Restriction Map in Cohomology

Motivated by the instances of Grothendieck’s generalized Hodge conjecture which have a purely algebraic expression, we study the restriction map in cohomology from a smooth projective variety X

to an affine U. Starting from X being defined over an affine S smooth over ℤ, we consider de Rham cohomology over a dense set of closed points in S. Then the restriction is ‘controlled’ by the global differential forms on X. When we go p-adically we prove the ’same’ result with values in a certain separated quotient of Bhatt-Scholze prismatic cohomology which maps to the separated quotient of derived p-complete de Rham cohomology.

We construct examples which show that one needs an extra assumption to lift this control to the whole cohomology.

Caro-D’Addezio showed that for derived p-complete de Rham cohomology modulo p-torsion it is controlled by H1 of differential forms in addition. We lift this to prismatic cohomology modulo I-torsion, which also has a corollary on étale cohomology of the rigid analytic fibre.