Let $L$ be a line bundle over $X$ and let there be a projection $p : X \times G \to X$, where $X$ is projective and $G$ is an algebraic group. I understand the isomorphism given above.
I have two questions. 1)Ginzburg refers me to the whole of Hartshorne chapter 3(on Cohomology) to show this equality when $L$ is a sheaf over $X$. Where can I find it?(Even better why is it true?)
2)This is just a remark: I know why this is true when $L$ is a line bundle:
I am not going to use at all that $G$ is a group. We have that the sequence of maps $X \to X \times G \xrightarrow{p} X$ which is the identity and $G \to X \times G \xrightarrow{p} X$ which is null. Thus given a section $s \in $ $\Gamma(X \times G, p^*L)$ one can pull $s$ back to a section of $\Gamma(G,\epsilon)=\mathbb{C}[G]$ and one can also pull $s$ back to get a section of $\Gamma(X,L)$. Hence one gets a map from $\Gamma(X, p^*(L)) \to \mathbb{C}[G] \otimes \Gamma(X,L)$.
My motivation is that I want to find a map from $H^i(X, p^*(L)) \to \mathbb{C}[G] \otimes H^i(X,L)$.