# Masayoshi Nagata

### Quick Info

Born
9 February 1927
Obu, Aichi Prefecture, Japan
Died
27 August 2008
Kyoto, Japan

Summary
Masayoshi Nagata was a Japanese mathematician who worked in the field of commutative algebra.

### Biography

Masayoshi Nagata's father ran a small factory in Obu and, for several years, served on the town council. In 1933, when he was six years old, Masayoshi began his schooling at the local primary school. The standard education for Japanese children at this time involved six years in primary school, followed by a number of years in middle school and finally three years in high school as a preparation for university studies. He had only just begun his studies in middle school in the nearby town of Kariya in 1939 when World War II started. Particularly as the years went by his education became disrupted with pupils being sent from the schools into the countryside to work in factories and on the land. It was also a difficult time with food shortages and bombing raids. In 1944 Nagata completed his middle school education and began his studies at the Eighth High School in Nagoya where he was a boarder. Certainly at this stage he did not think of himself as a mathematical star [1]:-
Though I did not think I was any good at mathematics, I liked mathematics only. ... Several of my friends seemed better at mathematics than I was. ... I do not think that I am a late bloomer. I was a usual boy who liked to think of problems in mathematics.
Immediately after the war ended in 1945 food shortages became even worse and high inflation made life incredibly difficult. Nagata entered Nagoya Imperial University in April 1947 and there he studied mathematics under Tadasi Nakayama. It was a few years before that, in 1942, that Nakayama had been appointed to Nagoya Imperial University, becoming a professor two years later. He was producing outstanding research on infinite dimensional algebras and he advised Nagata in his studies in algebra. Nagata graduated in 1950 but he had already undertaken research in algebra and, resulting from this, had a number of papers in print: (with Noboru Ito) Note on groups of automorphisms (1949), On the structure of complete local rings (1950), and On the theory of semi-local rings (1950). In the two ring theory papers he generalised results already obtained for Noetherian rings to rings which are not necessarily Noetherian. In doing so he answered a open question by I S Cohen.

Six months after graduating, Nagata was appointed as a research assistant in the Faculty of Science of Nagoya University. Despite the name change, this was the same university at which he had studied - Nagoya Imperial University had been renamed Nagoya University in 1947. He held this position for three years, then moved to Kyoto University in May 1953 when he was appointed as an instructor. During these three years he had published many articles on ring theory and valuations (and one on group theory), three articles appearing in each of 1951, 1952 and 1953. In Kyoto he joined the Kyoto School of Algebraic Geometry which was being developed by Yasuo Akizuki and attracting many talented young mathematicians. For example Heisuke Hironaka, who won a Fields Medal in 1970, was in his final undergraduate year at Kyoto when Nagata arrived and he continued to work in the Kyoto School of Algebraic Geometry for the following four years.

Nagata was promoted to associate professor at Kyoto University in 1957 and, in February 1963, was appointed to the Chair of Algebra at Kyoto succeeding Yasuo Akizuki. The authors of [2] write about his mathematical contributions:-
Nagata played outstanding roles, especially in the 1950s and 1960s, in the development of commutative algebra and algebraic geometry. Many of his contributions were through a result of producing crucial counterexamples.
It is worth commenting at this point that Nagata's skill in producing counterexamples led to his fellow mathematicians giving him the nicknamed "Mr Counterexample". The authors of [2] continue:-
The most famous among [these counterexamples] is a nonfinitely generated ring of invariants for a group acting on a polynomial ring, thereby negatively solving Hilbert's 14th problem in 1958.
In fact Nagata announced his negative solution to Hilbert's 14th problem in his invited lecture On the fourteenth problem of Hilbert at the International Congress of Mathematicians held in Edinburgh, Scotland, in August 1958. The authors of [2] continue:-
Another [counterexample] is a complete nonsingular 3-dimensional algebraic variety that cannot be embedded in any projective space. ... A series of papers in the late 1950s on algebraic geometry over Dedekind domains laid the foundation for later developments of algebraic geometry in terms of schemes. The concept of the Henselization of rings, developed in a series of papers in the 1950s, turned out to be fundamental for algebraic spaces and étale topology. The completion of algebraic varieties - that is, embedding of algebraic varieties as open subvarieties of complete varieties - published in his paper in 1962, remains one of the basic techniques in algebraic geometry.
As well as this outstanding research contribution, Nagata is famed for his outstanding books. With Yosikazu Nakai, he published Algebraic geometry (Japanese) in 1957. T Kambayashi writes:-
If you can read Japanese, know some algebraic geometry already and have enough nerves not to mind the five-page errata, then you will greatly enjoy reading this book.
Nagata's most famous book, Local rings, appeared in 1962. R C Hartshorne begins a review as follows:-
This authoritative work, by an expert in the field, gives a complete up-to-date account of the theory of local rings, starting from the beginning. The early part of the book contains basic results of commutative algebra, often with new proofs. The latter part of the book treats the advanced theory of local rings, reuniting most of the results of the author's many research papers, and containing some new results, notably concerning pseudo-geometric rings.
It is worth noting that the Noetherian rings which Nagata calls 'pseudo-geometric rings' in this book are now known as 'Nagata rings'. A number of Nagata's books are based of lecture courses he gave at different institutions. Lectures on the fourteenth problem of Hilbert (1965) resulted a course of lectures he gave at the Tata Institute of Fundamental Research, Bombay, in 1964. On flat extensions of a ring (1971) is a book in which the lectures he gave in the summer of 1970 at the University of Montreal as published. Polynomial rings and affine spaces (1978) is a written version of lectures given by Nagata in 1977 at Northern Illinois University.

Some of his books came about because of a research visits but were not lecture notes. On automorphism group of k[x, y] (1972) arose as a result of Nagata's visit to Purdue University as he explains in the Preface:-
During my stay at Purdue University in 1970, I discussed the problem of investigating the structure of the automorphism group of polynomial rings over a field. The one variable case is nearly obvious. Although it was known that in the two variable case the group of automorphisms is generated by automorphisms of the type $(x, y) \mapsto (x + f (y), y)$ and by linear transformations, the known proofs were not easy. Furthermore, the three variables case was completely open. For the two variables case, I could give a new proof. For the case of more variables, I understand that there are many difficulties in studying the structure of the automorphism group. I give here some comments on the general case.
Finally we mention his texbook Theory of commutative fields published in Japanese in 1985 and in English translation by the American Mathematical Society in 1993.

In [4] Nagata's other contributions to mathematics are mentioned:-
The mathematical influence of Masayoshi Nagata is enormous not only through his research works but also through his contributions to the domestic and international mathematical communities. He played a quite active role in the mathematical community in Japan by serving as trustee of the Mathematical Society of Japan and as a member of the Science Council of Japan. At the International Mathematical Union, he served as a member of the Executive Committee between 1975 and 1978 and as vice president from 1979 to 1982.
He was awarded the Chunichi Cultural Prize (1961), the Matsunaga Prize (1970) and the Japan Academy Prize (1986). The work which led to the award of the Academy Prize is discussed in [2]. He was honoured with the Order of the Sacred Treasure, Gold and Silver Star, in November 1998.

Among Nagata's students, we must mention Shigefumi Mori who studied for his doctorate with Nagata at Kyoto between 1975 and 1978. Mori was awarded a Fields Medal in 1990 at the International Congress of Mathematicians held in Kyoto. In fact Mori was appointed to a chair at Kyoto University in 1990, the year in which his former supervisor Nagata retired. Nagata died of cancer at the age of 81.

### References (show)

1. B Yandell, The honors class: Hilbert's problems and their solvers (A K Peters, 2002).
2. M Maruyama, M Miyanishi, S Mori and T Oda, Masayoshi Nagata (1927-2008), Notices Amer. Math. Soc. 56 (1) (2009), 58.
3. H Matsumura, The award of the Academy Prize to Professor Masayoshi Nagata (Japanese), Sugaku 38 (4) (1986), 342-344.
4. M Miyanishi, Masayoshi Nagata (1927-2008) and his mathematics, Kyoto J. Math. 50 (4) (2010), 645-659.