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The functor $f_*$ does extend to a functor between categories of chain complexes and so it makes perfect sense to talk about the cohomology of $f_*\mathcal{C}^{\cdot}$. Be careful though, while we can view the object $f_*\mathcal{C}^{\cdot}$ as an element of the derived category (just apply the functor from chain complexes to the derived category), the ...

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This is an application of a more general fact. Let $X,Y,Y'$ be noetherian schemes, let $f:X\to Y$ be of finite type and separated, let $u:Y'\to Y$ be arbitrary, and let $X'=X\times_YY'$ be the base change, organized in the following commutative diagram: $$\require{AMScd} \begin{CD} X' @>{v}>> X\\ @VV{g}V @VV{f}V \\ Y' @>{u}>> Y \end{CD}$$ ...

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$\newcommand{\cF}{\mathcal F}$On the second page, out of $H^i(X',R^jg_*'(g'_*\cF))$ for $i+j=n+1$, you have two possible nonzero terms: $$E^{n+1,0}_2 = H^{n+1}(X',g_*'(g'_*\cF)),\quad E^{0,n+1}_2=H^{0}(X',R^{n+1}g_*'(g'_*\cF)).$$ (I think you might be forgetting about the second one) The first one has no more nonero arrows in any pages, so like you say $E^{... 0 This is a special case of a general fact from EGA. Let$f:X\to S$be a projective morphism and$F$a coherent sheaf on$X$flat over$S$. Then, there exists a complex of vector bundles$0\to E_0\to E_1\to\cdots$which computes the direct images after any base change of$S$. Further, if$n\$ is the maximum of the relative dimensions of all fibers, then we may ...

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