Assume $ \pi:Y \rightarrow X$ is an unramified double cover of smooth projective varieties with Galois group $ G:=<\iota>$ , $ \iota^2=id_Y$ and $ X=Y/G$ , so $ G$ acts freely on $ Y$ (everything over $ \mathbb{C}$ ).

We have furthermore an additional sheaf of rings or algebras $ R_X$ on $ X$ , which gives us $ R_Y:=\pi^{*}R_X$ on $ Y$ . Then a left $ R_Y$ -module $ M$ defines the left $ \iota^{*}R_Y$ -module $ \iota^{*}M$ . But $ \iota^{*}R_Y=\iota^{*}\pi^{*}R_X\cong(\pi\circ\iota)^{*}R_X=\pi^{*}R_X=R_Y$ . So it makes sense to ask:

$ \textbf{Q:}$ If $ M$ is a quasi-coherent $ \mathcal{O}_Y$ -module which is a simple left $ R_Y$ -module, meaning $ Hom_{R_Y}(M,M)=\mathbb{C}$ , such that there is an isomorphism $ \phi: \iota^{*}M\rightarrow M$ of $ R_Y$ -modules, then does $ M$ and its $ R_Y$ -module structure descend to $ X$ ?

That is: can we find a quasi-coherent sheaf $ N$ on $ X$ , sucht that $ N$ is a left $ R_X$ -module with an isomorphism of left $ R_Y$ -modules $ \pi^{*}N\cong M$ ?

I know this argument is true if we take $ R_X=\mathcal{O}_X$ , because in this case we can take the isomorphism $ \phi: \iota^{*}M\rightarrow M$ and look at $ \iota^{*}\phi: (\iota^2)^{*}M \rightarrow \iota^{*}M$ . But $ \iota^2=id_Y$ so $ \phi\circ\iota^{*}\phi: M\rightarrow M$ , which by simplicity must be a multiple of $ id_Y$ . So by multiplication with a scalar we may assume $ \phi\circ\iota^{*}\phi=id_Y$ . But this is exactly the necessary descent datum for $ M$ , hence $ M=\pi^{*}N$ .

I think this proof shows that in general we have $ M=\pi^{*}N$ as $ \mathcal{O}_Y$ -modules for some $ \mathcal{O}_X$ -module $ N$ . But how to see if $ N$ is actually an $ R_X$ -module such that $ \pi^{*}N$ gives back $ M$ as left $ R_Y$ -modules? Or do we need more conditions for the module struture to also denscend to $ X$ ?