×

zbMATH — the first resource for mathematics

Vanishing cycles, ramification of valuations, and class field theory. (With the collaboration of Takeshi Saito.). (English) Zbl 0665.14005
Let A be an excellent henselian discrete valuation ring with field of fractions k and with algebraically closed residue field E. Let B be the henselization of A[T] with respect to the maximal ideal \(m_ AA[T]+TA[T]\). Let \(U\subset \text{Spec}(B\otimes_ Ak)\) be a non-empty open subset, u the corresponding inclusion, K a field of positive characteristic with \(\text{char}(K)\neq \text{char}(E)\), and F a K-module in the étale site \(U_{et}\) which is locally constant of finite rank. Let x be the closed point of \(\text{Spec}(B)\) and p the generic point of \(\text{Spec}(B\otimes_ AE).\)
Then the problem is to evaluate the dimension of \(R^ 1_ x(u_ F)= H^ 1_{et}(\text{Spec}(B\otimes_ A\bar k),u_(F)).\) In this sense Deligne had computed the dimension of this space under the hypothesis that F is unramified with respect to the valuation ring A by giving an explicit formula.
The main purpose of the paper under review is to prove a similar formula in general, i.e. when F may be ramified with respect to \(A_ p\). This formula contains some terms involving the Swan conductor, and these are related to the 2-dimensional class field theory.
Reviewer: L.Bădescu

MSC:
14F20 Étale and other Grothendieck topologies and (co)homologies
13A18 Valuations and their generalizations for commutative rings
11S31 Class field theory; \(p\)-adic formal groups
14C99 Cycles and subschemes
13J15 Henselian rings
13F30 Valuation rings
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. S. Abhyankar, Resolution of singularities of arithmetical surfaces , Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper and Row, New York, 1965, pp. 111-152. · Zbl 0147.20503
[2] H. P. Epp, Eliminating wild ramification , Invent. Math. 19 (1973), 235-249. · Zbl 0254.13008 · doi:10.1007/BF01390208 · eudml:142195
[3] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I , Inst. Hautes Études Sci. Publ. Math. (1964), no. 20, 259. · Zbl 0136.15901 · doi:10.1007/BF02684747 · numdam:PMIHES_1964__20__5_0
[4] A. Grothendieck, Groupes de monodromie en géométrie algébrique. I , Lecture Notes in Math., vol. 288, Springer-Verlag, Berlin, 1972. · Zbl 0237.00013
[5] O. Hyôdo, Wild ramification in the imperfect residence field case , to appear in Advanced Studies in Pure Math. 12.
[6] 1 K. Kato, A generalization of local class field theory by using \(K\)-groups. I , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), no. 2, 303-376. · Zbl 0428.12013
[7] 2 K. Kato, A generalization of local class field theory by using \(K\)-groups. II , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 3, 603-683. · Zbl 0463.12006
[8] 3 K. Kato, A generalization of local class field theory by using \(K\)-groups. III , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 1, 31-43. · Zbl 0503.12004
[9] K. Kato and S. Saito, Two dimensional class field theory , Galois groups and their representations (Nagoya, 1981), Advanced Studies in Pure Math., vol. 2, North-Holland, Amsterdam, 1983, pp. 103-152. · Zbl 0544.12011
[10] K. Kato and S. Saito, Global class field theory of arithmetical schemes , to appear in Proc. of Alg. \(K\)-theory Conference held at Boulder in June 1983. · Zbl 0614.14001
[11] G. Laumon, Semi-continuité du conducteur de Swan (d’après P. Deligne) , The Euler-Poincaré characteristic (French), Astérisque, vol. 83, Soc. Math. France, Paris, 1981, pp. 173-219. · Zbl 0504.14013
[12] G. Laumon, Caractéristique d’Euler-Poincaré des faisceaux constructibles sur une surface , Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque, vol. 101-102, Soc. Math. France, Paris, 1983, pp. 193-207. · Zbl 0533.14007
[13] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization , Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, 195-279. · Zbl 0181.48903 · doi:10.1007/BF02684604 · numdam:PMIHES_1969__36__195_0 · eudml:103893
[14] A. N. Paršin, Abelian coverings of arithmetic schemes , Soviet, Math. Dokl. 19 (1978), no. 6, 1438-1442. · Zbl 0443.12006
[15] P. Ribenboim, Théorie des Valuations , Deuxième édition multigraphiée. Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Univ. de Montréal, 1968. · Zbl 0139.26201
[16] J.-P. Serre, Corps Locaux , Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, Actualités Sci. Indust., No. 1296. Hermann, Paris, 1962. · Zbl 0137.02601
[17] J.-P. Serre, Représentations linéaires des groupes finis , Hermann, Paris, 1967, Collection Méthodes. · Zbl 0189.02603
[18] A. Weil, Basic Number Theory , Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag New York, Inc., New York, 1967. · Zbl 0176.33601
[19] 1 G. Whaples, Generalized local class field theory, I. Reciprocity law , Duke Math. J. 19 (1952), 505-517. · Zbl 0047.03702 · doi:10.1215/S0012-7094-52-01953-4
[20] 2 G. Whaples, Generalized local class field theory, II. Existence theorem , Duke Math. J. 21 (1954), 247-255. · Zbl 0058.03001 · doi:10.1215/S0012-7094-54-02125-0
[21] 3 G. Whaples, Generalized local class field theory, III. Second form of existence theorem. Structure of analytic groups , Duke Math. J. 21 (1954), 575-581. · Zbl 0058.03001 · doi:10.1215/S0012-7094-54-02125-0
[22] 4 G. Whaples, Generalized local class field theory, IV. Cardinalities , Duke Math. J. 21 (1954), 583-586. · Zbl 0058.03001 · doi:10.1215/S0012-7094-54-02125-0
[23] S. Bloch, Algebraic \(K\)-theory and classfield theory for arithmetic surfaces , Ann. of Math. (2) 114 (1981), no. 2, 229-265. JSTOR: · Zbl 0512.14009 · doi:10.2307/1971294 · links.jstor.org
[24] N. Bourbaki, Éléments de Mathématique, Algèbre commutative , Hermann, Paris, 1962.
[25] V. G. Lomadze, On the ramification theory of two-dimensional local fields , Math. U.S.S.R. Sbornik 37 (1980), 349-365. · Zbl 0442.12013 · doi:10.1070/SM1980v037n03ABEH001957
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.