That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R, \mathfrak m, \kappa= R/\mathfrak m )$ be a local ring with max ideal $m$.
We can obtain two new rings $R^h$ (the Henselization) and $\widehat{R}_m$ the completion wrt $m$. Consider $R$ as a stalk of a nice enough scheme $S$ we can use these two constructions to obtain new new objects stalkwise: $S^h$ (here we have to differ between strict and "weak" Henselization) and the completion $\widehat{S}$. (recall $\widehat{S}$ is not more a scheme but a ringed space: localizations and completions not behave well to each other).
in would like to compare the main differences & (dis)advantages of completions & Henselizations from viewpoint of commutative algebra and (as well possible) geometric intuition.
The main motivation is that I often read comments like "in pracise it's often nicer to work with Henselisations than with completions" in order to study the ring $R$ itself.
Question: Could anybody point out what are the advantages making Henselizations from certain viewpoint nicer to handle with then with completions?
In many comments the hand weavy arguments apearing in this context are like $\widehat{R}_m$ is much "bigger" that $R^h$ making it not "so easy handable like $R^h$". Could anybody bring more light in this formulation? When is mean by "bigger" (the added limits ofCauchy sequences I guess) but much more intersting what makes $R^h$ more "handable"?
The only point that I found out is that $Frac(R)=K \subset K^h$ stays algebraic and in many situations even finite. Is $R \to R^h$ also a finite $R$-module. In general that's not true for completions $ R \to \widehat{R}_m$.
Is this the only point making $R^h$ more handable than $ R \to \widehat{R}_m$?
What can we say about the geometrical part? The completion $\widehat{S}$ gives in certain way "analytic structure" to an (algebraic) scheme $S$ (very hand weavy; I know). About what kind of "geometry" one can think when one consider a henselization of a scheme (as for completion: local ring wise)? Some sources refer to "etale topology". It's a starting point of a hige mashinery cumulating in stack theory.
Is there an geometric intuition how one can draw comparisons between endowings of $S$ "analytical structure" (as for completions) and with "etale topology" for $S^h$?
I know that there are a couple of questions here with similar titles (eg henselization and completion , Completion versus henselization. , comparison of completion and Henselization in class field theory ) but none of them deal with question of pure comparison of tqo constructions in the way I explained above.
