= Top Ext detects finite projective dimension over a left Noetherian ring
{title2=$\operatorname{pd}_RM=n<\infty\implies\operatorname{Ext}_R^n(M,R)\ne0$}
For a nonzero left <finitely generated module> over a <left Noetherian ring>, take a finite <projective resolution> with <finite projective modules>. Vanishing of its top <Ext functor> would make the dual of its final injection surjective. These duals are right <finite projective modules>, so the surjection splits. Dualizing back and using finite projective <double dual modules> splits the original injection and shortens the resolution, contradicting the definition of <projective dimension>. When the dimension is zero, a coordinate map of a finite free module containing the nonzero projective summand supplies a nonzero map to $R$. The left/right distinction in the duals is essential for a noncommutative <ring>.
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