schemes. In: Applications of algebraic K-theory to algebraic geometry and number theory, Part I, Boulder, Colo., 1983, (Contemp. Math. 55), pp. 255–331. American Mathematical Society, Providence (1986)
Matsumi K.: A Hasse principle for three-dimensional complete local rings of positive characteristic. J. Reine Angew. Math. 542, 113–121 (2002)
Matsumi P.: Class field theory for \({\mathbb {F}_{q}[\hspace{-1.4pt}[{X_1, X_2, X_3}}]\hspace{-1.4pt}]\). J. Math. Sci. Univ. Tokyo 9, 689–746 (2002)
Matsumi, P.: Class field theory for \({\mathbb {F}_{q}[\hspace{-1.4pt}[{X_1, \ldots, X_n}}]\hspace{-1.4pt}]\), preprint (2002)
Merkur’ev A.S., Suslin A.A.: K-cohomology of Severi–Brauer varieties and the norm residue homomorphism. Math. USSR Izv. 21, 307–341 (1983)
Merkur’ev A.S., Suslin A.A.: On the norm residue homomorphism of degree 3. Math. USSR Izv. 36, 349–367 (1991)
Orgogozo, F.: Le théorème d’uniformisation de Ofer Gabber et un survol de quelques conséquences: finitude, Lefschetz affine et pureté. http://www.math.polytechnique.fr/~orgogozo/articles/trame.pdf
Panin I.A.: The equi-characteristic case of the Gersten conjecture. Proc. Steklov Inst. Math. 241, 154–163 (2003)
Pilloni, V., Stroh, B.: Le théorème de Lefschetz affine, d’après Gabber. Preprint (2006)
Raskind, W.: Abelian class field theory of arithmetic schemes. In: K-theory and algebraic geometry: connections with quadratic forms and division algebras. Santa Barbara, 1992, (Proc. Sympos. Pure Math., 58, Part 1), pp. 85–187. American Mathematical Society, Providence (1995)
Riou, J.: Dualité (d’après Ofer Gabber). preprint http://www.math.jussieu.fr/~riou/doc/dualite.pdf
Saito, S.: Class field theory for two-dimensional local rings. In: Galois Representations and Arithmetic Algebraic Geometry. Kyoto, 1985/Tokyo, 1986, (Adv. Stud. in Pure Math. 12), pp. 343–373. North-Holland, Amsterdam (1987)
Serre J.-P.: Cohomologie Galoisienne. 5e éd. Lecture Notes in Mathematics, vol. 5. Springer, Berlin (1992)
Shiho A.: A note on class field theory for two-dimensional local rings. Compos. Math. 124, 305–340 (2000)
Tate J.: Relations between K 2 and Galois cohomology. Invent. Math. 36, 257–274 (1976)
Voevodsky V.: Motivic cohomology with \({\mathbb {Z}/2}\)-coefficients. Inst. Hautes Études Sci. Publ. Math. 98, 59–104 (2003)