Finite complex computing cohomology in a proper flat family

For a proper morphism $X\to \mathrm{Spec}A$ 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 thatnaturally for every $A$-module $M$.

In a proper flat family with a coherent sheaf, the fiber dimension $s\mapsto {\dim }_{\kappa (s)}{H}^p(X_s,{\mathcal{F}}_s)$ is upper semicontinuous, and the fiberwise Euler characteristic is locally constant.