How can we show that the projective dimension of the $\mathbb{Z}/p^2 \mathbb{Z}$-module $\mathbb{Z}/p \mathbb{Z}$ is infinite?
How to show that the projective dimension is infinite
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commutative-algebra
1 Answers
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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 $0\to\mathbb Z/p\mathbb Z\to \mathbb Z/p^2\mathbb Z\to\mathbb Z/p\mathbb Z\to 0$