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An Example of Normal Local Ring which is Analytically Ramified

Published online by Cambridge University Press:  22 January 2016

Masayoshi Nagata
Masayoshi Nagata*
Affiliation:
Mathematical Institute, Kyoto University
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Previously the following question was offered by Zariski [6]:

Is any normal Noetherian local ring analytically irreducible?

In the present note, we will construct a counter-example against the question.

TERMINOLOGY. A ring (integrity domain) means always a commutative ring (integrity domain) with identity. A normal ring is an integrity domain which is integrally closed in its field of quotients. When 0 is an integrity domain, the integral closure of 0 in its field of quotients is called the derived normal ring of 0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 9 , October 1955 , pp. 111 - 113
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

References

[1] Chevalley, C., On the theory of local rings, Ann. of Math., vol. 44 (1943), pp. 690–708.Google Scholar
[2] Cohen, I. S., On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc., vol. 59 (1946), pp. 54–106.Google Scholar
[3] Nagata, M., Note on integral closures of Noetherian domains, Memoirs Kyoto, Ser. A., vol. 28 (1953), pp. 121–124.Google Scholar
[4] Nagata, M., On the theory of Henselian rings, II, Nagoya Math. Journ., vol. 7 (1954), pp. 1–19.Google Scholar
[5] Nagata, M., Basic theorems on general commutative rings, to appear in Memoirs Kyoto, Ser. A.Google Scholar
[6] Zariski, O., Analytical irreducibility of normal varieties, Ann. of Math., vol. 49 (1948), pp. 352–361.Google Scholar