# Dieudonné module

A module over a ring of Witt vectors (cf. Witt vector), where is a perfect field of characteristic , provided with two endomorphisms and which satisfy the following relationships:

Here , , . In an equivalent definition, is a left module over the ring (the Dieudonné ring) generated by and two variables and connected by the relations

For any positive integer there exists an isomorphism

where is the left ideal generated by and is the -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 of with respect to the topology generated by the powers of the two-sided ideal of .

#### References

[1] | J. Dieudonné, "Lie groups and Lie hyperalgebras over a field of characteristic . 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 |

**How to Cite This Entry:**

Dieudonné module.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Dieudonn%C3%A9_module&oldid=24027