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:

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..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).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.

commutativegroup schemes? Surely the category of all finite group schemes cannot be abelian. $\endgroup$ – R. van Dobben de Bruyn Aug 20 '18 at 0:41