C^*-equivariant modules on a vector bundle vs graded modules on the pushforward.

Question

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.

Answer 1

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 $\mathcal B$.