Let $f: X\to Y$ be a morphism of schemes and $\mathcal I$ a quasi-coherent sheaf of ideals of $\mathcal O_X$, then we have an exact sequence of $\mathcal O_X$-modules $$0\to \mathcal I\to \mathcal O_X\to \mathcal O_X/\mathcal I\to 0$$, it induces the sequence of $\mathcal O_Y$-modules $$0\to f_*\mathcal I\to f_*\mathcal O_X\to f_*(\mathcal O_X/\mathcal I)\to 0$$, is it exact?
If the answer is not, we suppose $X,Y$ are Noetherian, is the sequence of $\mathcal O_Y$-modules $0\to f_*\mathcal I\to f_*\mathcal O_X\to f_*(\mathcal O_X/\mathcal I)\to 0$ exact?