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A module $N$ over a ring of Witt vectors $V ( k )$ (cf. [[Witt vector|Witt vector]]), where $k$ is a perfect field of characteristic $p &gt; 0$, provided with two endomorphisms $F _ { M }$ and $V _ { M }$ which satisfy the following relationships:

\begin{equation} F _ { M } ( \omega m ) = \omega ^ { ( p ) } F _ { M } ( m ) \end{equation}

\begin{equation} \omega V _ { M } ( m ) = V _ { M } ( \omega ^ { ( p ) } m ) \end{equation}

\begin{equation} F _ { M } ( V _ { M } ( m ) ) = V _ { M } ( F _ { M } ( m ) ) = p m \end{equation}

Here $m \in M$, $\omega = ( a _ { 0 } , \ldots , a _ { n } , \ldots ) \in W ( k )$, $\omega ^ { ( p ) } = ( a _ { 0 } ^ { p } , \dots , a _ { n } ^ { p } , \dots )$. In an equivalent definition, $N$ is a left module over the ring $D _ { k }$ (the Dieudonné ring) generated by $V ( k )$ and two variables $H ^ { \prime }$ and $V$ connected by the relations

\begin{equation} F \omega = \omega ^ { ( p ) } F , \quad \omega V = V \omega ^ { ( p ) } , \quad F V = V F = p \end{equation}

\begin{equation} \omega \in W ( k ) \end{equation}

For any positive integer $12$ there exists an isomorphism

\begin{equation} D _ { k } / D _ { k } V ^ { n } \simeq \operatorname { End } _ { k } ( W _ { n k } \end{equation}

where $D _ { k } V ^ { n }$ is the left ideal generated by $V ^ { N }$ and $V _ { n }$ is the $k$-scheme of truncated Witt vectors. Dieudonné modules play an important part in the classification of unipotent commutative algebraic groups [[#References|[1]]]. Dieudonné modules is also the name given to left modules over the completion $\hat { D }$ of $D _ { k }$ with respect to the topology generated by the powers of the two-sided ideal $( F , V )$ of $D _ { k }$.

====References====
<table><tr><td valign="top">[1]</td> <td valign="top"> J. Dieudonné, "Lie groups and Lie hyperalgebras over a field of characteristic $p &gt; 0$. VI" ''Amer. J. Math.'' , '''79''' : 2 (1957) pp. 331–388</td></tr><tr><td valign="top">[2]</td> <td valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , Masson (1970) {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} </td></tr><tr><td valign="top">[3]</td> <td valign="top"> Yu.I. Manin, "The theory of commutative formal groups over fields of finite characteristic" ''Russian Math. Surveys'' , '''28''' : 5 (1963) pp. 1–83 ''Uspekhi Mat. Nauk'' , '''18''' : 6 (1963) pp. 3–90 {{MR|157972}} {{ZBL|0128.15603}} </td></tr></table>



====Comments====
Dieudonné modules also play a role in different cohomology theories of algebraic varieties over fields of positive characteristic, [[#References|[a1]]], and in the (classification) theory of formal groups [[#References|[3]]], [[#References|[a2]]]. Cartier duality [[#References|[a2]]], [[#References|[a3]]] (cf. [[Formal group|Formal group]]) provides the link between the use of Dieudonné modules in formal group theory (historically the first) and its use in the classification theory of commutative unipotent algebraic groups [[#References|[2]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> P. Berthelot, A. Ogus, "Notes on crystalline cohomology" , Princeton Univ. Press (1978) {{MR|0491705}} {{ZBL|0383.14010}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) {{MR|0506881}} {{MR|0463184}} {{ZBL|0454.14020}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> P. Cartier, "Groups algébriques et groupes formels" , ''Coll. sur la théorie des groupes algébriques. Bruxelles, 1962'' , CBRM (1962) pp. 87–111</td></tr></table>