p-Divisible Groups Revisited 1

13 Jul

A Mind for Madness

I’ve posted about $latex {p}&fg=000000$-divisible groups all over the place over the past few years (see: here, here, and here). I’ll just do a quick recap here on the “classical setting” to remind you of what we know so far. This will kick-start a series on some more subtle aspects I’d like to discuss which are kind of scary at first.

Suppose $latex {G}&fg=000000$ is a $latex {p}&fg=000000$-divisible group over $latex {k}&fg=000000$, a perfect field of characteristic $latex {p>0}&fg=000000$. We can be extremely explicit in classifying all such objects. Recall that $latex {G}&fg=000000$ is just an injective limit of group schemes $latex {G=\varinjlim G_\nu}&fg=000000$ where we have an exact sequence $latex {0\rightarrow G_\nu \rightarrow G_{\nu+1}\stackrel{p^\nu}{\rightarrow} G_{\nu+1}}&fg=000000$ and there is a fixed integer $latex {h}&fg=000000$ such that group schemes $latex {G_{\nu}}&fg=000000$ are finite of rank $latex {p^{\nu h}}&fg=000000$.

As a corollary to the standard connected-étale sequence for group…

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