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Difference between revisions of "Dieudonné module"

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A module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d0316401.png" /> over a ring of Witt vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d0316402.png" /> (cf. [[Witt vector|Witt vector]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d0316403.png" /> is a perfect field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d0316404.png" />, provided with two endomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d0316405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d0316406.png" /> which satisfy the following relationships:
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{{TEX|done}}
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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} $ .
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d0316407.png" /></td> </tr></table>
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d0316408.png" /></td> </tr></table>
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d0316409.png" /></td> </tr></table>
 
 
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164012.png" />. In an equivalent definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164013.png" /> is a left module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164014.png" /> (the Dieudonné ring) generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164015.png" /> and two variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164017.png" /> connected by the relations
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164018.png" /></td> </tr></table>
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164019.png" /></td> </tr></table>
 
 
For any positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164020.png" /> there exists an isomorphism
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164021.png" /></td> </tr></table>
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164022.png" /> is the left ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164024.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164025.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164027.png" /> with respect to the topology generated by the powers of the two-sided ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164029.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Dieudonné, "Lie groups and Lie hyperalgebras over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031640/d03164030.png" />. VI" ''Amer. J. Math.'' , '''79''' : 2 (1957) pp. 331–388</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></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Dieudonné, "Lie groups and Lie hyperalgebras over a field of characteristic d03164030.png. VI" ''Amer. J. Math.'' , '''79''' : 2 (1957) pp. 331–388</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></table>
  
  

Latest revision as of 18:35, 17 December 2019

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} $ .


References

[1] J. Dieudonné, "Lie groups and Lie hyperalgebras over a field of characteristic d03164030.png. 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=44298
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article