When I saw this paper of René Schoof, there are two questions on the first page and what confuses me is that how to reduce the first question to the second. This is expalined in the first paragraph on page 2, but I don't understand.
- First question, is every finite locally free group scheme over any base scheme $X$ killed by its order?
- Second question, Let $R$ be a local Artin ring with residue field of characteristic $p>0$. Is every finite free local group scheme G over R killed by its order?
Then the authur said that in order to answer the first question, it suffices to assume $X$ is the spectra of a local ring. Why?
If we assume this, then the authur replaces $R$ by the subring generated by the entries of the matrices and localize. We devote the subring generated by the entries of the matrices by $R'$, then we localize $R'$ at any prime ideal $p$ of $R'$? But $G$ is not a group scheme over $R'_p$, what's going on here?
Then we can assume $G$ is a group scheme over a local Noetherian ring $R$, and why Krull's theorem shows we may even assume that R is a local Artin ring?
Thanks!
