Let $f:Y\to X$ be a contraction of a rational curve between projective threefolds. Let $E$ be a sheaf on $Y$. Is it true that $Rf_*(E)=0$ iff $E$ is supported on the contracted curve? If so how to prove it?
Everything is clear except at the exceptional locus. I cannot see what happens there.
Thanks for the help!
Let us say that we work over the complex numbers. Take the closed immersion $i:C\hookrightarrow Y$, where $C$ is the rational curve. Then I claim that the sheaf $E=i_*\mathcal{O}_C$ is supported on $C$ but $Rf_*E$ is not zero in the derived category of X. It's enough to show that $f_*E$ is not zero. Let us say that the curve is contracted to a point $p$ in $X$, then if we take an open $U\subset X$ containing $p$ we have that $f_*E(U)$ is not zero, and actually is equal to $\mathbb{C}$, because $f^{-1}U$ contains $C$.
The other implication should be true: indeed the hypothesis implies that $f_*E=0$, and using the fact that $f:Y-C\rightarrow X-p$ is an isomorphism, then you see that the stalks of E at the points in $Y-C$ are all zero, thus the support of $E$ must be contained in $C$.