Hi,everyone.
I want to know that how to compute the henselization of some simple rings, for example: $k[x]_{(x)}$ and $R[X]_{(X)}$ where $k$ is a field and $R$ is a excellent DVR.
thank you very much!
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2$\begingroup$ I have the impression that in the first case, you can identify it with the subring of formal power series $p\in k[[x]]$ which are algebraic in the sense that $f(p)=0$ for some polynomial $f$. $\endgroup$– Donu ArapuraFeb 10, 2012 at 9:48
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2 Answers
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In both cases, the henselization $A^h$ of your ring $A$ is its algebraic closure in its completion $\hat{A}$.This follows for instance from Artin approximation (Algebraic approximation of structures over complete local rings, Publications Mathématiques de l'IHÉS, 36, 1969, p. 23-58): any system of polynomial equations that has a solution in $\hat{A}$ has a solution in $A^h$ (and you may find a solution as close as you want to the original one).
answered Feb 10, 2012 at 21:36
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Perhaps a better source for this result is Nagata's 1962 book, "Local Rings". Chapter VII begins with a section on constructing the henselization of a local ring and its properties. For a normal domain R the above result (note though that the separable algebraic closure, not the algebraic closure must be used in char p) follows since his definition is to take the separable closure of R in the algebraic closure of the function field k(R), intersect it with the fixed field for the decomposition group of the maximal ideal of R, and then localize the resulting subring.
