All varieties are over $\mathbb{C}$.


Let $X$ be a variety and $\pi \colon E \to X$ a geometric vector bundle. So $ \pi $ is affine. Then certainly the assignment $ M \mapsto \pi\_\*M $ defines an equivalence between quasi-coherent $\mathcal{O}\_E$-modules and quasi-coherent $\pi\_\*\mathcal{O}\_E$-modules.


Now $\mathbb{C}^{\times}$ acts on $E$ via dilation of the fibres of $\pi$. So $\pi\_\* \mathcal{O}\_E$ acquires a grading. Is it true that $M \mapsto \pi\_\* M$ gives an equivalence between $\mathbb{C}^{\times}$-equivariant quasi-coherent $\mathcal{O}\_E$-modules and graded quasi-coherent $\pi\_\*\mathcal{O}\_E$-modules?


If this is true, does it generalize to replacing $\pi$ being a vector bundle with $E$ just equipped with a $\mathbb{C}^{\times}$-action, $\pi$ affine and $\mathbb{C}^{\times}$-equivariant, $\mathbb{C}^{\times}$ acting on $X$ trivially?



**Added later** (in response to a-fortiori's comment): Perhaps I hadn't done my homework as conscientiously as I thought. Regardless, here are some thoughts. As candidate for the quasi-inverse (is there a more sensible choice?) take


$N\mapsto \mathcal{O}\_E \otimes\_{\pi^{-1}\pi\_\*\mathcal{O}\_E}\pi^{-1}N$


with $\mathbb{C}^{\times}$-equivariant structure given by


$z \cdot (f(x,v) \otimes n\_i) = f(x, z^{-1}v) \otimes z^{-i}n\_i$,


where $n\_i$ is in the $i$-th component of $N$ and the rest of the notation is (I hope) self-explanatory. Hitting the structure sheaf $\mathcal{O}\_E$ (with the trivial/evident equivariant structure) with these functors works fine, so this isn't completely ridiculous. But now I am not even sure whether there are other equivariant structures on $\mathcal{O}\_E$ that would make this breakdown.



C^*-equivariant modules on a vector bundle vs graded modules on the pushforward.
I think the picture is clearer if you do everything on $X$. With $\mathcal B=\pi\_\ast\mathcal O\_E$, the operation of $\mathbf G\_m$ becomes a homomorphism $\mathcal B\to\mathcal B\otimes\mathbf C[t^{\pm1}]$ which is a $\mathcal O\_X\otimes\mathbf C[t^{\pm1}]$-comodule structure; and $\mathbf G\_m$-equivariant quasi-coherent sheaves on $E$ correspond to quasi-coherent $\mathcal B$-modules $\mathcal N$ together with a homomorphism $\mathcal N\to\mathcal N\otimes\mathbf C[t^{\pm1}]$ which is a $\mathcal O\_X\otimes\mathbf C[t^{\pm1}]$-comodule structure such that the multiplication $\mathcal B\otimes\mathcal N\to\mathcal N$ is a comodule homomorphism. (For this part, $\mathbf G\_m$ may be replaced by any affine group.)


Now the comodule structures translate into gradings, and the last condition says that the grading on $\mathcal N$ is compatible with the grading on $\mathcal B$.


