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!



Free resolution dimension?
I'm sort of stealing the idea from t3suji, but here goes:


If the module is projective, i.e. $PD = 0$, then $FD \leq 1$.


If the module is not projective, i.e. $PD > 0$, then $PD = FD$.


The first statement is t3suji's comment. The second statement follows by first observing that if $F$ is the free module on infinitely many generators, there is an exact sequence of length $n+1$,


$0\rightarrow F\rightarrow F\rightarrow\ldots\rightarrow F\rightarrow 0$,


so long as $n\geq 1$. Then if we have a projective resolution


$0\rightarrow P\_n\rightarrow\ldots\rightarrow P\_0\rightarrow M\rightarrow 0$,


we can make the free resolution


$0\rightarrow P\_n\oplus F\rightarrow\ldots\rightarrow P\_0\oplus F\rightarrow M\rightarrow 0$.


