Abstract

The aim of the lectures is to present the Dieudonn$\acute{e}$ theory over perfect fields of positive characteristic. More precisely, if $k$ is a perfect field of characteristic $p > 0$, we will assign, functorially, to every $p$-torsion finite commutative group scheme over $k$ a finite length module over the ring of Witt vectors, called the Dieudonn$\acute{e}$ module, endowed with two endomorphisms $F$ (Frobenius) and $V$ (Verschiebung) satisfying certain identities. This assignment turns out to be an anti-equivalence of categories, from the category of finite commutative $p$-group schemes over $k$ and the category of modules over the Dieudonn$\acute{e}$ ring $\frac{W(k)[F,V]}{(FV =VF=p)}$, which are of finite length over the ring of Witt vectors, $W(k)$. We can also extend this equivalence between the category of $p$-torsion commutative formal group schemes over $k$ and the category of modules over the Dieudonn$\acute{e}$ ring of $k$, which are finitely generated over $W(k)$. If time permits, as an application, we will give a classification of $p$-divisible groups over $k$.



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