TITLE: Locally constant group schemes
QUESTION [5 upvotes]: Let $k$ be a field of characteristic $p$, then the group scheme $\mu_\ell$, if $p$ does not divide $\ell$, is étale-locally isomorphic to $\mathbb{Z}/\ell\mathbb{Z}$. I have two quick questions which I don't feel very certain. Is $\mu_p$ fppf locally isomorphic to $\mathbb{Z}/p\mathbb{Z} $? Also, is $\alpha_p $ fppf locally constant?

REPLY [3 votes]: Let me see if I've unwound the definitions properly: it sounds to me like to check whether two things are fppf-locally isomorphic over $\text{Spec } k$ it suffices to check whether or not there exists some finitely presented $k$-algebra $A$ (flatness and surjectivity of $\text{Spec } A \to \text{Spec } k$ being automatic) such that the extensions of scalars to $A$ are isomorphic. Do I have that right?
If so, we can show that $\mu_{p, A} \cong \text{Spec } A[x]/(x^p - 1)$ is never isomorphic to the constant group scheme $\text{Spec } A^{\mathbb{Z}/p\mathbb{Z}}$; in fact already they are never isomorphic as $A$-schemes, forgetting the group structure. I will be very explicit here for those reading along who, like me, are not true algebraic geometers! Since every finitely presented $A$ admits a map to $\bar{k}$, it suffices to show this for $A = \overline{k}$, and then we can just take $\bar{k}$-points and observe that $\mu_{p, \overline{k}}$ has only one $\bar{k}$-point but $\text{Spec } \bar{k}^{\mathbb{Z}/p\mathbb{Z}}$ has, of course, $p$ of them.
Similarly we can show that $\alpha_p$ is not locally constant by checking that $\alpha_{p, A} \cong \text{Spec } A[x]/x^p$ is never isomorphic to any constant group scheme, which is already true forgetting the group structure. Again it suffices to show this for $A = \bar{k}$ and then again taking $\bar{k}$-points shows that there's one of them, so if $\alpha_{p, A}$ were locally constant it would have to be locally isomorphic to a point and it's not. (And of course since $x^p - 1 \cong (x - 1)^p$ we have that $\mu_p$ and $\alpha_p$ have the same underlying scheme.)