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.)
Best Answer
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.