Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$:


(1) $\mbox{pd}(M)\leq n$ iff $\mbox{Ext}^{n+1}(M,-)=0$


(2) $\mbox{pd}(M)=0$ iff $M$ is projective


(3) $\mbox{grade}(M):=\infty$ if $M=0$


If one attempts to define $\mbox{pd}((0))$ by extending one of these results, (1), (2), (3) suggest $\mbox{pd}=-1, 0, \infty$, respectively.



Projective dimension of zero module
Although I agree that one can easily decide to not worry about the case of the zero module, but as *ashpool* points out, it happens that sometimes we end up with the zero module whether we want or not and then each time we need to say (using *ashpool*'s example) if $M/aM\neq 0$, then bluh and if $M/aM=0$ than something else happens.


So, I think there is actually something to be gained from making a definition that makes sense for the zero module (or the zero object in a more general situation). Of course, sometimes the definition that makes one (in)equality work does not work for another. However, one could still say in a paper (less likely in a book I suppose) that we are using the following definition for *whatever* which is the usual one if the object is not zero and gives *this* or *that* when it is zero and makes the following inequality work.


So having philosophized about this let me give a definition of *projective dimension* that gives $-\infty$ for the zero module. 



> 
> **Definition** Let $(R,\mathfrak m,k)$ be a noetherian local ring and $M$ a finite $R$-module. Define the *projective dimension* of $M$ as
>  $$
> \mathrm{proj\, dim}\_R M:=\sup \left\{ i\in \mathbb{Z} \ \vert \ \mathrm{Ext}\_R^i (M,k)\neq 0 \right\}, 
> $$
>  where $\sup$ is taken in $\mathbb{Z}\cup\{\pm\,\infty\}$.
> 
> 
> 


This is actually essentially *ashpool*'s definition (1), except that for $M=0$ it takes the $\sup$ of the empty set. (This may have been what *samantha*'s professor told her). It also makes the change of rings formula to work. 


In fact, I would argue that this is the "right" definition anyway, because the point is those Ext groups that are non-zero, not those that are.


Regarding adding the $\{\pm\,\infty\}$ possibilities: We definitely need to allow $+\infty$, so it makes sense to allow $-\infty$ as well, especially because we need it for $M=0$. 


*Comment* Of course one can start wondering what to do with non-local and/or non-noetherian rings, but I will leave that meditation to the reader.


