Is direct image of finite morphism exact?

Question

Let $f:X\to Y$ be a finite morphism of schemes (or algebraic varieties if you prefer). Then is the direct image functor $f_*:Mod(O_X)\to Mod(O_Y)$ exact?

If we restrict to quasi-coherent modules, then $f_*:Qch(X)\to Qch(Y)$ is exact by Serre vanishing.