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On the Brauer group of a two-dimensional local field. (English. Russian original) Zbl 1134.11043

Math. Notes 81, No. 6, 753-756 (2007); translation from Mat. Zametki 81, No. 6, 838-841 (2007).
Let \(K=\mathbb{F}_q((u))((t))\) be a two dimensional local field of characteristic \(p\) and let \(\text{Br}(K)\) be its Brauer group. There is a nondegenerate pairing \(K^*\times \text{Br}(K)\rightarrow \mathbb{Q}/\mathbb{Z}\) (defined in A. N. Parshin [Proc. Steklov Inst. Math. 183, 191–201 (1991; Zbl 0731.11064)]) and in the paper under review the author provides a proof of the nondegeneracy (not given in the original paper) by proving the equivalent assertion: \[ f\in K^*\;\text{verifies}\;(y,f| ut^{-p}]_K=0\;\text{for\;any}\;y\in K^* \iff f\in N^L_K(L^*) \] where \(L=K(x)\), \(x\) is a root of \(X^p-X=ut^{-p}\), \(N^L_K\) is the norm and \((\cdot,\cdot| ut^{-p}]_K\) is the residue given by Artin-Schreier theory.
The proof is obtained by evaluating componentwise the norm of \[ L^*={xt}\times {t}\times \mathbb{F}_q^* \times U_L \times U_{\overline L} \] (where \(U_L\) and \(U_{\overline L}\) are the units of \(L\) and of the residue field \(\overline L=\mathbb{F}_q((xt))\) respectively) and showing that the valuation coincides with the one of the annihilators of the pairing \((\cdot,\cdot| ut^{-p}]_K\,\).

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)

Citations:

Zbl 0731.11064
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References:

[1] A. N. Parshin, ”Galois cohomology and the Brauer group of local fields,” in Galois Theory, Rings, Algebraic Groups and Their Applications, Trudy Mat. Inst. Steklov (1990), Vol. 183, pp. 159–169 [in Russian].
[2] A. N. Parshin, ”Local class field theory,” in Algebraic Geometry and Its Applications, Trudy Mat. Inst. Steklov (1984), Vol. 165, pp. 143–170 [in Russian]. · Zbl 0535.12013
[3] J.-P. Serre, Corps Locaux, Deuxième édition. Publications de l’Université de Nancago (Hermann, Paris, 1968), Vol. VIII.
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