Local class field theory.

*(Russian)*Zbl 0535.12013By a local field in commutative algebra we mean usually a field of fractions of a complete discrete valuation ring. This notion has many applications in arithmetics and algebraic geometry. In the papers [Usp. Mat. Nauk 30, No. 1 (181), 253–254 (1975; Zbl 0302.14005); Izv. Akad. Nauk SSSR, Ser. Mat. 40, 736–773 (1976; Zbl 0358.14012)] the author has introduced a notion of local field of arbitrary dimension. From this point of view usual local fields are the local fields of dimension one. It was shown that this notion is useful for many problems of multidimensional algebraic geometry (duality theory, for example). More of that, for multidimensional local fields it is possible to construct an exact analogy of classical class field theory. This theory gives a full description of Abelian extensions in terms of higher \(K\)-functors introduced by J. Milnor [Invent. Math. 9, 318–344 (1970; Zbl 0199.55501)]. This aspect of the theory of local fields was discovered independently by K. Kato and developed in his papers [Proc. Jap. Acad., Ser. A 53, 140–143 (1977; Zbl 0436.12011); ibid. 54, 250–255 (1978; Zbl 0411.12013); J. Fac. Sci., Univ. Tokyo, Sect. I A 26, 303–376 (1979; Zbl 0428.12013); ibid. 27, 603–683 (1980; Zbl 0463.12006); ibid. 29, 31–43 (1982; Zbl 0503.12004)].

The paper under review contains a detailed exposition of local class field theory for the local fields of positive characteristic. It consists of four parts. The notion of local field is introduced in Part 1. The higher \(K\)-groups \(K_ m\!^{top}(K)\) are defined and computed in Part 2. Part 3 is devoted to the construction of Kummer and Artin-Schreier dualities. The fundamental almost-isomorphism between \(\text{Gal}(K^{ab}/K)\), where \(K\) is a local field of dimension \(n\), and the \(K\)-group \(K_ n\!^{top}(K)\) is established in Part 4. If \(\dim K=1\) then \(K_ n\!^{top}(K)\) equals the multiplicative group \(K^{\times}\) and we have the usual class field theory. If \(n=0\) then \(K\) is a finite field and \(K_ n\!^{top}(K)={\mathbb Z}\), and the Galois group \(\text{Gal}(K^{ab}/K)\) is its completion. Some results, in particular a computation of the Brauer group, will be published in a separate paper.

The paper under review contains a detailed exposition of local class field theory for the local fields of positive characteristic. It consists of four parts. The notion of local field is introduced in Part 1. The higher \(K\)-groups \(K_ m\!^{top}(K)\) are defined and computed in Part 2. Part 3 is devoted to the construction of Kummer and Artin-Schreier dualities. The fundamental almost-isomorphism between \(\text{Gal}(K^{ab}/K)\), where \(K\) is a local field of dimension \(n\), and the \(K\)-group \(K_ n\!^{top}(K)\) is established in Part 4. If \(\dim K=1\) then \(K_ n\!^{top}(K)\) equals the multiplicative group \(K^{\times}\) and we have the usual class field theory. If \(n=0\) then \(K\) is a finite field and \(K_ n\!^{top}(K)={\mathbb Z}\), and the Galois group \(\text{Gal}(K^{ab}/K)\) is its completion. Some results, in particular a computation of the Brauer group, will be published in a separate paper.

##### MSC:

11S31 | Class field theory; \(p\)-adic formal groups |

11S70 | \(K\)-theory of local fields |

14G20 | Local ground fields in algebraic geometry |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

11T99 | Finite fields and commutative rings (number-theoretic aspects) |

14F22 | Brauer groups of schemes |