nLab Dieudonné ring

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

$$FV=VF=p$$

$$Fw=w^\sigma F$$

$$w V=V w^\sigma$$

where

$$\sigma:\begin{cases} W(k)\to W(k) \\ (w_1,w_2,\dots)\mapsto (w_1^p,w_2^p,\dots) \end{cases}$$

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

$$\sum_{i\gt 0} a_{-i} V^i + a_0 + \sum_{i\gt 0} a_i F^i$$

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 $n\gt 0$

References

• 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