Let $f:X\to Y$ be a projective morphism of noetherian schemes and let $\mathcal{F}$ be a coherent sheaf on $X$ which is flat over $Y$. If you read Hartshorne chapter 3 section 12 or Vakil chapter 25, you can see the development of a wonderful theory which leads to Grauert's theorem:
If $Y$ is integral and $\dim_{k(y)} H^i(X_y,\mathcal{F}_y)$ is constant, then $R^if_*\mathcal{F}$ is locally free on $Y$ and the natural map $R^if_*\mathcal{F} \otimes k(y) \to H^i(X_y,\mathcal{F}_y)$ is an isomorphism.
This is great. Can the case $i=0$ (i.e. global sections) be done without quite so much of the machinery? What I mean is, if in the situation above, we know $f_*\mathcal{F}$ is locally free, is there an argument which shows that $f_*\mathcal{F}\otimes k(y) \to H^0(X_y,\mathcal{F}_y)$ is an isomorphism without introducing the Mumford complex (theorem 25.2.1 in Vakil or lemma 3.12.3 in Hartshorne).