I'm trying to translate the Proj construction as a kind of quotient by a $\mathbb{G}_m$ action. Here's what I have so far:

Let $X=Spec\,A$ be an affine scheme (after this case is setteled I imagine it would be relatively easy to generalize to relatively affine schemes and get global Proj). Let $\mathbb{G}_m = Spec\,\mathbb Z[x,x^{-1}]$ be the multiplicative group.

The data of an action $\rho :X \times \mathbb{G_m} \to X$ is equivalently a homomorphism $\rho^{\flat}: A \to A[x,x^{-1}]$ which is equivalently a grading on $A$ given by $A=\bigoplus_{d \in \mathbb{Z}} A_d = \bigoplus_{d \in \mathbb{Z}} (\rho^{\flat})^{-1}(Ax^d)$.

This is very similar to a decomposition into irreducible subrepresentations only it doesn't look like a representation but rather a "co-representation".

**How should I think of this "co-representation?"** Is the theory of comodules over a hopf algebra just "dual" in some sense to the that of modules over a hopf algebra? (meaning everything is practically the same upto flipping arrows).

As far as I understand at this point we'd like to consider the free locus of the action on $X$. This is reasonable (coming from differential geometry where free actions by compact groups always exist) although I wonder if this technique works for any action.

Question 1 : Does the quotient of a scheme by a free action of a group scheme always exist? (as a scheme).

Interpreting the "free locus" of the action in terms of algebra is pretty confusing for me. Having knowledge of the solution these are supposed to be prime submodules of $A$ (prime ideals) which contain the irrelevant ideal $A_+$. The irrelevant ideal corresponds geometrically to the maximal irreducible subset of $X$ on which $\mathbb{G}_m$ acts trivially. So primes containing it correspond to irreducible sets contained in it which we should throw away as well. **Is this true/enough?**

Having thrown out the problematic points we get a free action on $X\text{ \ }X_p$ which we can now hopefully quotient by the action and get a $\mathbb{G}_m$ torsor. $SpecA \text{ \ } Spec{A_0} \to Proj A$. **The fact that orbits of the action are in 1-1 with homogeneous prime ideals (not containing the irrelevant ideal) is a consequence of the fact that the orbit of every point in the free locus is itself a point (irreducible subscheme).**

Having a torsor we have an equivalence of categories between $\mathbb{G}_m$-equivariant sheaves on $SpecA \text{ \ } Spec{A_0}$ and sheaves on $Proj A$

Question 2:Is this correct?

Question 3:What's a good reference for equivariant sheaves, quotients by group schemes and quotient stacks in the context of geometric representation theory?

sufficientcondition is that A[1/f_i] contain an invertible element of degree 1. As Friedrich pointed out below, this is much weaker than being generated in degree ±1 (though it is equivalent for polynomial rings). $\endgroup$2more comments