# Weibel Exercise 2.4.3, dimension shift

Exercise 2.4.3., pg 47 If $$0 \rightarrow M \rightarrow P \rightarrow A \rightarrow 0$$ is exact with $$P$$ projective (or $$F$$-acyclic), then $$L_iF(A) \cong L_{i-1}F(M)$$ for $$i \ge 2$$ and that $$L_1F(A)$$ is the kernel of $$F(M) \rightarrow F(P)$$

I could do the first part: simply construction a resolution $$P_2 \rightarrow P_1 \rightarrow M$$ and connect it to $$P$$, via $$P_0 \rightarrow M \rightarrow P$$.

I have trouble showing the second. It would really help if someone spells out the details.

• Use the long exact sequence of a left derived functor for your short exact sequence. – Pedro Tamaroff Apr 9 at 10:37