# 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. Mar 1, 2020 at 16:47
• $\newcommand{\GG}{\mathbf{G}}$Let $W[F]$ be the kernel of $F: W \to W$, so that $W[F] \cong \GG_a^\sharp$, the PD-completion of $\GG_a$. The Cartier dual of $W$ is $\hat{W}$, and Cartier duality flips $F$ and $V$; so the Cartier dual of $F: W \to W$ is $V: \hat{W} \to \hat{W}$, and therefore the Cartier dual of $W[F] = \GG_a^\sharp$ is $\hat{W}/V = \hat{\GG}_a$. Similarly, if $W^\times$ denotes the group scheme of units in $W$, then $W^\times[F]$ is isomorphic to the PD-completion $\GG_m^\sharp$ of $\GG_m$. So the Cartier dual of $W^\times[F]$ is $\hat{W}^\times/V \cong \hat{\GG}_m$.
– skd
Sep 16 at 14:21