Analytically irreducible ring

In algebra, an analytically irreducible ring is a local ring whose completion has no zero divisors. Geometrically this corresponds to a variety with only one analytic branch at a point. (Zariski 1948) proved that if a local ring of an algebraic variety is a normal ring, then it is analytically irreducible. There are many examples of reduced and irreducible local rings that are analytically reducible, such as the local ring of a node of an irreducible curve, but it is hard to find examples that are also normal. Nagata (1958, 1962, Appendix A1, example 7) gave such an example of a normal Noetherian local ring that is analytically reducible.

Nagata's example

Suppose that K is a field of characteristic not 2, and K [[x,y]] is the formal power series ring over K in 2 variables. Let R be the subring of K [[x,y]] generated by x, y, and the elements z_n and localized at these elements, where

[math]\displaystyle{ w=\sum_{m\gt 0} a_mx^m }[/math] is transcendental over K(x)

[math]\displaystyle{ z_1=(y+w)^2 }[/math]

[math]\displaystyle{ z_{n+1}=(z_1-(y+\sum_{0\lt m\lt n}a_mx^m)^2)/x^n }[/math].

Then R[X]/(X ²–z₁) is a normal Noetherian local ring that is analytically reducible.

References

• Nagata, Masayoshi (1958), "An example of a normal local ring which is analytically reducible", Mem. Coll. Sci. Univ. Kyoto. Ser. A Math. 31: 83–85, http://projecteuclid.org/euclid.kjm/1250776950 
• Nagata, Masayoshi (1962), Local rings, Interscience Tracts in Pure and Applied Mathematics, 13, New York-London: Interscience Publishers 
• Zariski, Oscar (1948), "Analytical irreducibility of normal varieties", Ann. of Math., 2 49: 352–361, doi:10.2307/1969284 
• Zariski, Oscar; Samuel, Pierre (1975) [1960], Commutative algebra. Vol. II, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90171-8