How can I prove that the Cartier dual of $\alpha_p$ is again $\alpha_p$ (using the Yoneda lemma)? It should be something like $\alpha_p(R) \to (\alpha_p(R) \to \mu_p(R)), x \mapsto (y \mapsto \exp_{p-1}(x + y))$, where $\exp_{p-1}$ is the truncated exponential sequence. My problem is that this isn't a homomorphism.
Cartier dual of $\alpha_p$
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algebraic-geometry