Grothendieck in SGA 1 introduces a proposition in expose 5 (proposition 3.1) which states:

Let $X$ be etale, separated of finite type over $Y$, locally noetherian, and let $G$ be a finite group which operates on $X$ by $Y$-automorphisms. Then $G$ operates admissibly and the quotient scheme $X/G$ is etale over $Y$.

The hint he gives is that we may show this for $X$ quasi-projective, and to use proposition 1.8, which states that $G$ operates admissibly on $X$ iff $X$ is the union of open affines that are invariant under the action of $G$.

I am unsure how to show this. Help please?

EDIT: so I understand how to make the reduction to the quasi-projective case (since every etale morphism is quasi-finite and radicial, and then just apply Zariski's main theorem, and we get thus that this is quasi-projective), but I am unsure how to show that if a finite group operates on a quasi-projective scheme then it operates admissibly. I have a rough idea for how it could be done for a projective scheme, but I do not see how I could alter it for quasi-projective :( I could post this proof if it is of any help to readers??

Grothendieck in SGA 1 introduces a proposition in expose 5 (proposition 3.1) which states:

Let $X$ be etale, separated of finite type over $Y$, locally noetherian, and let $G$ be a finite group which operates on $X$ by $Y$-automorphisms. Then $G$ operates admissibly and the quotient scheme $X/G$ is etale over $Y$.

The hint he gives is that we may show this for $X$ quasi-projective, and to use proposition 1.8, which states that $G$ operates admissibly on $X$ iff $X$ is the union of open affines that are invariant under the action of $G$.

I am unsure how to show this. Help please?

EDIT: so I understand how to make the reduction to the quasi-projective case (since every etale morphism is quasi-finite, and then just apply Zariski's main theorem, and we get thus that this is quasi-projective), but I am unsure how to show that if a finite group operates on a quasi-projective scheme then it operates admissibly. I have a rough idea for how it could be done for a projective scheme, but I do not see how I could alter it for quasi-projective :( I could post this proof if it is of any help to readers??

Grothendieck in SGA 1 introduces a proposition in expose 5 (proposition 3.1) which states:

Let $X$ be etale, separated of finite type over $Y$, locally noetherian, and let $G$ be a finite group which operates on $X$ by $Y$-automorphisms. Then $G$ operates admissibly and the quotient scheme $X/G$ is etale over $Y$.

The hint he gives is that we may show this for $X$ quasi-projective, and to use proposition 1.8, which states that $G$ operates admissibly on $X$ iff $X$ is the union of open affines that are invariant under the action of $G$.

I am unsure how to show this. Help please?

Grothendieck in SGA 1 introduces a proposition in expose 5 (proposition 3.1) which states:

Let $X$ be etale, separated of finite type over $Y$, locally noetherian, and let $G$ be a finite group which operates on $X$ by $Y$-automorphisms. Then $G$ operates admissibly and the quotient scheme $X/G$ is etale over $Y$.

The hint he gives is that we may show this for $X$ quasi-projective, and to use proposition 1.8, which states that $G$ operates admissibly on $X$ iff $X$ is the union of open affines that are invariant under the action of $G$.

I am unsure how to show this. Help please?

EDIT: so I understand how to make the reduction to the quasi-projective case (since every etale morphism is quasi-finite and radicial, and then just apply Zariski's main theorem, and we get thus that this is quasi-projective), but I am unsure how to show that if a finite group operates on a quasi-projective scheme then it operates admissibly. I have a rough idea for how it could be done for a projective scheme, but I do not see how I could alter it for quasi-projective :( I could post this proof if it is of any help to readers??