We have met the following Grauert's theorem, say in Hartshorne's book.
Let $\pi: \mathcal X\to B$ be a propre flat morphism with $B$ reduced. Let $\mathcal F$ be a locally free coherent sheaf over $\mathcal X$. If for any (closed) points $b\in B$, $\dim H^i(X_b,\mathcal F|_{X_b})$ is constant, then $R^i\pi_*\mathcal F$ is locally free over $B$ and satisfies the base change: $R^i\pi_*\mathcal F|_b\to H^i (X_b,\mathcal F|_b)$ is an isomorphism.
I am wondering if an infinitesimal version of this theorem is true, i.e., when the base is artinian (so not reduced in general). Precisely, let $B=\mathrm{Spec}A$ be a local artin scheme of finite type over $\mathbb C$, with $0\in B$ the point. Let $\pi: \mathcal X\to B$ be a propre flat morphism defined over $\mathbb C$. Let $\mathcal F$ be a locally free coherent sheaf over $\mathcal X$. If we assume that the dimension of $H^I(\mathcal X,\mathcal F)$ is equal to the dimension of $H^i(X_0,\mathcal F|_{X_0})$ times the length of $A$ (which morally seems to me to be an analogue of "$\dim H^i(X_b,\mathcal F|_{X_b})$ is constant" in the previous paragraphe), I would like to show that $R^i\pi_*\mathcal F$ is locally free over $B$ and satisfies the base change: $R^i\pi_*\mathcal F|_0\to H^i (X_0,\mathcal F|_0)$ is an isomorphism.
Is it possibly true ?
This has nothing to do with $B$ being reduced. These are originally proved by Grothendieck using the following result of his.
Let $\pi:X\to B=\operatorname{Spec}A$, $A$ Noetherian, be a proper map of finite type and $F$ a coherent sheaf on $X$, flat over $B$. Then there is a complex $P_•$ of projective modules of finite type, $0\to P_0\to P_1\to\cdots\to P_n\to 0$ over $A$ such that for any morphism $g:Y\to B$ if we denote by $p:X\times_B Y\to Y$, the pull back and $h:X\times_B Y\to X$, then $R^ip_*h^*F=H^i(g^*P_•)$.
Using this, whatever you can get is the best.