Henselization of a local ring

Question

When is it true that a local ring is embedded into its henselization? In the cases when it is true, it means that to check that a presheaf is separated in Nisnevich topology is the same as to check that it is separated in Zariski topology, am I right?

Answer 1

If $A$ is a local ring, its henselization $i:A\hookrightarrow A^h$ is always injective and even faithfully flat.

The rings $A$ and $A^h$ have the same dimension and share many properties :
$A$ is noetherian (resp. reduced, resp. a normal domain) $\iff$ $A^h$ is noetherian (resp. reduced, resp. a normal domain).

And if $A$ (and thus $A^h$ ) is noetherian we can add that $depth(A)= depth(A^h)$ and that :
$A$ is regular (resp. Cohen-Macaulay) $\iff $ $A^h$ is regular (resp. Cohen-Macaulay).

Notice that a complete local ring is henselian and that in general a local ring and its henselization have the same completion : the morphism $\hat i:\hat A \stackrel {\cong}{\to} \hat {A^h}$ is an isomorphism.

Edit: Bibliography
The main source is (surprise, surprise! ) EGA IV,4,§18 which contains proofs of most of the above.

Raynaud wrote a rather elementary (scheme theory not assumed) self-contained monograph Anneaux Locaux Henséliens .
It is short (129 pages) but packed with highly non trivial results like a complete proof of Zariski's Main theorem.
I felt a feeling of loss and nostalgy when reading in the Introduction that Raynaud thanked N. Bourbaki for having allowed him access to his [Bourbaki's] book "in preparation" (!) on étale algebras ...

For a recent presentation, one can look at Section 38 of De Jong and collaborators' ever growing (as of today: 3150 pages =24.4 Raynauds ) Stacks Project