Nagata ring

In commutative algebra, an ${\displaystyle N-1}$ ring is an integral domain ${\displaystyle A}$ whose integral closure in its quotient field is a finitely generated ${\displaystyle A}$ module. It is called a Japanese ring (or an ${\displaystyle N-2}$ ring) if for every finite extension ${\displaystyle L}$ of its quotient field ${\displaystyle K}$, the integral closure of ${\displaystyle A}$ in ${\displaystyle L}$ is a finitely generated ${\displaystyle A}$-module (or equivalently a finite ${\displaystyle A}$-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 ${\displaystyle N-2}$ rings.) A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring (Danilov 2001) harv error: no target: CITEREFDanilov2001 (help), but this concept is not used much.

Examples[]

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 ${\displaystyle p}$ and an infinite degree field extension ${\displaystyle K}$ of a characteristic ${\displaystyle p}$ field ${\displaystyle k}$, such that ${\displaystyle K^{p}\subseteq k}$. Let the discrete valuation ring ${\displaystyle R}$ be the ring of formal power series over ${\displaystyle K}$ whose coefficients generate a finite extension of ${\displaystyle k}$. If ${\displaystyle y}$ is any formal power series not in ${\displaystyle R}$ then the ring ${\displaystyle R[y]}$ is not an ${\displaystyle N-1}$ ring (its integral closure is not a finitely generated module) so ${\displaystyle R}$ is not a Japanese ring.

If ${\displaystyle R}$ is the subring of the polynomial ring ${\displaystyle k[x_{1},x_{2},...]}$ in infinitely many generators generated by the squares and cubes of all generators, and ${\displaystyle S}$ is obtained from ${\displaystyle R}$ by adjoining inverses to all elements not in any of the ideals generated by some ${\displaystyle x_{n}}$, then ${\displaystyle S}$ is a 1-dimensional Noetherian domain that is not an ${\displaystyle N-1}$ ring, in other words its integral closure in its quotient field is not a finitely generated ${\displaystyle S}$-module. Also ${\displaystyle S}$ has a cusp singularity at every closed point, so the set of singular points is not closed.

References[]

• 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
• V.I. Danilov (2001) [1994], "geometric ring", Encyclopedia of Mathematics, EMS Press
• A. Grothendieck, J. Dieudonné, Eléments de géométrie algébrique Publ. Math. IHES, 20, section 23 (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

External links[]

• http://stacks.math.columbia.edu/tag/032E