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I am looking for a basic (understandable) reference on Dieudonné modules.

Specifically, I am trying to understand how it relates to certain cohomology groups of abelian varieties.

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There are a number of different technical frameworks that work for different purposes. You should give some more details of the context in which you are working. –  Neil Strickland May 9 '12 at 21:30
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There's a decent explanation in Waterhouse's Abelian varieties over finite fields. –  René May 9 '12 at 21:33
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Look at the list of references in the following syllabus (Eva Viehmann's seminar on p-divisible groups).

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