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?)