Projective Dimension and Schanuel's Lemma

Question

Let $R$ be a ring and $M$ a (say, left) $R$-module of projective dimension $n$. According to Noncommutative Noetherian Rings, any projective resolution of $M$ can be terminated at length $n$, and this is proved using an extended version of Schanuel's lemma: if $$0\to K \to P_0 \to \dots \to P_n \to M \to 0$$ $$0\to K' \to P_0' \to \dots \to P_n' \to M \to 0$$ are exact sequences of $R$-modules with each $P_j$, $P_j'$ projective then $$K' \oplus P_0 \oplus P_1' \dots \simeq K \oplus P_0' \oplus P_1 \cdots$$ I surprisingly managed to prove Schanuel's lemma and the extended version stated above, but I don't know how to put the pieces together and prove the claim about projective resolutions. Any thoughts? (Note: I don't want a full solution, just a nice hint would be great.)

Answer 1

Hint: The statement of the lemma is using the wrong $n$; you'll want to use it for $n-1$ rather than $n$ (so the $P$s go up to $P_{n-1}$). If $M$ has projective dimension $n$, then it has some such resolution in which $K$ is projective. Now what does this tell you about the $K'$ you get by truncating any other projective resolution?