Last time we saw that if we consider a $latex {p}&fg=000000$-divisible group $latex {G}&fg=000000$ over a perfect field of characteristic $latex {p>0}&fg=000000$, that there wasn’t a whole lot of information that went into determining it up to isomorphism. Today we’ll make this precise. It turns out that up to isomorphism we can translate $latex {G}&fg=000000$ into a small amount of (semi-)linear algebra.

I’ve actually discussed this before here. But let’s not get bogged down in the details of the construction. The important thing is to see how to use this information to milk out some interesting theorems fairly effortlessly. Let’s recall a few things. The category of $latex {p}&fg=000000$-divisible groups is (anti-)equivalent to the category of Dieudonné modules. We’ll denote this functor $latex {G\mapsto D(G)}&fg=000000$.

Let $latex {W:=W(k)}&fg=000000$ be the ring of Witt vectors of $latex {k}&fg=000000$ and $latex {\sigma}&fg=000000$ be the natural Frobenius map on $latex {W}&fg=000000$. There…

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