Nagata ring explained
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, 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 .
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 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
References
- Bosch, Güntzer, Remmert, Non-Archimedean Analysis, Springer 1984,
- A. Grothendieck, J. Dieudonné, Eléments de géométrie algébrique, Ch. 0IV § 23, Publ. Math. IHÉS 20, (1964).
- H. Matsumura, Commutative algebra, 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)
External links
- http://stacks.math.columbia.edu/tag/032E