I have to construct the quotient of a scheme. Let $G$ is a finite group of automorphisms of a ring.
1) Let $p: \operatorname{Spec} A \mapsto \operatorname{Spec}(A^G)$ the morphism induced by the inclusion $A^G \mapsto A$. It's clear that the action $Pg=g^{-1}(P)$ is natural on $\operatorname{Spec}(A)$. Is it true that $p(x_1)=p(x_2)$ if and only if there exists $g \in G$ such that $(x_1)g=x_2$? The if part is very simple, because the following relations hold
$p(x_2)=p(x_1 g)=p(g^{-1}(x_1))=p(x_1)$. I'm convinced that also the only if part is valid, because of the definition of $G-$orbits, but I can't formalize.
2) Let $a \in A$ and $P(T)=\prod_{g \in G} (T-ag)=T^d+\sum_{i \le d-1}b_iT^i$, with $b_i \in A^G$. If $D(a)$ denote $\operatorname{Spec}(A) \setminus V(a)$, I have to prove that $p(D(a))=\bigcup_i D(b_i)$ and that $p$ is open.
3) I have also to show that for any $b \in A^G$, we have $p^{-1}(D(b))=D(bA)$ and $(A^G)_b=(A_b)^G$.
4) Let $V \subset \operatorname{Spec}(A^G)$ open. Clearly $G$ acts on the subscheme $p^{-1}(V)$ by restriction of the action on $\operatorname{Spec}(A)$. Is it true that $\mathcal{O}_{\operatorname{Spec}(A^G)}(V)=\mathcal{\operatorname{Spec}(A)}(p^{-1}(V))^G$, similarly as the case of ringed spaces?
5) Identify $G$ with a group of automorphisms of the scheme $\operatorname{Spec}(A)$. Does the quotient scheme $\operatorname{Spec}(A)/G$ exist in this conditions?
6) Let $U$ subscheme of $\operatorname{Spec}(A)$ stable under $G$. I have to show that the quotient scheme $U/G$ exists and is isomorphic to $p(U)$.