Timeline for Quotient of an affine scheme by an étale finite group

7 events

when toggle format	what		by	license	comment
Jun 12, 2019 at 6:15	comment	added	naf		Let $K$ be a finite Galois extension over which the group scheme $G$ splits. Then $G_K$ acts on $A_K := A \otimes_k K$ and $Gal(K/k)$ also acts on $A_K$ via its action on $K$. What you need to show is that the action of $Gal(K/k)$ preserves $(A_K)^{G_K}$; the invariants of this action will then give the quotient over $k$ that you want.
Jun 11, 2019 at 12:55	review	Close votes
Jun 16, 2019 at 3:05
Jun 11, 2019 at 11:34	comment	added	Gaussian		@ulrich: I am still getting familiar with it. Could you please make a more detailed answer? This would be very helpful! :)
Jun 11, 2019 at 11:32	comment	added	LSpice		A minor TeX point: you want $\mathrm{Spec}(A)$ `\mathrm{Spec}(A)` or ${\rm Spec}(A)$ `{\rm Spec}(A)`, not $\rm{Spec}(A)$ `\rm{Spec}(A)`. Even better is $\operatorname{Spec}(A)$ `\operatorname{Spec}(A)`. I have edited accordingly.
Jun 11, 2019 at 11:31	history	edited	LSpice	CC BY-SA 4.0	TeX fixes
Jun 11, 2019 at 10:34	comment	added	naf		If you are comfortable with descent, then you can take the quotient over a finite Galois extension of $k$ over which the group splits and then show that it has descent data with respect to the field extension.
Jun 11, 2019 at 10:24	history	asked	Gaussian	CC BY-SA 4.0