How does one define projective dimension in terms of the Ext functor?

Question

Would it be $pd_{R}M:=sup\lbrace i\vert Ext^{i}_{R}(M,N)\neq 0:N$ is an $R$-module$\rbrace$ or $pd_{R}M:=sup\lbrace i\vert Ext^{i}_{R}(M,N):N\neq 0$ is a free $R$-module$\rbrace$ or something else entirely?

Answer 1

We can use the following equivalence to establish the definition:

$\text{Ext}_R^{n}(M,N) = 0$ for all $N$ iff $\text{pd}_R M < n$

This is probably in any textbook discussing homological dimensions, also conveniently in these notes.

From here, a definition easily pops out:

definition: $\text{pd}_R M = \inf\big\{n \mid \text{Ext}_R^{n+1}(M,N) =0 \text{ for all $N$}\big\} = \sup\big\{n \mid \text{Ext}_R^{n}(M,N) \not=0 \text{ for some $N$}\big\}$