The Dieudonné ring $D_k$ of a field $k$ of prime characteristic $p$ is defined to be the ring generated by two objects $F,V$ subject to the relations
where
denotes the endomorphism of the Witt ring $W(k)$ of $k$ given by raising each component of the Witt vectors to the $p$-th power; this means that $\sigma$ is component-wise given by the Frobenius endomorphism of the field $k$.
More precisely an element of $D_k$ can uniquely be written as a finite sum
The Dieudonné ring is a $\mathbb{Z}$-graded ring where the degree $n$-part is the $1$-dimensional free module generated by $V^{-n}$ if $n\lt 0$ and by $F^n$ if $n\gt 0$
Pierre Cartier, Groupes algébriques et groupes formels, in Théorie des Groupes Algébriques (Bruxelles, 1962) pdf
Jean Dieudonné, Lie groups and Lie hyperalgebras over a field of characteristic $p\gt 0$. IV, American Journal of Mathematics
Manin, Ju. I., Theory of commutative formal groups over fields of finite characteristic, 1963
Last revised on August 24, 2016 at 12:04:57. See the history of this page for a list of all contributions to it.