As far as I understand (and tbh for my purposes), one of the main points of strict henselisation of a local ring is that it computes the stalk at a point of a scheme in the étale topology. In the smooth case using étale coordinates, these are rather easy to understand. However, I don't think I know how to do any sort of computation for any sort of singularity. My hope would be that the strict henselisation of complete intersections have a nice description, but I'd be interested in any class where we have a good description of them.
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$\begingroup$ I'm not sure what you want exactly. If a local ring $R$ is a quotient of a regular local ring $R = S/I$, then $S^{\mathrm{sh}}/IS^{\mathrm{sh}}$ is a strict Henselization of $R$, that is its isomorphic to $R^{\mathrm{sh}}$. See for instance stacks.math.columbia.edu/tag/05WS Are you assuming that your scheme is not locally embeddable, so being a complete intersection means that, after completion, it is a complete intersection in a regular local ring? $\endgroup$– Karl SchwedeCommented May 19, 2022 at 16:30
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$\begingroup$ @KarlSchwede my apologies, I was not aware of that. I think my question was just the hope that "things become easier at the stalk", but that doesn't seem to be the case. $\endgroup$– curious math guyCommented May 19, 2022 at 20:40
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