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) |
[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 |
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) |
[a2] | M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) |
[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://encyclopediaofmath.org/index.php?title=Dieudonn%C3%A9_module&oldid=22351
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article