Difference between revisions of "Dieudonné module"
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><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> |
Revision as of 17:26, 31 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:
 |
 |
 |
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) 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 |
Dieudonné module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dieudonn%C3%A9_module&oldid=24027