When Grauert's Theorem is presented in Hartshorne, the statement goes as follows:
Let $f:X\rightarrow Y$ be a projective morphism of noetherian schemes, and $\mathcal F$ a coherent sheaf on $X$, flat over $Y$. Assume that $Y$ is integral and for some $i$, the function $h^i(y,\mathcal F)=\dim_{k(y)} H^i(X_y,\mathcal F_y)$ is constant on $Y$. Then $R^if_*(\mathcal F)$ is locally free on $Y$ and for every $y$ the natural map
$$R^if_*(\mathcal F)\otimes k(y)\rightarrow H^i(X_y,\mathcal F_y).$$
I was wondering if the assumption that $Y$ be integral can be removed. Certainly the proof in Hartshorne uses the integrality condition.
I know that the projective requirement can also be made more general in allowing proper morphisms. I have seen a statement to this effect (in allowing properness and any Noetherian scheme $Y$ as a base) in Nitsure's notes "Construction of Hilbert and Quot schemes" (Part 2 of "FGA:Explained"), and indeed part (3) of Theorem 5.10 there says precisely the statement above with properness replacing projective and without the requirement that the base be integral, but I was wondering if that was accurate. The reference given for the proof is the above result and proof in Hartshorne, which doesn't cover the general case.
In short, I would like to know whether this generalization (1) is indeed true, (2) if Hartshorne's proof can be easily modified, and (3) if not, is there a good reference (EGA is acceptable, but not preferred).
Best Answer
There is a very clear exposition of the base change theory in Mumford's book on abelian varieties.