TITLE: Free resolution dimension?
QUESTION [9 upvotes]: Is there a notion of a dimension associated to free resolutions like projective and injective dimensions associated to projective and injective resolutions? My guess is that it coincides with projective dimension but couldn't find any reference talking about this. Any help or tips will be appreciated!

REPLY [8 votes]: When requiring finitely-generatedness of the resolution, then the free dimension of a projective module can be infinite.
As a simple example, take the ring $R=k\oplus k$, and let $e_1=(1,0)$ and $e_2=(0,1)$ be the obvious idempotents.  Then the module $ke_1$ is projective.  However, any surjection
$$ R^{\oplus n}\rightarrow ke_1 $$
will have a kernel $S_1$ which is isomorphic to $R^{n-1}\oplus ke_2$.  Therefore, any surjection
$$ R^{\oplus m}\rightarrow S_1 $$
will have a kernel $S_2$ which is isomorphic to $R^{m-n}\oplus ke_1$.  Thus, any f.g. free resolution of $ke_1$ must infinite.
Of course, as noted above, $ke_1$ has an infinitely-generated free resolution of length 1, as it is projective.