User:Maximilian Janisch/latexlist/Algebraic Groups/Dieudonné module

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A module $N$ over a ring of Witt vectors $V ( k )$ (cf. Witt vector), where $k$ is a perfect field of characteristic $p > 0$, 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 [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

[1]	J. Dieudonné, "Lie groups and Lie hyperalgebras over a field of characteristic $p > 0$. VI" Amer. J. Math. , 79 : 2 (1957) pp. 331–388
[2]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970) MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401
[3]	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 MR157972 Zbl 0128.15603

Comments

Dieudonné modules also play a role in different cohomology theories of algebraic varieties over fields of positive characteristic, [a1], and in the (classification) theory of formal groups [3], [a2]. Cartier duality [a2], [a3] (cf. 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 [2].

References

[a1]	P. Berthelot, A. Ogus, "Notes on crystalline cohomology" , Princeton Univ. Press (1978) MR0491705 Zbl 0383.14010
[a2]	M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) MR0506881 MR0463184 Zbl 0454.14020
[a3]	P. Cartier, "Groups algébriques et groupes formels" , Coll. sur la théorie des groupes algébriques. Bruxelles, 1962 , CBRM (1962) pp. 87–111