Let $A$ be a Noetherian integral domain, and $G$ a finite group of automorphisms acting on $A$. Let $B = A^G$, the ring of invariants. The inclusion $B \hookrightarrow A$ induces a surjective morphism of affine schemes: $\pi : Spec(A) \rightarrow Spec(B)$.
EDIT: I will add the assumptions that $A$ is a finitely generated $k$-algebra for some field $k$, and that $Spec A$ is a curve.
Let $\mathcal{L}$ be a $G$-equivariant invertible sheaf on $Spec(A)$. Then one can define the $G$-invariant pushforward $\mathcal{F} = \pi_{\ast}^G \mathcal{L}$ which is a sheaf of $\mathcal{O}_{Spec(B)}$-modules.
David Mumford in Abelian Varieties shows that if the action of $G$ is free, then $\mathcal{F}$ will also be an invertible sheaf on $Spec(B)$. I'm trying to think about the situation where $G$ does not act freely.
What I expect is that for certain invertible sheaves $\mathcal{L}$, the $G$-invariant pushforward $\mathcal{F}$ will also be free. For others, $\mathcal{F}$ will just be a coherent sheaf, with the torsion part supported at the primes that ramify. But I'm having trouble seeing how it works. So suppose the case is that there is a prime ideal $\mathfrak{q} \in Spec(A)$ such that $g(\mathfrak{q}) = \mathfrak{q}$ for all $g \in G$, and put $\mathfrak{p} = \mathfrak{q} \cap A^G$, the prime under $\mathfrak{q}$.
Here's what I was trying. The $G$-equivariant invertible sheaf $\mathcal{L} = \widetilde{M}$ for $G$-$A$-module $M$, that is, a module $M$ and a right action of $G$ such that, for all $x \in M$, $g,h \in G$, and $a \in A$, we have: $x \cdot (gh) = (x\cdot g)\cdot h,$
$(ax) \cdot g = g(a) (x \cdot g)$.
The sheaf $\mathcal{F} = \pi_{\ast}^G \mathcal{L}$ then corresponds to $\widetilde{N}^G$ where $N = (_BM)$, that is, $M$ viewed as a $B$ module. So now I'm trying to look $(N^G)_{\mathfrak{p}}$. But isn't it true that $(N^G)_{\mathfrak{p}} = (M_\mathfrak{q})^G$? And doesn't that imply that $(N^G)_{\mathfrak{p}}$ cannot contain any torsion elements?
I hope someone can explain the situation to me. I know that this is a special case of descent theory, but I feel that all references about that are too complicated for my currents purposes, and I want to keep the arguments as simple as possible. In particular, I want to understand statements of the sort " $\mathcal{L}$ does not descend to the quotient, there exists an $n$ such that $\mathcal{L}^{\otimes n}$ descends".
Thanks very much in advance!