One of characterizations of projective modules over noetherian ring of finite global dimension

Question

Let $A$ be noetherian ring of finite global dimension, $M$ be finitely generated module. Then i want to prove that $\mathop{Ext^i}(M,A) = 0, i>0 \implies M -$ projective. Since in this case projectivity is the local property and localization is interchanble with $\mathop{Ext}$ then it's enough to prove this in the case when $A$ is local. But i still have no idea how to prove it..

Answer 1

Hint: Use induction to reduce to $\text{pdim}_A(M)\leq 1$. In this case pick a presentation $0\to P\to Q\to M\to 0$ with $P,Q$ finitely generated and projective and consider the associated extension class.

Remark: Dropping the finiteness of the global dimension, but assuming instead that the injective dimension of $_AA$ is finite, one can still deduce that $M$ has an infinite projective resolution $0\to M\to P^0\to P^1\to ...$ to the right. This generalizes the claim above since if $\text{gldim}(A)<\infty$, such an infinite syzyzygy is necessarily projective. In general, the modules which admit an infinite projective resolution to the right are called Gorenstein projective (or (maximal) Cohen-Macaulay); the quotient $\text{G-Proj}(A)/\text{Proj}(A)$ can be taken as a 'measure' for the singularity of $A$ and is therefore called Singularity Category of $A$.