# Dieudonné module

A module $M$ over a ring of Witt vectors $W (k)$ ( 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: $$F _{M} ( \omega m ) = \omega ^{(p)} F _{M} (m),$$ $$\omega V _{M} (m) = V _{M} ( \omega ^{(p)} m ) ,$$ $$F _{M} ( V _{M} (m) ) = V _{M} ( F _{M} (m) ) = p m .$$ Here $m \in M$ , $\omega = ( a _{0} \dots a _{n} , . . . ) \in W (k)$ , $\omega ^{(p)} = ( a _{0} ^{p} \dots a _{n} ^{p} , . . . )$ . In an equivalent definition, $M$ is a left module over the ring $D _{k}$ ( the Dieudonné ring) generated by $W (k)$ and two variables $F$ and $V$ connected by the relations $$F \omega = \omega ^{(p)} F , \omega V = V \omega ^{(p)} , F V = V F = p ,$$ $$\omega \in W (k) .$$ For any positive integer $n$ there exists an isomorphism $$D _{k} / D _{k} V ^{n} \mathop{\rm End}\nolimits _{k} ( W _{nk} ) ,$$ where $D _{k} V ^{n}$ is the left ideal generated by $V ^{n}$ and $W _{nk}$ 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 $\widehat{D} _{k}$ 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 d03164030.png. 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