# Ext group for commutative finite group schemes

EDIT: all group schemes are commutative. Thanks @R. van Dobben de Bruyn!

Could anyone provide a reference request about extensions of finite group schemes / Ext groups.

As far as I know the category of finite group schemes is abelian over a field, but I am working over an arbitrary base (can assume characteristic $p$, but can be non-reduced etc), so not sure even how to define $\text{Ext}$. I am not even sure if these $\text{Ext}$ would be sheaves or just abelian groups.

A few things I thought of:

1. maybe we can define $\text{Ext}^i$ as the right derived functors of $\text{Hom}$ in the category of fppf sheaves of groups? Is the category of sheaves of groups on the fppf site abelian? (Seems like it should, but I am afraid to just claim it). If so, we then view everything just as sheaves, then we do have an ambient abelian category and I can happily define $\text{Ext}$ as such? Of course, then this would classify extensions by sheaves, and these extensions need not be representable, i.e. finite group schemes..

2. I would like $\text{Ext}^1$ to classify extensions - Maybe it can be defined this way. It would be helpful if I could see this spelled in more detail in the particular case of finite group schemes).

3. If we start with an exact sequence of finite flat group schemes: $0\to G'\to G\to G''\to0$, is there a sense in which it defines a section of some $\text{Ext}$ sheaf? For example fppf locally, we do have exact sequences of actual abelian groups $0\to G'(T)\to G(T)\to G''(T)\to0$, so this gives an element in $\text{Ext}^1(G''(T), G'(T))$. As $T$ varies, can these be patched to give a "section" of some $\text{Ext}$ sheaf?

I am mainly interested in extensions (hence in $\text{Ext}^1$) but in the absence of an "ambient" abelian category", I am not sure how much things carry over. For example, in the category of abelian groups, we have $\text{Ext}^1(\mathbb Z/p\mathbb Z,B)=B/pB$. Is there an equivalent such statement for finite group schemes? (i.e. $\mathbb Z/p\mathbb Z$ viewed as the constant group scheme, and $B$ replaced by some finite group $G$; again, over arbitrary base).

I am mostly interested in finite flat group schemes, but since kernels need not be flat, I phrased the above question in a more general setting.

Apologies for some of the vagueness, but hence the reference request. Thank you.

• Do you mean finite commutative group schemes? Surely the category of all finite group schemes cannot be abelian. – R. van Dobben de Bruyn Aug 20 '18 at 0:41
• @R.vanDobbendeBruyn, yes definitely! Edited the question - thanks for bringing this up! – aytio Aug 20 '18 at 0:43
• The category of abelian sheaves on a small site is abelian; see e.g. Tag 03CN. Already with things like the [big] fppf site there are some mild set-theoretical issues, but let's assume you've solved those (e.g. using universes, or like the Stacks project by bounding all objects in sight; see Tag 000H). – R. van Dobben de Bruyn Aug 20 '18 at 0:46
• I think this goes like this. You define your $\mathrm{Ext}^i$ using derived functors in the category of abelian fppf sheaves. Then it is a generality about abelian categories that $\mathrm{Ext}^1(G'', G')$ is the set of iso. classes of extensions of, in your case, abelian fppf sheaves $0 \rightarrow G' \rightarrow G \rightarrow G'' \rightarrow 0$. Then the key point is that $G$ is a $G'_{G''}$-torsor, so by fppf descent for relatively affine morphisms $G$ is representable by a scheme. Hence $G$ is a finite locally free group scheme, and you have identified $\mathrm{Ext}^1(G'', G')$ concretely. – Kestutis Cesnavicius Aug 20 '18 at 11:20
• I agree with Kestutis, expcept it is not clear whether our finite group schemes are assumed locally free. (This, however, does not affect the representability of extensions of affine group schemes). – Laurent Moret-Bailly Aug 20 '18 at 18:59