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**Record on an introductory lecture on the works of Kazuya Kato.**
*(Japanese)*
Zbl 0711.11044

The work of Kazuya Kato, a prize winner of the Mathematical Society of Japan 1988 Spring, is described. Kato has been the central figure in the development of higher class field theory, along with S. Bloch, A. Parshin, Shunji Saito, among others.

Let k be a finitely generated field over a prime field. The aim of higher class field theory is to describe the Galois group, \(Gal(k^{ab}/k)\), of the maximal abelian extension \(k^{ab}\) of k and its number theory in terms of the ground field k. Kato’s approach is to bring in K-theoretic considerations to the problem.

Kato first considered local class field theory in higher dimension, and the results can be summarized as follows: Theorem L. Let k be a local field of dimension n (\(\geq 0)\). Then there is a homomorphism (canonical), which is almost an isomorphism \(K_ n(k)\to Gal(k^{ab}/k)\) where \(K_ n(k)\) is the Milnor K-group of k.

The proof of Theorem L rests on the following Key Lemma. Let k be as in Theorem L. Then there is an isomorphism (canonical) \(H^{n+1}(k,{\mathbb{Q}}/{\mathbb{Z}}(n))\cong {\mathbb{Q}}/{\mathbb{Z}}\) where the cohomology is Galois cohomology.

For higher global class field theory, Kato in collaboration with Shunji Saito obtained the following beautiful result.

Theorem G. Let X be a projective integral scheme over \({\mathbb{Z}}\) and let k be its function field. Then there is a homomorphism (canonical) \((*)\quad C_ X\to Gal(k^{ab}/k).\) here \(C_ X\) is an abelian group defined by \(C_ X=\lim_{I} H^ n(X,K_ n({\mathcal O}_ X,I))\) where \(H^ n(X,)\) is Zariski cohomology, and I runs over non-zero coherent \({\mathcal O}_ X\)- ideals.

In particular, if \(char(k)=0\), then (*) is an isomorphism, and if \(char(k)=p\neq 0\), (*) is almost an isomorphism with possible difference at most \({\mathbb{Z}}\) or \({\hat {\mathbb{Z}}}\). Corollary. \(CH_ 0(X)\) is finitely generated.

Several applications of higher class field theory are also described, e.g., p-adic Hodge-Tate theory, Fontaine-Messing theory. Higher ramification theory is touched briefly at the end: proof in general case by higher class field theoretic consideration.

Let k be a finitely generated field over a prime field. The aim of higher class field theory is to describe the Galois group, \(Gal(k^{ab}/k)\), of the maximal abelian extension \(k^{ab}\) of k and its number theory in terms of the ground field k. Kato’s approach is to bring in K-theoretic considerations to the problem.

Kato first considered local class field theory in higher dimension, and the results can be summarized as follows: Theorem L. Let k be a local field of dimension n (\(\geq 0)\). Then there is a homomorphism (canonical), which is almost an isomorphism \(K_ n(k)\to Gal(k^{ab}/k)\) where \(K_ n(k)\) is the Milnor K-group of k.

The proof of Theorem L rests on the following Key Lemma. Let k be as in Theorem L. Then there is an isomorphism (canonical) \(H^{n+1}(k,{\mathbb{Q}}/{\mathbb{Z}}(n))\cong {\mathbb{Q}}/{\mathbb{Z}}\) where the cohomology is Galois cohomology.

For higher global class field theory, Kato in collaboration with Shunji Saito obtained the following beautiful result.

Theorem G. Let X be a projective integral scheme over \({\mathbb{Z}}\) and let k be its function field. Then there is a homomorphism (canonical) \((*)\quad C_ X\to Gal(k^{ab}/k).\) here \(C_ X\) is an abelian group defined by \(C_ X=\lim_{I} H^ n(X,K_ n({\mathcal O}_ X,I))\) where \(H^ n(X,)\) is Zariski cohomology, and I runs over non-zero coherent \({\mathcal O}_ X\)- ideals.

In particular, if \(char(k)=0\), then (*) is an isomorphism, and if \(char(k)=p\neq 0\), (*) is almost an isomorphism with possible difference at most \({\mathbb{Z}}\) or \({\hat {\mathbb{Z}}}\). Corollary. \(CH_ 0(X)\) is finitely generated.

Several applications of higher class field theory are also described, e.g., p-adic Hodge-Tate theory, Fontaine-Messing theory. Higher ramification theory is touched briefly at the end: proof in general case by higher class field theoretic consideration.

Reviewer: N.Yui

### MSC:

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

11R37 | Class field theory |

19F05 | Generalized class field theory (\(K\)-theoretic aspects) |

01A70 | Biographies, obituaries, personalia, bibliographies |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11-03 | History of number theory |

11S25 | Galois cohomology |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

14-03 | History of algebraic geometry |

19D45 | Higher symbols, Milnor \(K\)-theory |