Parshin, A. N. Local class field theory. (Russian) Zbl 0535.12013 Tr. Mat. Inst. Steklova 165, 143-170 (1984). By 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. Cited in 9 ReviewsCited in 24 Documents 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 Keywords:local field of arbitrary dimension; multidimensional algebraic geometry; duality theory; analogy of classical class field theory; description of Abelian extensions in terms of higher K-functors; Artin-Schreier dualities; fundamental almost-isomorphism Citations:Zbl 0302.14005; Zbl 0358.14012; Zbl 0199.55501; Zbl 0436.12011; Zbl 0411.12013; Zbl 0428.12013; Zbl 0463.12006; Zbl 0503.12004 PDFBibTeX XMLCite \textit{A. N. Parshin}, Tr. Mat. Inst. Steklova 165, 143--170 (1984; Zbl 0535.12013)