Potentially semi-stable deformation rings for representations with values in $PG{L}_n$

We study the potentially semi-stable deformation rings for Galois representations taking their values in $PG{L}_n$, by comparing them to the deformation rings for $G{L}_n$. As an application, we state an analogue of the Breuil-Mézard conjecture, and we show that the case of $PG{L}_n$ follows from the case of $G{L}_n$.