# When is the henselization ('the smallest etale neighbourhood') of the intersection of locally closed subvarieties $Z_1,Z_2\subset X$ isomorphic to the product of the henselizations of $Z_i$ over the one of $Z_1\cup Z_2$?

For a locally closed subscheme $Z\subset X$ (I am interested in the case when $X$ is a variety) one can consider its henselization in $X$ i.e. the 'smallest' pro-etale morphism $U\to X$ such that $Z$ lifts to $U$. We denote the corresponding scheme by $Z_h$.

Now, for $Z_1,Z_2\subset X$ one certainly has a canonical morphism $(Z_1\cap Z_2)_h\to Z_{1h}\times_{(Z_1\cup Z_2)_h} Z_{2h}$. My question is: when can one be sure that this morphism is pro-open? An isomorphism?

It seems that Theorems 2.4 and 2.5 of GRECO S.,STRANO R., Quasi-coherent sheaves over affine hensel schemes (http://www.ams.org/journals/tran/1981-268-02/S0002-9947-1981-0632537-7/S0002-9947-1981-0632537-7.pdf) yield that we have an isomorphism when $X$ and $Z_i$ are affine. I am interested in the case when $Z_i$ are affine, but $X$ is (smooth) projective. So, I have the following questions.

1. What is currently known about this setting? Actually, I am somewhat confused by Theorems 2.4 and 2.5 mentioned, since their assumptions coincide, whereas their conclusions are very much distinct.

2. It seems that the argument of Greco and Strano could be carried over to my situation if instead of AIC rings one considers those (irreducible?) pro-varieties whose fraction fields are algebraically closed (see also my previous question Do etale neighhbourhoods of a subvariety descend along base field extensions; does normalization commute with etale base change?). Could this be true, or do I miss something?

3. Whom could I ask about these matters (in a letter?)? Greco and Strano themselves are not very young at the moment.

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Greco and Strano are not very young, but they are not decrepit either (Greco recently turned 70, Strano is younger). – Angelo Nov 17 '11 at 10:45
The (main) question is whether they are willing to answer my letters.:) Possibly, I will write them. – Mikhail Bondarko Nov 17 '11 at 10:53
Well, I'm 75, and I wouldn't think of ignoring an inquiry about a paper of mine. – Lubin Nov 17 '11 at 16:51
Possibly, responding to letters does not depend very much on age. Still, the paper was written 30 years ago. – Mikhail Bondarko Nov 17 '11 at 21:34
Hi Mikhail - I've fixed the formula by adding backticks. The subscript underscore was being interpreted as 'turn stuff italic', and then MathJax got confused. – David Roberts Nov 18 '11 at 6:24