The Dieudonné Module

14 Jul

I’m not sure how much of this Witt sheaf stuff to keep talking about. There is this beautiful invariant associated to any variety in positive characteristic that doesn’t come up in characteristic $latex {0}&fg=000000$. It is called the height, and the way it is defined is by attaching a $latex {p}&fg=000000$-divisible (formal) group to your variety and looking at the height of that. This will tie together all these things we’ve been talking about since it turns out that the Dieudonné module of this formal group is exactly $latex {H^n(X, \mathcal{W})}&fg=000000$, and the non-finite generatedness of this module corresponds to the variety being “supersingular” which just means it has infinite height.

So anyway, this means at some point I should talk about formal groups, and $latex {p}&fg=000000$-divisible groups, and height, and Dieudonné modules if I ever want to get there, which means we should cut off the discussion on Witt…

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