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Let $G$ be a finite étale group scheme over a field $k$ and $X=\mathrm{Spec}(A)$ be an affine scheme on which $G$ acts. The categorical quotient $X/G$ exists and may be described as $\mathrm{Spec}(A^H)$ where $H$ is the Hopf algebra associated to $G$ and $A^H$ is the coequalizer of the coaction of $H$ on $A$ and $a\mapsto a\otimes 1$.

Nevertheless, I feel uncomfortable with this description all the more so as $G$ is a finite étale group scheme and is well understood using the equivalence of categories between finite étale group schemes and finite $\mathrm{Gal}(\overline{k}/k)$-groups.

Is there a way to use the latter equivalence in order to describe the quotient? I am especially interested in the case where $X$ is a closed subscheme of the affine space.

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    $\begingroup$ 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. $\endgroup$
    – naf
    Jun 11, 2019 at 10:34
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    $\begingroup$ 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. $\endgroup$
    – LSpice
    Jun 11, 2019 at 11:32
  • $\begingroup$ @ulrich: I am still getting familiar with it. Could you please make a more detailed answer? This would be very helpful! :) $\endgroup$
    – Gaussian
    Jun 11, 2019 at 11:34
  • $\begingroup$ 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. $\endgroup$
    – naf
    Jun 12, 2019 at 6:15

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