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.

Template:Harvtxt 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. Template:Harvs 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 Template:Brackets is the formal power series ring over K in 2 variables. Let R be the subring of K Template:Brackets generated by x, y, and the elements z_n and localized at these elements, where

$w=\sum _{m>0}{a}_m{x}^m$ is transcendental over K(x)

$z_1=(y+w{)}^2$

$z_{n+1}=(z_1-(y+\sum _{0<m<n}{a}_m{x}^m{)}^2)/x^n$.

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

References

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