Finite complex computing cohomology in a proper flat family
= Finite complex computing cohomology in a proper flat family
For a proper morphism $X\to\operatorname{Spec}A$ with $A$ Noetherian and an $A$-flat coherent sheaf $\mathcal F$, locally on the base there is a bounded complex of finite free modules $K^\bullet$ such that
$$
H^p(X,\mathcal F\otimes_AM)\cong H^p(K^\bullet\otimes_AM)
$$
naturally for every $A$-module $M$.