Pull-back of the structure sheaf of a flat family

Question

Let $\pi:\mathcal{X} \to B$ be a flat family of projective schemes over $\mathbb{C}$. Assume that $B$ is smooth. Take a point $b \in B$ and $k(b)$ its residue field. Consider the natural closed immersion $f:\mbox{Spec} k(b) \to B$. As far as I understand, $f^*\pi_*\mathcal{O}_{\mathcal{X}}$ is a $k(b)$-vector space. Is it isomorphic to the space of global section of $\mathcal{O}_{\pi^{-1}(b)}$?

Answer 1

Assume $B$ is integral. Then, for all $i\geq 0$, if the function $e_i(b)=\dim_{k(b)}H^i(\mathcal X_b,\mathcal O_{\mathcal X_b})$ is constant on $b\in B$, we have that the natural map $$f^\ast(R^i\pi_\ast\mathcal O_{\mathcal X})=R^i\pi_\ast\mathcal O_{\mathcal X}\otimes k(b)\to H^i(\mathcal X_b,\mathcal O_{\mathcal X_b})$$ is an isomorphism (cfr. also Hartshorne, III.12.9). (In this result, flatness of $\mathcal O_{\mathcal X}$ over $B$ is important, smoothness of $B$ I am not sure.) So the answer is yes, if $e_0(b)$ is constant on $b\in B$.