Equivariant sheaves over affine schemes

Let $k$ be a field, let $G$ be a linear algebraic group over $k$ and
let $A$ be a commutative $k$-algebra which is acted on by $G$.

We say that an $A$-module $M$ is a $(G,A)$-module if it satisfies the following two properties:

1) $M$ is a rational $G$-module (over $k$)

2) The multiplication map $A \otimes M \rightarrow M$ is a morphism of
rational $G$-modules.

If we let $X=\text{Spec}(A)$ and view $M$ as a quasi-coherent sheaf on $X$,
then giving $M$ the structure of a $(G,A)$-module should be equivalent to giving it the structure of a $G$-equivariant sheaf on $X$
(see https://en.wikipedia.org/wiki/Equivariant_sheaf). Is there an easy, explict way to see why these two definitions coincide?

Answer

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Source : Link , Question Author : gustav101 , Answer Author : Community