Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 26 / 6 / ii

Assume $L_{2}F=0$ for every additive right exact functor into an abelian category. Given a subobject $K\hookrightarrow P$ of a projective, let $X$ be its cokernel in a category. The sequence $0\to K\to P\to X\to 0$ is exact, and dimension shifting gives $L_{1}F(K)\cong L_{2}F(X)=0$ for every $F$. The projectivity criterion proved in the root solution makes $K$ projective. This gives (ii)$\Rightarrow$(i).

For any short exact sequence $0\to X\to Y\to Z\to 0$, the long exact sequence now starts at the relevant place withIt is exact because the preceding term $L_{2}F(Z)$ vanishes. This is precisely left exactness of $L_{1}F$, proving $\boxed{\text{(ii)}\Rightarrow \text{(iii)}}$.