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 AMM is a morphism of
rational G-modules.

If we let X=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

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