= Top Ext detects finite projective dimension
{title2=$\operatorname{pd}_A M=n<\infty\implies\operatorname{Ext}_A^n(M,A)\ne0$}
Let $M$ be a nonzero <finitely generated module> over a commutative <Noetherian ring>. Take a length-$n$ resolution with finite projective terms. If its top <Ext functor> with coefficients in $A$ vanished, the dual of its last injection would be surjective. The dual of a finite projective module is projective, so that surjection splits. Finite projective modules are isomorphic to their double duals, so dualizing gives a splitting of the original injection. The resolution then shortens, contradicting minimality of $n$. When $n=0$, a nonzero coordinate of a finite free module containing $M$ as a summand gives a nonzero map $M\to A$.
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