Let $V$ be a $\Bbbk$-variety such that $\Bbbk^\times$ (as an algebraic group) acts algebraically on $V$. Given any $f\in\Bbbk[V]$, let us call $f$ **homogeneous of degree $d$** if for all $v\in V$ and all $\lambda\in\Bbbk^\times$, we have $f(\lambda.v)=\lambda^d f(v)$. 


My question is: Does this define a grading on $\Bbbk[V]$? 


I was convinced that it is true, but I am running into difficulties. Let us first assume $\Bbbk=\mathbb{C}$, the ground field should not be an obstruction. The linear span of $\Bbbk^\times f$ decomposes since $\Bbbk^\times$ is reductive, but I don't see how to turn this into a grading on all of $\Bbbk[V]$. 


If it is true, I would really like to see a proof - it should use as little machinery as possible.



Action of k* on a variety induces grading?
Turning the action map of varieties into a map of rings, we get a ring map $\phi$ from $k[V]$ to $k[V][t,t^{-1}] $, the coordinate ring with an extra invertible variable (the coordinate on $k^\*$) adjoined. Now, for any function $\phi(f)=\sum\_{i\in \mathbb{Z}}f\_it^i$ for some $f\_i$'s, almost all of which are 0. Note that $f=\sum f\_i$, which we obtain by restricting the function to $t=1$. Using associativity, applying $\phi$ again to the $f\_i$'s is the same as applying pull-back by the multiplication map to t. Thus, as functions on $V\times k^\*\times k^\*$ (letting $t,u$ be the two coordinates)


$$\sum\_{i\in \mathbb{Z}}\phi(f\_i)u^i=\sum\_{i\in \mathbb{Z}} f\_i t^i u^i$$ 


since the pull-back of the coordinate by multiplication is just the product of the coordinates . Thus, $\phi(f\_i)=f\_it^i$. 


We can define the grading by letting $f$ be homogeneous of degree $i$ if $\phi(f)=ft^i$. We have already seen that every element can be written uniquely as a sum of such elements (the $f\_i$'s), and this is multiplicative since $\phi$ is a ring homomorphism.


Alternatively, we can note that we have proven that the span of the $f\_i$'s is an finite-dimensional invariant subspace containing $f$, so we can apply your argument. In general, essentially the same argument shows that the action of any affine algebraic group on the coordinate ring of any affine variety by pull-back is a locally finite action: any function is contained in a finite-dimensional invariant subspace.


