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3 removed extraneous $Others can do this much better than I, but here's what's happening: to describe a group scheme of any kind, you need to talk about not only the underlying space, but also the law of composition on the group. In this case, the kernel of$[p]$in the muliplicative group, you describe the law of composition by writing down the the comultiplication on the affine ring$k[X]/(X^p)$. This is simply$X\mapsto $1 \otimes X + X \otimes 1 + X \otimes X$.
Others can do this much better than I, but here's what's happening: to describe a group scheme of any kind, you need to talk about not only the underlying space, but also the law of composition on the group. In this case, the kernel of $[p]$ in the muliplicative group, you describe the law of composition by writing down the the comultiplication on the affine ring $k[X]/(X^p)$. This is simply $x\mapsto X\mapsto$1 \otimes x X + x X \otimes 1 + x X \otimes x$X$.
Others can do this much better than I, but here's what's happening: to describe a group scheme of any kind, you need to talk about not only the underlying space, but also the law of composition on the group. In this case, the kernel of $[p]$ in the muliplicative group, you describe the law of composition by writing down the the comultiplication on the affine ring $k[X]/(X^p)$. This is simply $x\mapsto$1 \otimes x + x \otimes 1 + x \otimes x\$.