This is from Hartshrone exercise 6.6 part (a).

Let $A$ be a regular local ring and $M$ be a finitely generated $A$-module, prove the following

$M$ is projective $\iff$ $\operatorname{Ext}^{i}(M,A)=\{0\}$ for all $i>0$

The hint is to use the following

**Proposition (6.11 A)**
If $A$ is a regular local ring, then

(1) for every $M$, pd$(M)\le \dim(A)$ where pd(M) is the projective dimension and dim(A) is the Krull dimension

(2) If $K=A/m$ then $\operatorname{pd}(K)=\dim(A)$

and to use the descending induction to prove that $\operatorname{Ext}^i(M,N)=\{0\}$ for all $i>0$ and all finitely generated $A$-module $N$. Then finally show that $M$ is a direct summand of a free module.

I really don't know how to put together all this information. Any help is appreciated