TITLE: What is an example of a smooth variety over a finite field F_p which does not embed into a smooth scheme over Z_p?
QUESTION [15 upvotes]: Such an example of course could not be projective and would not itself lift to Z_p. The context is that one can compute p-adic cohomology of a variety X over a finite field F_p via the cohomology of an embedding of X into a smooth Z_p scheme. 
This is similar in spirit to my questions here and here (but a different question than the second link).

REPLY [7 votes]: A theorem of Wlodarczyk in "Embedding varieties in toric varieties" says that any smooth variety such that any two points are contained in an affine open set can be embedded in a smooth toric variety. Toric varieties can be lifted to Z_p so any variety over F_p with the above property can be embedded in a smooth scheme over Z_p.
Unfortunately, not all smooth varieties have this property; the example in Hartshorne of a smooth proper 3-fold which is not projective appears to be one where this fails (though for suitable choices these could lift to Z_p).