Analytically Reducible

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.

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. 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''  is the formal power series ring over ''K'' in 2 variables. Let ''R'' be the subring of ''K''  generated by ''x'', ''y'', and the elements ''z''_''n'' and localized at these elements, where
:$w=\sum_ a_mx^m$ is transcendental over ''K''(''x'')
:$z_1=(y+w)^2$
:$z_=(z_1-(y+\sum_a_mx^m)^2)/x^n$.
Then ''R'' 'X''(''X'' ²–''z''₁) is a normal Noetherian local ring that is analytically reducible.

References

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*{{Citation , last1=Zariski , first1=Oscar , author1-link=Oscar Zariski , last2=Samuel , first2=Pierre , author2-link=Pierre Samuel , title=Commutative algebra. Vol. II , origyear=1960 , publisher=Springer-Verlag , location=Berlin, New York , isbn=978-0-387-90171-8 , mr=0389876 , year=1975

Commutative algebra