= Projective dimension
{title2=$\operatorname{pd}_RM$}
For a nonzero <module>, the projective dimension $\operatorname{pd}_RM$ is the shortest length of a <projective resolution> of the <module> $M$, or infinity if no finite <projective resolution> exists. Equivalently, it is the supremum of degrees with nonzero $\operatorname{Ext}_R^n(M,N)$ as $N$ ranges over all <modules>. This equivalence follows by dimension shifting and the characterization of <projective modules> by vanishing $\operatorname{Ext}^1$.
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