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Let $G$ be an algebraic group, and $X$ a scheme on which $G$ acts: i.e the $S$ points of $G \times X \to X$ is a group for each affine $S$.

Let $F$ be a quasicoherent sheaf on $X$. There is a notion of a $G$ equivariant sheaf here:

I am wondering if it is equivalent to the following definition (which in my eyes is simper, being more geometric):

Let $T$ be the total space of $F$, by which I mean relative $\operatorname{Spec_X}$ of the symmetric algebra on $F$. $\pi : T \to X$ is the natural affine map. (Maybe there is some subtlety in the extent to which this is related to $F$. I think $F$ is still the degree 1 piece of $\pi_* O_T$.)

Let $F$ is a $G$ equivariant sheaf on $X$ if the action of $G$ on $X$ extends to an action of $G$ on $T$. In symbols:

There is some $G \times T \to T$ which is a group-scheme action. Moreover, this action commutes with $\pi : T \to X$, so that the natural diagram involving the actions and $G \times T \to G \times X$ and $T \to X$ commutes.

Is this definition different for some obvious reason that I am not aware of?

Assume $X$ is affine. Let $O(Y)$ denote the total sections of the structure sheaf of some scheme $Y$. Then the $O(G)$ comodule structure on $F(X)$ should be seen by dualizing the equation $G \times T \to T$. In particular, we have $O(T) \to O(G) \otimes O(T)$, and I expect that this can be restricted to degree 1 to reconstruct the comultiplication $F(X) \to F(X) \otimes O(G)$.

So for instance, from this definition it is easy to feel how the category of $G$ equvariant quasicoherent sheaves on $X$ equivalent to the category of $H$ comodules which are $A$ modules, such that the multiplication $A \otimes M \to M$ is a map of comodules (is there a snappy name for this, better than (H-co,A)-modules?)

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    $\begingroup$ One geometric way of saying what a $G$-equivariant sheaf on $X$ is is that it's a sheaf on the stacky quotient $X/G$. This tells you what sort of things you have to do to push forward or pull back a $G$-equivariant sheaf: provide a map from or to $X/G$ respectively. $\endgroup$ – Qiaochu Yuan Jan 26 '16 at 1:57
  • $\begingroup$ @QiaochuYuan I was wondering if that was true, after nlab suggested to me that I could think of G equivariant bundles on a set as bundles on their quotient groupoid. Unfortunately, I don't know anything about stacks. This seems like a nice opportunity to learn a bit about them - maybe you can suggest a reference? $\endgroup$ – Lorenzo Najt Jan 26 '16 at 1:58

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