effective descent of coherent sheaves

Question

I am new to stacks and algebraic spaces. I have the following question:

Let $X$ be a scheme and $G$ be a group scheme action on $X$. Then $X/G$ exists as an algebraic space. Let $\pi: X \to X/G$ be the natural map. Then by pullback and pushforward by $\pi$ we have $$Qcoh(X/G) = Qcoh(X)^G.$$ Do we have such an equality for coherent sheaves? If not then do we have the equality when we assume $G$ is finite or reductive or any other relevant properties?

In particular suppose action of $G$ on $X$ is free and $\pi$ is flat then any $G$-equivariant coherent sheaf descends to a coherent sheaf on $X/G$. My question is when is this descent effective?

Thanks in advance.

Answer 1

The answer depends on the action: If you take an action which is free, then the morphism $\pi$ is étale and, by Etale descent (cf. Bosch-Lutkebohmer-Raynaud "Neron Models page. 139) and you have equality.

But:

Take the action of $\mu_2$ over the affine line which sends $x$ in $-x$. In this case, the morphism $\pi:{\mathbb A}^1\to {\mathbb A}^1$ is the morphism $x\mapsto x^2$.

In this case, you can see that the line bundle $(x)$ (ideal sheaf of the origin) is $\mu_2$-invariant but it is not pull back of a line bundle (its square is).