Consider a morphism of topological spaces $f:X\to Y$. The direct image functor takes a sheaf $\mathcal{F}$ on $X$ to the sheaf defined by $f_*\mathcal{F}(U)=\mathcal{F}(f^{-1}(U))$. It's a right adjoint to the inverse image functor, which means it is automatically left-exact (but usually not right exact).
Here are some general situations I know when $f_*$ is exact:
1) $f$ is a closed immersion (true for any mapping of topological spaces, not just schemes)
2) $f$ is affine and we consider quasicoherent sheaves (a generalization of 1, but this time requiring $X,Y$ be schemes and the sheaves quasicoherent)
and then my knowledge of this situation kind of teeters off in to the distance.
Question: What are some more scenarios where we know that $f_*$ is exact (are there any)? I'm curious about what adjectives one can stick on $X,Y,f$ which would let me know without extra calculation that $f_*$ is exact. I'm mostly concerned with $X,Y$ schemes, I'm somewhat interested in the case where $X,Y$ are analytic spaces over any of $\mathbb{Q}_p,\mathbb{R},\mathbb{C}$, and I'd definitely take other interesting scenarios if any of you want to tell me about them.