# Cartier duality and Frobenius on Witt vector schemes

Suppose for simplicity we are working over $$\mathbb{F}_p$$. Cartier duality is an antiequivalence between formal groups and affine group schemes over $$Spec(\mathbb{F}_p)$$. Let $$\mathbb{W}_p(-)$$ denote the Witt vector affine group scheme. It is well known that the Cartier dual $$Map_{Grp}(\mathbb{W}, \mathbb{G}_m)$$ to this is the Witt vector formal group $$\widehat{\mathbb{W}_p}$$. This can be defined by taking the colimit of all formal completions at the identity of the truncated Witt schemes and comes endowed with natural formal group structure.

I know that Cartier duality should be viewed as exchanging the Frobenius and Verschiebung maps. Does this mean that the Cartier dual of the natural Frobenius map $$F: \mathbb{W}_p \to \mathbb{W}_p$$ will be the Verschiebung map defined on the Witt vector formal scheme? This seems to be incompatible with the statement that the Cartier dual of the formal multiplicative group $$\widehat{\mathbb{G}_m}$$ is the subgroup scheme of "fixed points" of $$F: \mathbb{W}_p(-) \to \mathbb{W}_p(-)$$.

• Can you say what duality does to F and V in the case of truncated Witt vectors in a bit more detail? Also, I thought the Cartier dual of $\widehat{\mathbb{G}_m}$ is $\mathbb{G}_a$. At least, that seems to be what you get from the topological dual Hopf algebra. – S. Carnahan Mar 1 '20 at 16:47