The long exact sequence for left derived functors in Eisenbud's Commutative algebra

Question

Could anyone say any details about 3.17c (author only writes that it's immediate from 3.15 and 3.16)? (The photos below are from "Commutative algebra with a view toward algebraic geometry" by David Eisenbud).

Answer 1

You will benefit from reading a book that explains this in more detail. Recall that if $F$ is some left exact functor in an abelian category with enough projectives, if $M$ is an object there and $P_* \to M\to 0$ is a projective resolution, then $L_*FM$ is computed as the homology of $FP_*$. One of the results of homological algebra is that $L_*FM$ is independent of the choice of resolution $P_*$.

Now suppose you have an exact sequence $0\to M'\to M\to M''\to 0$. Then you can resolve $M'$ by $P_*'$ and $M''$ by $P_*''$, and the Horseshoe lemma shows that you can resolve $M$ by a complex whose underlying objects are of the form $P_*=P_*'\oplus P_*''$, but the differential is not the direct sum of the differentials, and these resolutions fit into an exact sequence $0\to P_*'\to P_* \to P_*''\to 0$.

Since this sequence is split if we forget the differentials, and since $F$ is additive, the sequence remains exact when applying $F$, so what we get is a SES of the form $0\to FP_*'\to FP_* \to FP_*''\to 0$. The complexes appearing here compute the respective derived functors of $F$ at the arguments $M'$,$M$ and $M''$, so the long exact sequence of homology gives the result you're after.

Add Note that naturality follows from the fact the Horseshoe lemma can be applied to a diagram of SESs as in the book, first by resolving all four corners, next in using the lifting property of projective resolutions to obtain maps between the resolutions of the corners, and then by filling in the middle and using the liftings obtained to lift the middle map.