Nagata ring

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In commutative algebra, an N-1 ring is an integral domain whose integral closure in its quotient field is a finitely generated -module. It is called a Japanese ring (or an N-2 ring) if for every finite extension of its quotient field , the integral closure of in is a finitely generated -module (or equivalently a finite -algebra). A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, or a pseudo-geometric ring if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a prime ideal are N-2 rings). A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring,[1] but this concept is not used much.

Examples

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Fields and rings of polynomials or power series in finitely many indeterminates over fields are examples of Japanese rings. Another important example is a Noetherian integrally closed domain (e.g. a Dedekind domain) having a perfect field of fractions. On the other hand, a principal ideal domain or even a discrete valuation ring is not necessarily Japanese.

Any quasi-excellent ring is a Nagata ring, so in particular almost all Noetherian rings that occur in algebraic geometry are Nagata rings. The first example of a Noetherian domain that is not a Nagata ring was given by Akizuki (1935).

Here is an example of a discrete valuation ring that is not a Japanese ring. Choose a prime   and an infinite degree field extension   of a characteristic   field  , such that  . Let the discrete valuation ring   be the ring of formal power series over   whose coefficients generate a finite extension of  . If   is any formal power series not in   then the ring   is not an N-1 ring (its integral closure is not a finitely generated module) so   is not a Japanese ring.

If   is the subring of the polynomial ring   in infinitely many generators generated by the squares and cubes of all generators, and   is obtained from   by adjoining inverses to all elements not in any of the ideals generated by some  , then   is a 1-dimensional Noetherian domain that is not an N-1 ring, in other words its integral closure in its quotient field is not a finitely generated  -module. Also   has a cusp singularity at every closed point, so the set of singular points is not closed.

Citations

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References

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  • Akizuki, Y. (1935), "Einige Bemerkungen über primäre Integritätsbereiche mit teilerkettensatz", Proceedings of the Physico-Mathematical Society of Japan, 3rd Series, 17: 327–336
  • Bosch, Güntzer, Remmert, Non-Archimedean Analysis, Springer 1984, ISBN 0-387-12546-9
  • Danilov, V.I. (2001) [1994], "geometric ring", Encyclopedia of Mathematics, EMS Press
  • A. Grothendieck, J. Dieudonné, Eléments de géométrie algébrique, Ch. 0IV § 23, Publ. Math. IHÉS 20, (1964).
  • H. Matsumura, Commutative algebra ISBN 0-8053-7026-9, chapter 12.
  • Nagata, Masayoshi Local rings. Interscience Tracts in Pure and Applied Mathematics, No. 13 Interscience Publishers a division of John Wiley & Sons, New York-London 1962, reprinted by R. E. Krieger Pub. Co (1975) ISBN 0-88275-228-6
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