What does it mean for the coordinate ring of an affine variety to be graded?

Question

My question is relatively simple, assume that $X$ is an affine variety such that its coordinate ring $A:=\Bbbk[X]:=H^0(\mathcal O_X,X)$ is $\Lambda$-graded for some monoid $\Lambda$.

Now if $\Lambda=(\mathbb Z,+)$ and the grading is in some sense 'compatible' with the one on the polynomial ring after embedding $X$ into affine space, then this would imply that $X$ is a cone, i.e. cut out by homogeneous polynomials. More precisely, assume $\phi:\Bbbk[X_0,\ldots,X_n]\twoheadrightarrow A$ induces a closed immersion $X\hookrightarrow\mathbb A^n$. Then if $\phi$ is a morphism of graded algebras, we know that $\ker(\phi)=I(X)$ is generated by homogeneous polynomials. However, I would like to be able to see this kind of thing from the grading on $A$ in a more intrinsic manner, i.e. without talking about compatibility to some specific embedding into affine space.

Is there a result that makes this vague notion of compatibility precise? What other kind of geometrical information can I obtain from properties of a grading on $A$? I'd be very thankful for any kind of insight on this, admittedly, very broad matter.

Answer 1

(Nitpick: "is graded" is a mild type error. A grading is a structure, not a property.)

One geometric interpretation is as follows. Suppose $A$ is an abelian group which acts on $X$. Then $k[X]$ naturally becomes a linear representation of $A$. If this action splits up into a direct sum of one-dimensional representations, then $k[X]$ naturally becomes graded by the group of characters $A \to k^{\times}$. In the previous we may also internalize $A$ to an algebraic group and then talk about the algebraic characters. For example, if $A = k^{\times}$ then the algebraic characters $k^{\times} \to k^{\times}$ are precisely the power maps $x \mapsto x^n$, and if $k^{\times}$ acts on $X$ then $k[X]$ naturally acquires a $\mathbb{Z}$-grading.