# zbMATH — the first resource for mathematics

Arithmetic on two dimensional local rings. (English) Zbl 0609.13003
A henselian two-dimensional local field $$\Lambda$$ is, by definition, an excellent henselian discrete valuation field whose residue field is a henselian discrete valuation field with finite residue field. If $$A$$ is a two-dimensional excellent normal henselian local ring with finite residue field, then, for any prime $$\mathfrak p$$ of height one in $$A$$, the field of fractions $$K_{\mathfrak p}$$ of the henselization of $$A$$ at $$\mathfrak p$$ is a two-dimensional local field. And, conversely, every two-dimensional local field arises in this way.
The Brauer group of a two-dimensional local field has been extensively studied by K. Kato in his three papers “A generalization of local class field theory by using $$K$$-groups” [(I) J. Fac. Sci., Univ. Tokyo, Sec. I A 26, 303–376 (1979; Zbl 0428.12013); (II) ibid. 27, 602–683 (1980; Zbl 0463.12006); (III) ibid. 29, 31–43 (1982; Zbl 0503.12004)]. Using the results of Kato, the author describes the Brauer group $$\mathrm{Br}(K)$$ of the field of fractions $$K$$ of a henselian ring $$A$$ as above. Among several interesting results, he constructs a canonical pairing $$\mathrm{Br}(X)\times\mathrm{Pic}(X)\to\mathbb Q/\mathbb Z$$ where $$X$$ is the punctured spectrum of $$A$$ and shows that it is in fact a perfect pairing of finite abelian groups.
In an appendix he defines a similar perfect pairing when $$X$$ is a projective smooth geometrically connected curve over a complete discrete valuation field with finite residue field. This, in the case of a field $$k$$ of positive characteristic, improves upon a result of S. Lichtenbaum in [Invent. Math. 7, 120–136 (1969; Zbl 0186.26402)].

##### MSC:
 14F22 Brauer groups of schemes 14H25 Arithmetic ground fields for curves 13J15 Henselian rings 12J10 Valued fields
Full Text:
##### References:
 [1] Abhyankar, S.: Resolution of singularities for arithmetical surfaces. In: Arithmetical Algebraic Geometry. New York: Harper and Row, 1963, pp. 111-152 [2] Artin, M.: Algebraic approximation of structures over complete local rings. Publ. Math. I.H.E.S.36, 23-58 (1969) · Zbl 0181.48802 [3] Stein, M.R., Dennis, R.K.:K 2 of radical ideals and semilocal rings revisited. Lect. Notes Math.342, 281-303 (1973) [4] Gabber, O.: A lecture at I.H.E.S. on March in 1981 [5] Greenberg, M.: Rational points in henselian discrete valuation rings. Publ. Math. I.H.E.S.23 (1964) · Zbl 0142.00901 [6] Grothendieck, A.: Le groupe de Brauer III. In: Dix exposé sur la cohomologie des schémas. Amsterdam: North-Holland 1968 [7] Illusie, L.: Complexe de De Rham-Witt et cohomologie cristalline. Ann. Sci. Ec. Norm. Sup.12, 501-661 (1979) · Zbl 0436.14007 [8] Kato, K.: A generalization of local class field theory by usingK-groups, I. J. Fac. Sci., Univ. Tokyo, Sec. IA26, 303-376 (1979); II, ibid. Kato, K.: A generalization of local class field theory by usingK-groups, I. J. Fac. Sci., Univ. Tokyo, Sec. IA27, 602-683 (1980); III, ibid. Kato, K.: A generalization of local class field theory by usingK-groups, I. J. Fac. Sci., Univ. Tokyo, Sec. IA29, 31-43 (1982) · Zbl 0428.12013 [9] Kato, K., Saito, S.: Unramified class field theory of arithmetical surfaces. Ann. Math.118, 241-275 (1983) · Zbl 0562.14011 · doi:10.2307/2007029 [10] Kato, K., Saito, S.: Global class field theory of arithmetic schemes (Preprint) · Zbl 0614.14001 [11] Lichtenbaum, S.: Duality theorems for curves overp-adic fields. Invent. Math.7, 120-136 (1969) · Zbl 0186.26402 · doi:10.1007/BF01389795 [12] Mercurjev, A.S., Suslin, A.A.:K-cohomology of Severi-Brauer varieties and norm residue homomorphism (Preprint) [13] Saito, S.: Class field theory for two dimensional local rings. (to appear) · Zbl 0672.12006 [14] Saito, S.: Class field theory for curves over local fields. Journal Number Theory21, 44-80 (1985) · Zbl 0599.14008 · doi:10.1016/0022-314X(85)90011-3 [15] Serre, J.-P.: Cohomologie galoisienne. Lect. Notes Math.5, 1965 · Zbl 0136.02801 [16] Serre, J.-P.: Corps locaux. Paris: Hermann 1962 · Zbl 0137.02601 [17] Shafarevich, I.R.: Lectures on minimal models and birational transformations of two dimensional schemes. Tata Institute of Foundamental Research, Bombay, 1966 · Zbl 0164.51704 [18] Shatz, S.S.: Cohomology of artinian group schemes over local fields. Ann. Math.79, 411-449 (1964) · Zbl 0152.19302 · doi:10.2307/1970403 [19] Tate, J.: WC-groups over ?-adic fields. Séminaire Bourbaki; 10e année. 1957/1958 [20] Hironaka, H.: Desingularization of excellent surfaces. Lectures at Advanced Science Seminar in Algebraic Geometry. Bowdoin College, Summer 1967, noted by Bruce Bennett [21] Bourbaki: Algèbre commutative. Chapitre 7. Paris: Hermann 1965 [22] Kurke, H., Pfister, G., Popescu, D., Roczen, M., Mostowski, T.: Die Approximationseigenschaft lokaler Ringe. Lect. Notes Math.634, 1978 · Zbl 0401.13013 [23] Brown, M.L.: A class of 2-dimensional local rings with Artin’s approximation property. The Journal of the London Mathematical Society27 Ser. 2, 29-34 (1983) · Zbl 0533.13012 · doi:10.1112/jlms/s2-27.1.29 [24] Pfister, G., Popescu, D.: Die strenge Approximationseigenschaft lokaler Ringe. Invent. Math.30, 947-977 (1979)
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.