$\mathbb Z$-graded coordinate ring $\iff$ $\mathbb C^*$ action on affine variety

Question

Let $X$ be an affine variety with coordinate ring $\mathcal O$. I want to show that a $\mathbb C^*$ action on $X$ is equivalent to a $\mathbb Z$-grading on $\mathcal O$. I have problems with both directions.

1. Let's say we have an action $\phi:\mathbb C^*\times X\to X$ (we see it as a morphism between varieties), we have an algebra morphism $\phi^*:\mathcal O\to \mathcal O\otimes \mathbb C[t,t^{-1}]$ ($O(\mathbb C^*)=\mathbb C[t,t^{-1}]$).

So we can write $\phi^*(f)=\sum_{j=0}^{m}g_j\otimes(\sum_{n\in \mathbb Z}\lambda _{n,j}t^n)$. I want to say that the homogeneous components of $\mathcal O$ are of the form $\mathcal O_n=\{f\in \mathcal O|\lambda_{k,j}=0\quad \forall k\neq n,\forall j\}$. So I think we can see it as elements with image in the tensor product which can be written as $g\otimes t^n$.

My problem comes when I have to show that $\mathcal O=\sum_{n\in \mathbb z}\mathcal O_n$. Yes $\phi^*(f)=\sum h_i\otimes t^i$ by rearranging, but can I show that $h_i\otimes t^i$ is in the image of $\phi^*$ ?

2. Let's say we have a $\mathbb Z$-grading $\mathcal O\cong \bigoplus_{n\in \mathbb Z}\mathcal O_n$. We want an action $\mathbb C^*\times X\to X$. I'm not sure what to do for this one. Any ideas ?

Answer 1

For part 1) you are not using the fact that it is a group action. There are two maps $C^*\times C^*\times X\to X$.$(a,b,x)\mapsto \phi(a,\phi(b,x))$ and $\phi(ab,x)$. Group action would say that these are the same. You should be able to finish the rest.

For part 2), define $\phi^*$ as follows. Writing $f=\sum f_i$ as you have, $\phi^*(f)=\sum f_it^i$ and check that it is a group action.