I am reading a book titled "Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group" by J.P. Murre. I am in the chapter 4 titled "Fundamental groups". Here he fixes a base locally noetherian scheme $S$ and defines a category $\mathscr{C}$ taking the objects to be locally noetherian schemes with finite etale maps to $S$. Morphisms with in the objects are finite etale maps between them.

At the beginning of this chapter he is stating the basic properties of this category, like existence of fibre product and disjoint union. I am reading the proof of the existence of the quotient by a finite subgroup $\mathfrak{g}$ of the automorphisms of an object $X$. (I am unable to write proper notation for $\mathfrak{g}$, because I do not know what it is. What is the notation?) He states that the problem is enough to be solves locally and goes on to define a finite subgroup of automorphisms of $\operatorname{Spec}(A)$ corresponding to $\mathfrak{g}$. I am unable to understand what this means. It is not even clear to me why the problem is enough to be solved for affine open subschemes. I have attached the snapshot below.

I am reading a book titled "Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group" by J.P. Murre. I am in the chapter 4 titled "Fundamental groups". Here he fixes a base locally noetherian scheme $S$ and defines a category $\mathscr{C}$ taking the objects to be locally noetherian schemes with finite etale maps to $S$. Morphisms with in the objects are finite etale maps between them.

At the beginning of this chapter he is stating the basic properties of this category, like existence of fibre product and disjoint union. I am reading the proof of the existence of the quotient by a finite subgroup $\mathfrak{g}$ of the automorphisms of an object $X$. (I am unable to write proper notation for $\mathfrak{g}$, because I do not know what it is. What is the notation?) He states that the problem is enough to be solved locally and goes on to define a finite subgroup of automorphisms of $\operatorname{Spec}(A)$ corresponding to $\mathfrak{g}$. I am unable to understand what this means. It is not even clear to me why the problem is enough to be solved for affine open subschemes. I have attached the snapshot below.

I am reading a book titled "Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group" by J.P. Murre. I am in the chapter 4 titled "Fundamental groups". Here he fixes a base locally noetherian scheme $S$ and defines a category $\mathscr{C}$ taking the objects to be locally noetherian schemes with finite etale maps to $S$. Morphisms with in the objects are finite etale maps between them.

At the beginning of this chapter he is stating the basic properties of this category, like existence of fibre product and disjoint union. I am reading the proof of the existence of the quotient by a finite subgroup $\mathfrak{g}$ of the automorphisms of an object $X$. (I am unable to write proper notation for $\mathfrak{g}$, because I do not know what it is. What is the notation?) He states that the problem is enough to be solves locally and goes on to define a finite subgroup of automorphisms of $Spec(A)$ corresponding to $\mathfrak{g}$. I am unable to understand what this means. It is not even clear to me why the problem is enough to be solved for affine open subschemes. I have attached the snapshot below.

I am reading a book titled "Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group" by J.P. Murre. I am in the chapter 4 titled "Fundamental groups". Here he fixes a base locally noetherian scheme $S$ and defines a category $\mathscr{C}$ taking the objects to be locally noetherian schemes with finite etale maps to $S$. Morphisms with in the objects are finite etale maps between them.

At the beginning of this chapter he is stating the basic properties of this category, like existence of fibre product and disjoint union. I am reading the proof of the existence of the quotient by a finite subgroup $\mathfrak{g}$ of the automorphisms of an object $X$. (I am unable to write proper notation for $\mathfrak{g}$, because I do not know what it is. What is the notation?) He states that the problem is enough to be solves locally and goes on to define a finite subgroup of automorphisms of $\operatorname{Spec}(A)$ corresponding to $\mathfrak{g}$. I am unable to understand what this means. It is not even clear to me why the problem is enough to be solved for affine open subschemes. I have attached the snapshot below.