View source for Dieudonné module

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A module  $  M $ 
over a ring of Witt vectors  $  W (k) $ (
cf. [[Witt vector|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 [[#References|[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, [[#References|[a1]]], and in the (classification) theory of formal groups [[#References|[3]]], [[#References|[a2]]]. Cartier duality [[#References|[a2]]], [[#References|[a3]]] (cf. [[Formal group|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 [[#References|[2]]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> 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}}</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , Masson (1970) {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> 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 {{MR|157972}} {{ZBL|0128.15603}} </TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Berthelot, A. Ogus, "Notes on crystalline cohomology" , Princeton Univ. Press (1978) {{MR|0491705}} {{ZBL|0383.14010}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) {{MR|0506881}} {{MR|0463184}} {{ZBL|0454.14020}} </TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> P. Cartier, "Groups algébriques et groupes formels" , ''Coll. sur la théorie des groupes algébriques. Bruxelles, 1962'' , CBRM (1962) pp. 87–111</TD></TR>
</table>