# Homomorphisms between Oort–Tate group schemes

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.)

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.
• 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? Jan 20 at 20:09
• 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. Jan 21 at 1:38