Let $f:X \to \mathbb{P}^1$ be a projective flat morphism, $X$ is a projective scheme. Let $\mathcal{F}$ be a locally free sheaf on $X$. Are the higher direct image sheaves $R^if\_\*\mathcal{F}$ locally free for all $i>0$?



Is the higher direct image sheaf of a locally free sheaf over $\mathbb{P}^1$ locally free?
No. For instance there is a flat, projective morphism $f:X\rightarrow \Bbb{P}^1$ such that $X\_t:=f^{-1}(t)$ is a smooth rational curve for $t\neq 0$, but $X\_0$ is a nodal plane cubic curve with an embedded point (see Hartshorne, III.9.8.4). Then $H^1(X\_t,\mathcal{O}\_{X\_t})$ is zero for $t\neq 0$, but $\ \dim H^1(X\_0,\mathcal{O}\_{X\_0})=1$ . By base change it follows that $R^1f\_\*\mathcal{O}\_X$ is the skyscraper sheaf with 1-dimensional fiber at 0.


