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

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 $$\phi^*$$ ?

1. 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 ?

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.
• Thank you for 1) I'm not able to finish the rest. I agree with you but I'm not seeing what the group action imply at the level of algebras. The induced morphisms of algebras are the same, if $\phi^*(f)=\sum g_j\otimes t^j$ do we have like $\varphi^*(f)=\sum g_j\otimes t^j\otimes s^j$ ? Similarly for 2) I don't see what to show at the level of algebras to show that we have a group action could you elaborate on this ? @Mohan Apr 28 at 8:38