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I wonder if the base change for a flat complex of coherent sheaves works. Namely, let $f : X \rightarrow Y$ be a proper morphism of varieties over $k$. Let $\mathcal{F}^{\bullet}$ be a bounded complex of coherent sheaves on $X$ such that all the cohomology sheaves $\mathcal{H}^{\bullet}(\mathcal{F}^{\bullet})$ are flat over $Y$. Assume that the dimension of $\mathcal{H}^i (\mathrm{R} \Gamma(\mathcal{F}^{\bullet} \otimes k(y))$ is constant when $y$ varies in $Y$. Do we have a base-change formula:

$$ \mathcal{H}^i (\mathrm{R} \Gamma(\mathcal{F}^{\bullet} \otimes k(y)) = \mathcal{H}^i (\mathrm{R}f_* (\mathcal{F}^{\bullet})) \otimes k(y), $$ for all $y \in Y$ ?

Many thanks in advance.

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This write-up of Keerthi Madapusi Pera possibly answers your question (apologies if not): – Piotr Achinger Jul 21 '13 at 17:00
The comments to this question, especially ulrich's reference to EGA III, should be helpful: – Keerthi Madapusi Pera Jul 21 '13 at 17:38
@ Keerthi Madapusi Pera : Ah great, thanks! I guess that if the upper semi-continuity works then the base-change should also be fine... hopefully! – Libli Jul 22 '13 at 21:55

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