$pd(M) \leq n$ implies $\ker(P_n \to P_{n-1})$ projective

Question

Let $M$ be a finitely generated $A$-module with $A$ Noetherian. Suppose $pd(M) \leq n$. Then given any projective resolution $$\ldots \to P_n \to P_{n-1} \to \ldots \to P_0 \to M \to 0$$ why is the kernel of $P_n \to P_{n-1}$ projective? I can see this to be true if $n=1$. However, for the general induction I can't seem to be able to do it.

Answer 1

Actually, we can show that the kernel $K$ of $P_{n-1} \to P_{n-2}$ is projective. Consider any projective resolution $Q_\bullet \to K \to 0$. Then we have the following resolution of $M$:

$\dots \to Q_m \to \dots \to Q_0 \to P_{n-1} \to \dots \to P_0 \to M \to 0$.

Using this resolution we can compute $\operatorname{Ext}^1_A(K,N) = \operatorname{Ext}^{n+1}_A(M,N) = 0$ for any $A$-module $N$. Because of this the functor $\operatorname{Hom}_A(K,-)$ is exact, which is equivalent to $K$ being projective.