What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
1 Answer
Same as its importance in commutative algebra. Just to be clear about the definition, for a valued field $K$ with valuation ring $R$, the henselization $K^{\rm{h}}$ is defined to be the valued extension Frac($R^{\rm{h}}$) for the henselization $R^{\rm{h}}$ of $R$ in the sense of commutative algebra (and $R^{\rm{h}}$ is equipped with a preferred valuation extending the one on $R$).
This satisfies good properties as if it were a "completion" of $K$ even though it is (separable) algebraic over $K$, and it can be "approximated" using localetale extensions of $R$; that is really the point. It satisfies Hensel's Lemma and every finite extension $F$ of $K^{\rm{h}}$ admits a unique valuation (necessarily henselian...) extending the one on $K^{\rm{h}}$ (with associated valuation ring that is the integral closure of $R^{\rm{h}}$ in $F$).

$\begingroup$ Thanks Ayanta. I am trying to get the idea about the Henselization and its important in valuation theory. Do you know any text book or any other material where I can read about Henselization? $\endgroup$– RajnishDec 29, 2012 at 5:38

$\begingroup$ @Rajnish: To answer your reference question in a useful way it would be helpful to know the reason you are specifically interested in this rather specialized aspect of valuation theory (especially beyond the rank1 setting). $\endgroup$ Dec 29, 2012 at 5:55

$\begingroup$ Thanks Ayanta. I was reading the extension of valuation and suddenly appears the henselization as an immediate extension. That makes kind of hard time to get the idea for me. $\endgroup$– RajnishDec 30, 2012 at 0:00

$\begingroup$ @Rajnish: If nothing is being done with it (beyond as an example of an immediate extension) then I recommend focusing on the discretelyvalued case (where it's the fixed field of a decomposition group at a place on a separable closure) and ignore the topic until you have a real need to work with it. But if something is being done with it (especially beyond the rank1 case) then what is that? Anyway, the theory of henselization of local rings beyond dvr's requires a lot of hard work to set up; Raynaud's "Anneaux locaux henseliens" Springer LNM 169 is on exactly this topic. $\endgroup$ Dec 30, 2012 at 0:47