Nagata ring
In commutative algebra, an N-1 ring is an integral domain [math]\displaystyle{ A }[/math] whose integral closure in its quotient field is a finitely generated [math]\displaystyle{ A }[/math]-module. It is called a Japanese ring (or an N-2 ring) if for every finite extension [math]\displaystyle{ L }[/math] of its quotient field [math]\displaystyle{ K }[/math], the integral closure of [math]\displaystyle{ A }[/math] in [math]\displaystyle{ L }[/math] is a finitely generated [math]\displaystyle{ A }[/math]-module (or equivalently a finite [math]\displaystyle{ A }[/math]-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 (Danilov 2001), 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 [math]\displaystyle{ p }[/math] and an infinite degree field extension [math]\displaystyle{ K }[/math] of a characteristic [math]\displaystyle{ p }[/math] field [math]\displaystyle{ k }[/math], such that [math]\displaystyle{ K^p\subseteq k }[/math]. Let the discrete valuation ring [math]\displaystyle{ R }[/math] be the ring of formal power series over [math]\displaystyle{ K }[/math] whose coefficients generate a finite extension of [math]\displaystyle{ k }[/math]. If [math]\displaystyle{ y }[/math] is any formal power series not in [math]\displaystyle{ R }[/math] then the ring [math]\displaystyle{ R[y] }[/math] is not an N-1 ring (its integral closure is not a finitely generated module) so [math]\displaystyle{ R }[/math] is not a Japanese ring.
If [math]\displaystyle{ R }[/math] is the subring of the polynomial ring [math]\displaystyle{ k[x_1, x_2, ...] }[/math] in infinitely many generators generated by the squares and cubes of all generators, and [math]\displaystyle{ S }[/math] is obtained from [math]\displaystyle{ R }[/math] by adjoining inverses to all elements not in any of the ideals generated by some [math]\displaystyle{ x_n }[/math], then [math]\displaystyle{ S }[/math] 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 [math]\displaystyle{ S }[/math]-module. Also [math]\displaystyle{ S }[/math] 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, https://www.jstage.jst.go.jp/article/ppmsj1919/17/0/17_0_327/_article
- Bosch, Güntzer, Remmert, Non-Archimedean Analysis, Springer 1984, ISBN:0-387-12546-9
- Hazewinkel, Michiel, ed. (2001), "geometric ring", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=G/g044300
- 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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