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 if for every
finite extension L of its quotient field K, the integral closure of A in L is a finitely generated 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, 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.

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 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.
Here is an example of a discrete valuation ring that is not a Japanese ring. Choose a prime p and an infinite degree field extension K of a characteristic p field k, such that Kp⊆k. Let the discrete valuation ring R be the ring of formal power series over K whose coefficients generate a finite extension of k. If y is any formal power series not in R then the ring R is not an N−1 ring so R is not a Japanese ring.
If R is the subring of the polynomial ring k in infinitely many generators generated by the squares and cubes of all generators, and S is obtained from R by adjoining inverses to all elements not in any of the ideals generated by some xn, then S 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 S-module. Also S has a cusp singularity at every closed point, so the set of singular points is not closed.