$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. I'm not quite sure about the relation between finiteness of projective and injective dimensions of $M$. Does the finiteness (or infiniteness) of one necessarily imply the finiteness (or infiniteness) of another?



Projective & injective dimensions
To complement Mariano's answer: If finite projective dimension implies finite injective dimension for any module $M$, then $R$ better have finite injective dimension (the converse is also quite easy).


The local rings $R$ which have finite inj. dim. over themselves are also known as [Gorenstein rings](http://arxiv.org/abs/math/0209199). In fact, a theorem by Foxby says that $R$ possesses a module of both finite proj. and inj. dim. if and only if $R$ is Gorenstein. 


