How to show that the projective dimension is infinite

Question

How can we show that the projective dimension of the $\mathbb{Z}/p^2 \mathbb{Z}$-module $\mathbb{Z}/p \mathbb{Z}$ is infinite?

Answer 1

If $M=\mathbb Z/p\mathbb Z$ had finite projective dimension, the kernel of every surjective map $P\to \mathbb Z/p\mathbb Z$ from a $\mathbb Z/p^2\mathbb Z$-projective module $P$ onto $M$ would have strictly smaller projective dimension. Yet there is a short exact sequence $$0\to\mathbb Z/p\mathbb Z\to \mathbb Z/p^2\mathbb Z\to\mathbb Z/p\mathbb Z\to 0$$