Let $l$ be a prime number, $\mathbb{Z}\_l$ be the ring of $l$-adic integers, then what is the projective dimension of the ring $A:=\mathbb{Z}\_l[T,T^{-1}]$? 


Is it two?



What is the projective dimension of the ring $\mathbb{Z}_l[T,T^{-1}]$ ? l is a prime number 
$\mathbb Z\_p$ is a principal ideal domain, so it is Noetherian and its global dimension is $1$. Now, there is a general theorem that tells you that for all right Noetherian rings $R$ one has $$\operatorname{gldim}R[X,X^{-1}]=\operatorname{gldim}R+1.$$ So your answer is indeed $2$.


You'll find that theorem proved pretty much anywhere where global dimension is discussed. For example, J. C. McConnell and J. C. Robson's bible *Noncommutative Noetherian Rings* (which should be called, as usual, "non-necesarily commutative noetherian rings"...)


