# Cohomology and Base Change - Degree 0 Sanity Check

Both Vakil and Hartshorne describe Cohomology and Base Change in the following way:

Suppose $f:X \rightarrow Y$ is a projective (in Vakil, proper) morphism of Noetherian schemes, $F$ a coherent sheaf on $X$ flat over $Y.$ If the natural map $\phi^i(y): R^if_*(F) \rightarrow H^i(X_y,F_y)$ is surjective, then $\phi^{i-1}(y)$ is also surjective if and only if $R^if_*(F)$ is locally free near $y$.

An important special case would be when $i=0$. I'm nervous about the fact that neither author singles out this special case, and my silly question is: does the hypothesis "$\phi^{-1}(y)$ is surjective" hold vacuously when $i=0$? In other words, is surjectivity of $\phi^0(y)$ enough to conclude local freeness of $f_*F$ near $y$? Thanks.

28.1.6 Cohomology and Base Change Theorem. Suppose $$\pi:X\to Y$$ is proper, $$Y$$ is locally Noetherian, $$\mathcal{F}$$ is coherent over $$X$$ and flat over $$Y$$, and $$\phi_q^p: R^p\pi_*\mathcal{F}\otimes\kappa(q)\to H^p(X_q,\mathcal{F}|_{X_q})$$ is surjective. Then the following hold.
(i) There is an open neighborhood $$U$$ of $$q$$ such that for any $$\psi:Z\to U$$, $$\phi_Z^p$$ is an isomorphism. In particular, $$\phi_q^p$$ is an isomorphism.
(ii) Furthermore, $$\phi_q^{p-1}$$ is surjective (hence an isomorphism by (i)) if and only if $$R^p\pi_*\mathcal{F}$$ is locally free in some open neighborhood of $$q$$ (or equivalently, $$(R^p\pi_*\mathcal{F})_q$$ is a free $$\mathcal{O}_{Y,q}$$-module, Exercise 13.7.F). This in turn implies that $$h^p$$ is constant in an open neighborhood of $$q$$.
(Proofs of Theorems 28.1.5 and 28.1.6 will be given in §28.2. Note in (ii) that if $$p=0$$, $$\phi_q^{p-1}$$ is automatically surjective, as the codomain of $$\phi_q^{-1}$$ is $$H^{-1}(X_q,\mathcal{F}|_{X_q})$$, which is $$0$$ by definition.)