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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 $F: \mathbb{W}_p(-) \to \mathbb{W}_p(-)$.

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  • $\begingroup$ 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. $\endgroup$
    – S. Carnahan
    Mar 1, 2020 at 16:47
  • $\begingroup$ $\newcommand{\GG}{\mathbf{G}}$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 $\hat{W}^\times/V \cong \hat{\GG}_m$. $\endgroup$
    – skd
    Sep 16 at 14:21

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