Fontaine showed in Asterisque 47-48 that the category of finite dimensional smooth formal $p$-groups over the ring $A=W(k)$ of the Witt vectors over a finite field $k$ is anti-equivalent to the category of triples $(L,M,r)$ where $L$ is a free $A$-module of finite rank, $M$ is a profinite $A[F]$-module such that $F$ is injective on $M$ and $pM\subset FM$, $r$ is an $A$-morphism from $L$ to $M$ which induces an isomorphism from $L/pL$ to $M/FM$. Moreover, the connected groups correspond under this anti-equivalence to the triples where the action of $F$ on $M$ is topologically nilpotent. In addition, Fontaine obtained similar results for finite dimensional smooth formal $p$-groups over the ring $A’$ of integers in a finite extension of $Q_p$ with the ramification index less than $p-1$.
Do these results allow us to determine the categorical properties of these categories, i.e. if these categories admit kernels, cokernels, if they are semi-abelian etc.? I would appreciate if somebody could recommend me a reference where such questions are discussed.
