nLab
Dieudonné ring

The Dieudonné ring D kD_k of a field kk of prime characteristic pp is defined to be the ring generated by two objects F,VF,V subject to the relations

FV=VF=pFV=VF=p
Fw=w σFFw=w^\sigma F
wV=Vw σw V=V w^\sigma

where

σ:{W(k)W(k) (w 1,w 2,)(w 1 p,w 2 p,)\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)W(k) of kk given by raising each component of the Witt vectors to the pp-th power; this means that σ\sigma is component-wise given by the Frobenius endomorphism of the field kk.

More precisely an element of D kD_k can uniquely be written as a finite sum

i>0a iV i+a 0+ i>0a iF i\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 nn-part is the 11-dimensional free module generated by V nV^{-n} if n<0n\lt 0 and by F nF^n if n>0n\gt 0

References

Last revised on August 24, 2016 at 08:04:57. See the history of this page for a list of all contributions to it.