Let $f: X \rightarrow B$ be a smooth family of complex varieties ($X$ and $B$ are also smooth).
If $f$ is proper, then the direct image of the relative de Rham complex, $Rf_*\Omega_{X/B}^{\bullet}$, is a complex of finite-rank vector bundles that computes the (singular) cohomology of the fibers of $f$ in the following sense:
(*) The fiber at $b \in B$ of $R^kf_*\Omega_{X/B}^{\bullet}$ is isomorphic to $\mathbb{H}^k(f^{-1}(b), \Omega_{X/B}^{\bullet}|_{f^{-1}(b)})$.
(cf Voisin volume I, Thm 10.10).
What if $f$ is affine and not proper? Is the same statement (*) true? The reason I ask is because of something I read in Etingof's notes on D-modules here: http://www-math.mit.edu/~etingof/dmodwien.pdf
In Example 2.6 of that link, Etingof seems to assert that the statement holds for the quotient map $\text{SL}_2(\mathbb{C}) \rightarrow \mathbb{P}^1$ (which has affine fibers). Have I misunderstood?