Nagata ring
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In commutative algebra, an integral domain A is called an N-1 ring if its integral closure in its quotient field is a finite 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 finite A module. 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, but this concept is not used much.)
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 Y. Akizuki (1935).
[edit] References
- Y. Akizuki Proc. Phys-Math Soc. Japan 17 (1935) 327-366.
- V.I. Danilov, "geometric ring" SpringerLink Encyclopaedia of Mathematics (2001)
- 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
