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P-algebras over a multidimensional local field. (Russian) Zbl 0716.12005

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 175, 121-127 (1989).
[For the entire collection see Zbl 0698.00023.]
Let p denote a prime natural number, we call F an n-dimensional local field if there is a sequence of discrete valuation fields \(F_ s\) such that the residual field is \(F_{s-1}\), \(F=F_ n\), and \(F_ 0\) is a finite field of characteristic p. The set \(\omega_ 1,...,\omega_ n\) is called a generalized set of parameters if the value of the canonical image of \(\omega_ i\) in \(F_ i\) takes under the corresponding discrete valuation a value which is prime to p. Let Br(F), \(K_ 2(F)\), and X(F) denote, respectively, the Brauer group of F, the \(K_ 2\) Milnor group, the group of continuous characters of the Galois group of the separable closure of F. For an abelian group A denote by \({}_ mA\) the subgroup of elements of period m. Let \(\{a,b\}_ k\) denote the class in the Brauer group of the corresponding cyclic algebra of dimension \(p^{2k}\) and \(\chi\cup a\) the class corresponding to the cyclic algebra canonically constructed by the pairing \(X(F)\oplus F^*\to Br(F).\)
Extending the results of a previous paper by the same author describing \(K_ 2(F)/_ pK_ 2(F)\) this paper describes the central simple algebras over F of exponent \(p^ n\) where \(char(F)=p\). Here are the results:
1) if \(char(F)=0\) and F contains the \(p^ mth\) roots of unity then for each \(u\in_{p^ m} Br(F)\) there are some units such that \(u=\{\epsilon_ 1,\omega_ 1\}_ m+\{\epsilon_ 2,\omega_ 2\}_ m+...+\{\epsilon_ m,\omega_ n\}_ m.\)
2) if \(char(F)=0\) and \(u\in_ pBr(F)\) then there are some characters \(\chi_ 1,...,\chi_ n\in_ pX(F)\) such that \(u=\chi_ 1\cup \omega_ 1+...+\chi_ n\cup \omega_ n.\)
3) if \(char(F)=p\) and \(u\in_{p^ m}Br(F)\) then there are some \(\chi_ 1,...,\chi_ n\in_{p^ m}X(F)\) such that \(u=\chi_ 1\cup \omega_ 1+...+\chi_ n\cup \omega_ n\).
Reviewer: H.Pop

MSC:

12G05 Galois cohomology
14F22 Brauer groups of schemes
19C30 \(K_2\) and the Brauer group
11S45 Algebras and orders, and their zeta functions
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
11S70 \(K\)-theory of local fields

Citations:

Zbl 0698.00023
Full Text: EuDML