Let us fix a base ring $k$. The category of $\mathbb{Z}$-graded $k$-algebras is equivalent to the category of $\mathbb{G}\_m$ equivariant affine $k$-schemes. The following 2 properties often come up when constructing the $Proj$ of a graded ring: Let $A$ be a graded $k$-algebra.


1. The zeroth graded piece is the base ring $A\_0 =k$
2. The algebra $A\_\bullet$ is positively graded. i.e. $A\_j=0$ for all $j<0$.
3. The canonical multiplication homomorphism $Sym^\bullet A\_1 \to A\_\bullet$ is surjective.


I'm trying to find equivalent characterization of the above properties which don't use the aforementioned equivalence. In other words I want the above 3 properties expressed in terms of properties of $\mathbb{G}\_m$-equivariant schemes (orbits, stabilizers, fixed points etc.).



Expressing properties of graded algebras in terms of the $\mathbb{G}_m$action
Let $V$ be an affine variety with a $\mathbb{G}\_m$-action. Then you should think of $A\_0$ as a good model for $V/\mathbb{G}\_m$. So being connected says something like $\mathbb{G}\_m$ acts transitively- at least from the point of view of functions. Being positively graded is saying something like the action of $\mathbb{G}\_m$ extending to an action of the monoid scheme $\mathbb{A}^1$ (under multiplication). The last condition is more involved: The module $A\_1$ is the set of $\mathbb{G}\_m$-equivariant functions to $\mathbb{A}^1$ where we use the scaling action on $\mathbb{A}^1$. (So it's kind of like "linear" functions on $V$). This is a vector space over $k$, call its *dual* $W$ and view it as a variety. There's a natural equivariant map $V \to W$ given by evaluation (that's why we took the dual of the vector space of functions). We're asking that this equivariant map be an embedding. So, while $V$ is not a vector space with the standard linear action of $\mathbb{G}\_m$, it can be embedded in one in some natural way.


