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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
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References:
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