I would like to study the cohomology of the Henselization $H_X(Z)$ of a closed subvariety $Z$ of a variety $X$. I would like the following facts to be true (and to make sense!:)).
a.) The motivic cohomology of $Z$ with finite coefficients (prime to the base field characteristic) is isomorphic to that of $H_X(Z)$; cf. On motivic cohomology with compact support b.) For the natural morphism $f:Z\to H_X(Z)$ I would like $Rf_\ast f^\ast$ to be isomorphic to the identity functor on a certain triangulated category of 'nice' (constructible??) complexes of etale sheaves $H_X(Z)$. I would not like to assume that $X$ is affine or that $Z$ is smooth (in particular, I need $Z$ being a smooth normal crossing divisor).
So, I have several questions.
- Could the statements above be true; what additional restrictions are needed?
- Which references could help me?
- What could be the plan in order to deduce the statements I need from the known facts? What are the main difficulties? In particular, (as far as I understand) Henselizations are 'very far from being of finite type' (in the non-affine situation). So, in order to define the cohomology of $H_X(Z)$ I should pass to certain limits and sheafify? How does one usually study $f^\ast$ and $Rf_\ast$ in such an 'infinite' setting?
- Actually, I would be satisfied by any sort of 'algebraic tubular neigbourhood' of $Z$ in $X$ such that a.) and b.) hold (and make sense!!) for it. Is the henselization more appropriate than other existing constructions here?