Difference between revisions of "Dieudonné module"
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Revision as of 18:51, 24 March 2012
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:
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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
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For any positive integer there exists an isomorphism
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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 ![]() |
[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 |
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) |
[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 |
Dieudonné module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dieudonn%C3%A9_module&oldid=16568