Given $F$ is a covariant additive functor from left R-module to a left S-module, show that
$\mathscr{L}_n(\mathscr{L_m}(F))=0$ if $m>0$ (where $\mathscr{L}$ refers to the derived functor).
I am trying to show this from induction, but I can't think of a projective resolution for $\mathscr{L}(F(B))$.
Given a projective resolution of $B$
$P_n \to \ldots \to P_1 \to P_0 \to B \to 0$ we get the n-th derived functor by taking the homology of
$F(P_n) \to \ldots \to F(P_1) \to F(P_0) \to 0$
I am now wondering - how do I form a projective resolution for $\mathscr{L}_m F(B)$?
The problem I can see is that $\mathscr{L}_0 F(B)$ is only right exact and $\mathscr{L}_n F$ is half exact.
(I think I should set this up as an induction of $n$).
Any hints? (On how to form the projective resolution)?