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} $ .

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

[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 Zbl 0086.02604
[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
[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