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Let $R$ be a complete local $\mathbf{Z}_p$-algebra, for some prime $p$. In the 1970 paper Group schemes of prime order by Oort and Tate, they write down an explicit finite flat group scheme $G_R(a, b)$ of rank $p$, for each pair of elements $a,b \in R$ satisfying $ab = p$, and show that every FFGS over $R$ of rank $p$ is isomorphic to one of these.

What are the homomorphisms (of $R$-group schemes) from $G_R(a, b)$ to $G_R(a', b')$?

The underlying scheme of $G_R(a, b)$ is $\{ X : X^p - aX = 0\}$. If we look for homomorphisms which are "linear", $X \mapsto \lambda X$ for some $\lambda$, then (after a bit of unravelling) we conclude that $\lambda$ has to be a point of the $R$-scheme $$\{ Z : Z(aZ^{p-1} - a') = Z(b - b'Z^{p-1}) = 0\}.$$ (This is a group scheme, but neither finite nor flat over $R$ in general, although it is a FFGS if $\{a, b'\}$ generate the unit ideal of $R$.)

Are these all the homomorphisms $G_R(a, b) \to G_R(a', b')$?

(The answer is clearly "yes" for $p = 2$. It is also "yes" for $p = 3$ via a deeply nasty polynomial computation.)

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Yes. The Tate-Oort description is an equivalence of categories between finite flat group schemes (over a $\Lambda$-scheme $S$ where $\Lambda$ is a certain ring decribed in the paper) and the category of triples $(L , a, b)$ where $L$ is an invertible sheaf over $S$ and $a\in \Gamma(S,L^{\otimes (p−1)})$, $b\in\Gamma(S,L^{\otimes (1− p)})$ satisfy $a\otimes b=w_p\cdot 1$. The morphisms between $(L,a,b)$ and $(L',a',b')$ are the morphisms of invertible sheaves $f:L\to L'$, viewed as global sections of $L^{\otimes -1}\otimes L'$, such that $a\otimes f^{\otimes p}=f\otimes a'$ and $b'\otimes f^{\otimes p}=f\otimes b$. In the case where the base is a local scheme, this is exactly what you wrote. All I wrote is almost verbatim from the Tate-Oort paper.

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  • $\begingroup$ Oort and Tate just write this down as a bijection on automorphism classes of objects; they don't explicitly state that $G \to (L, a, b)$ is a functor. I'm sure you're correct and it's functorial, but do you know a reference where this is written down explicitly? $\endgroup$
    – David Loeffler
    Commented Jan 20, 2022 at 20:09
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    $\begingroup$ The classification of the group schemes in Oort-Tate uses the decomposition of the Hopf algebra (more precisely the augmentation ideal of this) of the group scheme by eigenspaces for the natural action of $\mathbf{F}_p^*$. So of course any morphism $G \to G'$ gives a corresponding map between the eigenspaces which is what $f$ in the answer above is (by picking the correct eigenspace). This is pretty clear from Lemma 2 of their paper. So I agree with Matthieu that you can't say this isn't in Oort-Tate IMHO. $\endgroup$
    – Johan
    Commented Jan 21, 2022 at 1:38
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    $\begingroup$ In "Moduli of Galois p-covers in mixed characteristics" with Dan Abramovich (Algebra Number Theory 2012) we write this in Definition A.5 + comments after it. But we say hardly more than that! $\endgroup$
    – Matthieu Romagny
    Commented Jan 21, 2022 at 15:14
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    $\begingroup$ Also even though it doesn't matter much, note that the Tate-Oort paper is authored by John Tate and Frans Oort in that order. $\endgroup$
    – Matthieu Romagny
    Commented Jan 21, 2022 at 15:14
  • $\begingroup$ Yes, very good Tate-Oort indeed. $\endgroup$
    – Johan
    Commented Jan 21, 2022 at 15:31

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