What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
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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 local-etale 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 $F$).
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$\begingroup$ 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? $\endgroup$– RajnishCommented Dec 29, 2012 at 5:38
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$\begingroup$ @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 rank-1 setting). $\endgroup$ Commented Dec 29, 2012 at 5:55
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$\begingroup$ 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. $\endgroup$– RajnishCommented Dec 30, 2012 at 0:00
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$\begingroup$ @Rajnish: If nothing is being done with it (beyond as an example of an immediate extension) then I recommend focusing on the discretely-valued 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 rank-1 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. $\endgroup$ Commented Dec 30, 2012 at 0:47