@AllenKnutson Thanks, I fixed this. I did not know these words before.

@DavidSpeyer Yes, this is what I meant. I am sorry, I did not make it clear.

Ah, thank you. So I definitely need local, but not necessarily complete.

Yes, that is exactly the book that I looked at first. I didn't find the formula there (Of course I might have missed it). But there is this other formula $\mbox{depth}(X)=\min\{i\ |\ \mbox{Ext}^i_R(k,X)\not=0\}$. Thus there is probably some duality involved.