Suppose $G$ is an affine group scheme over a perfect field $k$ of characteristic $p>0$. Let $G^{(p)}$ be the base change of $G$ with respect to the Frobenius map of $k$ (i.e. $p$-th power map). Is there any easy way to see $G$ and $G^{(p)}$ are canonically isomorphic?
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$\begingroup$ No, since they're not generally isomorphic at all. $\endgroup$– user30379Commented Apr 6, 2013 at 9:30
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$\begingroup$ [I assume you intend for the isomorphism to be of group schemes over $k$.] $\endgroup$– user30379Commented Apr 6, 2013 at 9:32
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$\begingroup$ In Waterhouse's book Intro. to Affine Group schemes, Chapter 11 Exercise 12 (c), it says that they are canonically isomorphic, but the given map is not k-linear, so I am not sure about the result. $\endgroup$– XingtingCommented Apr 6, 2013 at 15:47
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$\begingroup$ I recommend ignoring that exercise, since such an assertion is rather misleading. (The point is that the Frobenius is an automorphism of $k$, from which the isomorphism claim in his weak sense is obvious.) $\endgroup$– user30379Commented Apr 6, 2013 at 16:06
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