$\mathbb{C}^*$ action on a ring gives grading

Question

I'm reading a paper which makes the following claim:

"Let’s now assume that the singularity is defined by an affine ring $R = \mathbb{C}[x_1, x_2, . . . , x_n]/I$, and the first condition means that the ring is graded, i.e. there is a $\mathbb{C}^∗$ action on the ring."

I haven't been able to find any references to show why this is the case. Does anyone know of a reference that would explain this, or even just an explanation for why this is true?

Answer 1

If your ring $R$ is graded, there is an action the action of your group on $R$ such that $t\in\mathbb C^*$ acts on the homogeneous component of degree $n$ by multiplication by $t^n$.

Conversely, if the group acts on a ring of the form $C[x_1,\dots]/I$, then there is a grading whose component of degree $n$ is the subspace of elements onwhich every group element $t$ acts as multiplication by $t^n$.