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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.

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

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