$p$-adic étale sheaf Here is the context : I'm trying to understand Katz-Deligne theory of false modular forms as exposed in the third appendix of the Katz's paper Higher Congruences between modular forms : https://web.math.princeton.edu/~nmk/old/highercong.pdf
I haven't gone very far yet in fact I'm still trying to understand the first paragraph, and my question is just a question of algebraic geometry. So here is the setup
$(W,\pi,k)$ is mixed characteristic DVR of residue chararcteristic $p$. For all $m \geq 1$ we write $W_m$ for $W/\pi^mW$.
Let $S_m$ be a compatible sequence of affine flat affine $W_m$-schemes (i.e. $S_m = S_{m+1} \otimes_{W_{m+1}} W_m$).
Let $P$ be a rank one $p$-adic étale sheaf on the $S_m$ which, according to the paper, means the following : 
1) $P$ on $S_m$ is the unique $p$-adic étale sheaf on $S_m$ which induces $P$ on $S_1$ and $P$ 
2) $P$ is and inverse of system $(P_n)$ of étale sheaves which are twisted form of the constant étale sheaf $\mathbf{Z}/p^n\mathbf{Z}$
My question : They say that $ \omega_m := P \otimes_{\mathbf{Z}_p} \mathcal{O}_{S_m}$ is an invertible sheaf on $S_m$. I understand that an étale sheaf is a zariski sheaf but I don't see how $P$ is a $\mathbf{Z}_p$ sheaf (since it is only étale locally isomorphic to $\mathbf{Z}_p$) and especially I don't understand why it gives you an invertible sheaf ?
 A: This was too long for a comment, so I put this as an answer. I am speculating as much as you are unfortunately.
I also find this very confusing. OK, so let's see if we can even understand what $P$ is. So, if I'm understanding correctly, $P$ is just a projective system $P_m$ of abelian LCC $W_m$-sheaves. I guess the unicity he mentions is just because $\text{Spec}(W_m)\to\text{Spec}(W_{m+1})$ is a universal homemorphism (in particular an infinitesimal thickening). In particular, up to the identification of 
$$\text{Spec}(W_m)_{é\text{t}}\cong\text{Spec}(W_{m+1})_{é\text{t}}$$
one should be able to think about $P$ as just being a lisse $\mathbb{Z}_p$-sheaf on $\text{Spec}(W_m)$ for any fixed $m$. 
Let us take this perspective from the end of the last paragraph and see if we can make sense of why $P\otimes_{\mathbb{Z}_p}\mathcal{O}_{\text{Spec}(W_m)}$ is an invertible sheaf. As you most likely know, (quasi)coherent sheaves on $X_{é\text{t}}$ agree with those on $X_\text{zar}$. In other words, if $\mathcal{F}$ is an $\mathcal{O}_{X_{é\text{t}}}$-module locally the cokernel of a map of free-modules, then $\mathcal{F}$ is equal to $\mathcal{G}_{é\text{t}}$ for some (quasi)coherent $\mathcal{O}_X$-module on $X_\text{zar}$ (here for $f:Y\to X$ in $X_{é\text{t}}$ one has that $\mathcal{G}_{é\text{t}}\left(Y\xrightarrow{f} X\right)=f^\ast\mathcal{G}$).
So, it suffices to determine why $P\otimes_{\mathbb{Z}_p}\mathcal{O}_{\text{Spec}(W_m)}$ is a line bundle on the étale site of $\text{Spec}(W_m)$. But, this is almost clear. Namely, if $U\to X$ is a trivialization cover for the lisse sheaf $P$ then we should have that 
$$(P\otimes_{\mathbb{Z}_p}\mathcal{O}_{\text{Spec}(W_m)})\mid_U=(\mathbb{Z}_p\otimes_{\mathbb{Z}_p}\mathcal{O}_{\text{Spec}(W_m)})\mid_U=\mathcal{O}_U$$
(here, really, I think we're thinking of $P$ as the pro-LCC sheaf $\varprojlim P_m$), which would show that this sheaf is invertible.
My reservation? One needs to be a little careful how you construct $U$. Essentially, since each $P_m$ is LCC there is a finite étale cover $U_m\to X$ for which $P_m$ is trivialized over. I guess one then just cooks up these $U_m$ to lie in a sequence and considers $\varprojlim U_m$ (which exists since the transition maps are affine, and is étale for obvious reasons).
This is my best guess—hopefully someone else can say something enlightening.
