Finite flat group schemes of order 2

Stream: FLT

[Kevin Buzzard]

I remember Richard Borcherds once telling me that the moment he heard that over an algebraically closed field of characteristic p there were three group schemes of order p he decided that the theory must be very strange. The theory sounds very exotic, but it really is not. As an exercise I will explain here how to compute all the group schemes of order 2 over any base scheme. First there is some algebraic geometry mumbo-jumbo which says that base schemes are covered by affine schemes and we can reduce to the Noetherian case by EGA4 and locally we must be free rank 2, reducing us to a classification problem in algebra (classifying finite free rank 2 commuative Hopf algebras over a ring) and a gluing question which is solved in the paper and which I'll say no more about because I don't need it.

We are left with classifying commutative Hopf algebras $H$ over $R$ which are free R-algebras of rank 2. This is where the actual work happens. As well as the inclusion $R\to H$ coming from the algebra structure, there's a counit $R$-algebra map $H\to R$, and its kernel is an ideal $I$. One now proves that $H=R\oplus I$, chooses a basis $x$ of the free rank 1 $R$-module $I$ and just writes down formulas for how it behaves. For example $x^2$ is definitely in $I$ so it's $ax$ for some $a\in R$. Similarly comultiplication applied to $x$ will be of the form $1\otimes x+x\otimes 1+bx\otimes x$, and a cool calculation using the axioms of Hopf algebras shows that $ab=-2$. Changing the choice of $x$ by a unit $u$ just turns $a$ to $u^{-1}a$ and $b$ to $ub$ (or the other way around). Conversely given any $a$ and $b$ wih $ab=-2$ you can just construct an explicit commutative Hopf algebra structure on $R^2$ (with $(0,1)\times(0,1)=(0,a)$ for example). So the free group schemes of rank 2 over $R$ just correspond to sets of solutions to $ab=-2$ modulo the equivalence relation generated by the units sending $a$ to $ua$ and $b$ to $u^{-1}b$. In fact this shows that the moduli space of rank 2 Hopf algebras over R is represented by the stack corresponding to quotienting out the affine scheme Spec($R[a,b]/(ab+2)$) by the algebraic group $\mathbb{G}_m$ where $a$ has weight +1 and $b$ has weight -1, but we don't need all that nonsense, we just need the construction in both directions (from the Hopf algebra to the equivalence class, plus "proof of well-definedness" if you like, and conversely from a solution of $ab=-2$ to a Hopf algebra, plus proofs that going one way then the other gives you something isomorphic to what you started with). I mean, we don't need it for $p=2$, we need the theory for general $p$ (as developed in the Oort-Tate paper), but $p=2$ would be a fun place to practice.

Exercise: show that over an algebraically closed field of characteristic 2 there are 3 group schemes of order 2.