- Nagata ring
-
In commutative algebra, an integral domain A is called an N-1 ring if its integral closure in its quotient field is a finitely generated A module. It is called a Japanese ring (or an N-2 ring) if for every finite extension L of its quotient field K, the integral closure of A in L is a finitely generated A module (or equivalently a finite 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. 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.
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 PID or even a DVR 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 in (Akizuki 1935).
References
- Y. Akizuki Proc. Phys-Math Soc. Japan 17 (1935) 327–366.
- Bosch, Güntzer, Remmert, Non-Archimedean Analysis, Springer 1984, ISBN 0387125469
- V.I. Danilov (2001), "geometric ring", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/G/g044300.htm
- A. Grothendieck, J. Dieudonne, 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 0882752286
Categories:
Wikimedia Foundation. 2010.
