# 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. Aug 20, 2018 at 0:41
• @R.vanDobbendeBruyn, yes definitely! Edited the question - thanks for bringing this up! Aug 20, 2018 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). Aug 20, 2018 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. Aug 20, 2018 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). Aug 20, 2018 at 18:59