Is there a functor $\mathcal{F}$ on an abelian category $\mathcal{C}$ which is not exact but there is a natural number $n$ such that $\mathcal{F}^n$ is an exact functor? What about the same question when we replace exactness by half exactness? Namely a functor which is not left exact but it has a left exact power (or the same for right case)?
Remark: We exclude the case that $\mathcal{F}^n$ is the identity functor for some $n\in \mathbb{N}$.
Best Answer
Here's a simple example. Let $\mathcal{C}$ be the category of $\mathbb{Z}[x]/(x^2)$-modules, and let $\mathcal{F}$ be the functor that takes a module $M$ to the submodule $xM$. Then $\mathcal{F}$ is not half exact (consider $0\to xM\to M\to M/xM\to 0$ which $\mathcal{F}$ sends to $0\to 0\to xM \to 0\to 0$ which is not exact in the middle unless $xM=0$) but $\mathcal{F}^2=0$ is exact.