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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.

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Could you please explain what problems arise if you just try to verify that the obvious candidate for the inverse functor actually works? – user2035 Jun 21 '12 at 18:41
up vote 0 down vote accepted

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 $\mathcal B$.

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Ah, I think I understand. I haven't checked all the details but correct me if I am wrong: 1. First consider the case $E = Spec A \times \mathbf{A}^n$ and $X = Spec A$. This is proved exactly the same way as showing that a $G_m$ action on an affine variety is equivalent to a grading on the coordinate ring and that equivariant sheaves are graded modules (easiest method here is to use the language of comodules as you suggest). 2. Now the general case is a local to global argument. This also generalizes to just affine maps, as long as the affine covers can be chosen to be $G_m$-stable. – Reladenine Vakalwe Jun 22 '12 at 22:56
Yes. If you already know the argument for the affine case, you can also just check that it works identically for the relatively affine case. – user2035 Jun 23 '12 at 10:24

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