Resolution of a family of vector bundles

Question

I am studying this notes of Faltins http://www.mathe2.uni-bayreuth.de/stoll/lecture-notes/vector-bundles-Faltings.pdf and in the page 40 he give the following bold statement. If $E\rightarrow C\times S$ is a family of vector bundles being $C$ an algebraic projective smooth curve over an algebraically closed field $k$ of char $0$, then there exists a complex of locally free sheaves (over $S$) $E_{i}$, such that $E_{i}=0$ for all $ i\geq 2$ $$\mathcal{C}_{E}\equiv E_{0}\rightarrow E_{1}\rightarrow 0\rightarrow\cdots$$ This complex has the properties that for any morphism of $k$-schemes $f:Z\rightarrow S$ the formula $$\operatorname{R}^{i}\left(\left(\pi_{Z}\right)_{\ast}(Id\times f)^{\ast}E\right)= \operatorname{H}^{i}(f^{\ast}\mathcal{C}_{E})$$ being $\pi_{Z}:C\times Z\rightarrow Z$ the projection onto $Z$, holds. Faltings said that this result is in EGAIII but he doesn't give any further details. I assumme that this is a well known fact in algebraic geometry but can you tell me an exact reference to find it?

Answer 1

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 further assume $E_m=0$ for all $m>n$. If you do not want to read EGA, you can find a proof in many other places, like Mumford's Abelian varieties, as well as Curves on a surface. Look in the section on semicontinuity theorems.