Equivalent definitions of equivariant sheaves on affine schemes

Question

Let's work in a category of affine $k$-schemes, suppose $G = \mathrm{Spec}(H)$ is an algebraic group, and that $X = \mathrm{Spec}(A)$ has a $G$-action. Write $s: G \times X \to X$ for the action and $p: G \times X \to X$ for the projection onto the second factor. I am told that there are two equivalent definitions one could give of an equivariant (quasicoherent) sheaf $\mathcal{F}$ on $X$:

• A qc sheaf $\mathcal{F}$ equipped with an isomorphism $\theta: s^\ast \mathcal{F} \to p^\ast \mathcal{F}$ satisfying the usual cocycle condition
• $\mathcal{F} = \widetilde{M}$, where $M$ is a comodule for the Hopf algebra $H$ such that the $A$-module action $A \otimes M \to M$ is a map of $H$-comodules. (Here $A \otimes M$ is given the usual $H$-comodule structure as a tensor product of $H$-comodules, since $A$ gets a comodule structure from the group action $G \times X \to X$.)

I understand how to construct the correspondence but I struggle to verify all the necessary details. Given a qc sheaf $\mathcal{F}$ in the first definition, the comodule structure comes from taking the adjoint $\mathcal{F} \to s_\ast p^\ast \mathcal{F}$ (let's write this as the map $c: M \to H \otimes M$ on global sections), and the fact that this is an $A$-module homomorphism gives us the fact that $A \otimes M \to M$ is a map of $H$-modules, due to the diagram $$\require{AMScd} \begin{CD} A \otimes M @>{1 \otimes c}>> A \otimes (H \otimes M)\\ @VVV @V{\mu \otimes 1}VV \\ @. (H \otimes A) \otimes (H \otimes M) \\ @V{m}VV @VVV \\ @. (H \otimes H) \otimes (A \otimes M) \\ @VVV @V{m \otimes m}VV \\ M @>{c}>> H \otimes M \end{CD}$$ Here I'm using $\mu: A \to H \otimes A$ to denote the comultiplication on $A$ given by the group action and by an abuse of notation $m$ to denote both the multiplication $H \otimes H \to H$ and $A \otimes M \to M$. The commutativity of this diagram shows that $\mathcal{F} \to s_\ast p^\ast \mathcal{F}$ is an $A$-module homomorphism iff $m: A \otimes M \to M$ is an $H$-comodule homomorphism.

My question is this:

• The comodule structure map $c: M \to H \otimes M$ gives (by this diagram) an $\mathcal{O}_X$-module morphism $\mathcal{F} \to s_\ast p^\ast \mathcal{F}$, but why is the induced map $\theta: s^\ast \mathcal{F} \to p^\ast \mathcal{F}$ an isomorphism?

I welcome any reference which spells out this construction in detail. The corresponding question on MO doesn't have an official answer and the linked source only considers the case where $G$ acts trivially, in which case the proof is easy.

Answer 1

I was trying to pull back the cocycle condition along the map $G \times X \to G \times X$ given by $(g, x) \mapsto (g^{-1}, g, x)$ to reverse the direction of the map on the fibers but as far as I can tell this doesn't work. Direct computation shows that the map $s^\ast \mathcal{F} \to p^\ast \mathcal{F}$ is given by the composition $$H \otimes M \xrightarrow{1 \otimes c} H \otimes (H \otimes M) \xrightarrow{m \otimes 1} M$$ and this suggests that we should define the inverse by $$H \otimes M \xrightarrow{1 \otimes c} H \otimes (H \otimes M) \xrightarrow{1 \otimes S \otimes 1} H \otimes H \otimes M \xrightarrow{m \otimes 1} H \otimes M$$ and this works.