In the standard references for prismatic cohomology, most theorems are proved in a local context (i.e. with completeness assumptions), and the devissage to the global case (i.e. smooth proper varieties) is often left to the reader. My first (vague) question is: how do these devisagges work? Are there references where these arguments have been written up (possibly in other contexts, e.g. crystalline cohomology)? For example, in Bhatt's Columbia notes he says that he will work in the local $p$-adically complete setting, because standard arguments then yield the general case: I would like to know what these arguments are.
To make things concrete, let me ask a more specific question. How to compute the mod $p$ prismatic cohomology of $\mathbb{P}^n_k$ (I would expect the answer to be $\mathbb{F}_p[[T]][h]/h^{n+1}$)? Etale comparison gives us the torsion-free part, but how to show that there is no torsion?
