A form of cohomology and base change

Question

Let $f \colon X \to Y$ be a proper morphism of (Noetherian) schemes, $\mathcal{F} \in \mathop{Coh}(X)$. Let $i_Z \colon Z \hookrightarrow Y$ be a closed subscheme and take the inverse image $W := X \times_Y Z \overset{i_W}{\hookrightarrow} X$.

If we assume that $\dim X_y \leq k - 1$ for all $y \in Y \setminus Z$, then the sheaf $R^k f_{*}(\mathcal{F})$ is concentrated on $Z$ by flat base change for the open inclusion $Y \setminus Z \to Y$.

Is it true in this case that $$R^k f_{*}(\mathcal{F}) \cong i_{Z*} R^k (f|_W)_{*}(\mathcal{F}|_{W})$$ where $\mathcal{F}|_{W} = i_W^{*}(\mathcal{F})$?

I was not able to derive this from the standard results about base change, nor to find any counterexamples.

Special case: What I actually need is a rather special case of the former. Namely in the case I am insterested in:

• all objects are varieties over an algebraically closed field
• $X$, $Z$ and $W$ are smooth
• $\mathcal{F}$ is a line bundle
• $k = 1$
• the map $X \setminus W \to Y \setminus Z$ is an isomorphism.

It interesting to know something about the general case, though.

Answer 1

Here are more details for the second example from comments. Set $Y={\mathbb A}^2$, $Z=0$ (the origin), $X=$ blow-up of $Y$ and $Z$, so that $W$ is the exceptional divisor. Take $F_n=O_X(nW)$. Since the self-intersection of $W$ is $-1$, we see that $F_n/F_{n-1}=F_n|_W\simeq O_W(-n)$ (more properly, the direct image of this sheaf to $X$). Looking at the short exact sequence $$0\to F_{n-1}\to F_n\to F_n/F_{n-1}\to 0\qquad(n>0),$$ we see that $R^0f_*(F_n)=R^0f_*(F_{n-1})$, while $R^1f_*(F_n)$ is an extension $$0\to R^1f_*(F_{n-1})\to R^1f_*(F_n)\to R^1f_*(F_n/F_{n-1})\to 0.$$ The right term of this exact sequence is $i_{Z*}R^1f_*(F_n|_W)$, so your condition fails as long as $R^1f_*(F_{n-1})\ne0$. But this happens for $n>2$ (again, follows from the same sequence).