I am reading Simpson's paper on The Hodge filtration on nonabelian cohomology. In particular Chapter 5 (p.24) and I am confused about the notion of a group acting on an equivariant sheaf.

The set up is as follows:

$\mathbb{A}^1=Spec(\mathbb{C}[z])$ admits an obvious action by the multiplicative group $\mathbb{G}_m=Spec(\mathbb{C}[z,z^{-1}])$ \begin{equation} a:\mathbb{G}_m\times \mathbb{A}^1\to \mathbb{A}^1. \end{equation} which is equivalent to giving the canonical grading on $\mathbb{C}[z]$ (cf. here).

Now what is a $\mathbb{G}_m$-equivariant vector bundle over $\mathbb{A}^1$?

For me this is a $\mathbb{G}_m$-equivariant locally free sheaf $W$. In the affine case this reduces to a locally free module $B=W(\mathbb{A}^1)$ with a grading that respects the grading of $\mathbb{C}[z]$ or equivalently a comodule map $B\to B\otimes_\mathbb{C} \mathbb{C}[z,z^{-1}]$.

So I was quite surprised when I found the following comment in a more written out version of the paper*:

It seems like there is an identification $\mathbb{G}_m$ with $\mathbb{C}^*$, which is confusing to me.

- How does this action of $\mathbb{G}_m$ (or rather $\mathbb{C}^*$ ?) on $\mathbb{C}[z,z^{-1}]\otimes_\mathbb{C} V$ induce an equivariant structure on the sheaf or equivalently a comodule structure?
- Is my approach for the definition correct to begin with?

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