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 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 (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[edit]

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 (1935).

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 Kpk. 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[y] is not an N−1 ring (its integral closure is not a finitely generated module) so R is not a Japanese ring.

If R is the subring of the polynomial ringk[x1,x2,...] 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.

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