I am learning about group schemes at the moment. When it comes to finite group schemes, every author I have read so far restricts himself to the case of schemes which are also flat over the base, sometimes with the remark that it does not make much sense to consider a finite group scheme which is not flat. On the other hand the base scheme is usually assumed at least locally noetherian and in this case finite flat and locally free of finite rank are equivalent. I have thus two questions:
- Why does it not make sense to consider a finite group scheme that is not flat? Why do we assume the scheme to be flat and not only to be locally free of finite rank over the base?
- Why do we need the noetherian hypothesis on the base scheme? Could we not just consider schemes that are locally free of finite rank over an arbitrary base?
I am grateful for any insightful remark on this topic.
See Group schemes of prime order by J. Tate and F. Oort, first paragraph. A group scheme $G$ over an arbitrary base $S$ is said to be of finite order if $G = \textrm{Spec}(\mathscr{A})$ where $\mathscr{A}$ is an $\mathscr{O}_S$-Algebra which is locally free of constant finite rank. Then is stated, without proof (but it is an easy exercise), the following : if $S$ is locally noetherien and connected, $G$ is of finite order if and only if it is flat and finite over $S$.
Now, why the noetherian or locally noetherian hypothesis on the base ? Why locally noetherian if noetherian being obvious, why noetherian ? I would guess that it is historical, as first "concrete" group schemes that have been studied were group schemes over fields, rings of integers of number fields, or $p$-adic rings, all those rings being noetherian.
Remark. My answer tried to emphasize that "locally noetherian" was the real deal here (as almost everywhere in scheme theory), not "flat". Because it's generally easier to prove things under the locally noetherian hypothesis. About this, search "élimination des hypothèses noethériennes" in EGA IV, part 3, paragraph 8, section (8.9), paragraph 11, section 11.3, more generally, EGA IV part 3 paragraphs 8 up to and included 11. There's some kind of balance in fact : you prove very general results about $S$-morphisms of schemes with any finiteness hypothesis on the morphisms but with the hypothesis of local noetherianess on the base $S$, or you remove this hypothesis on $S$, and try to still prove prove general results on $S$-morphisms, but then you need some kind of finiteness hypothesis on morphisms, and the right one is usually "being a morphism of finite presentation". The advantage being that even if local noetherianess is not stable by base change, being of finite presentation is. This balance is really seen in EGA IV, where in paragraphs 5, 6 and 7 you always have local noetherianess hypothesis on the base but no hypothesis on morphisms, whereas in paragraphs 8, 9, 10 and 11 you remove the hypothesis made on the base, and make the hypothesis "of finite presentation" on morphisms. The introduction of EGA IV (after the end of the chapter 0 in it) gives a lot of details on that.