I know that a $\mathbf Z$-grading of a $k$-algebra $S$ is the same thing as a group-action of $(\mathbf G_m)_k$ on $Spec(S)$. I was wondering if there is a similar nice geometric description of a $\mathbf Z$-grading of an $S$-modules $M$. I was hoping that this might give me a geometric explanation of the Serre twisting sheaves $\mathcal O_{\mathbb P^n}(m)$.
Let $S$ be a fixed $\mathbf Z$-graded $k$-algebra. An invertible $S$-module $M$, graded or not, can be turned into a line bundle over $Spec(S)$. One way to do this is to send $M$ to the bundle $Spec(Sym\, M)$ or $Spec(Sym\, M^\vee)$. If we extend our category of spaces to $Sch_{/k} \subset Zar_{/k}$, then a second way is the construction described here in the stacks project. Let me denote the resulting space by $\underline{M}\to Spec(S)$.
Question: Given an invertible $S$-module $M$, where $S$ is a $\mathbf Z$-graded $k$-algebra, what its the geometric meaning of a $\mathbf Z$-grading of $M$? Is it maybe the same thing as a $\mathbf G_m$-linearization of one of the line bundles $L\to Spec(S)$ associated to $M$?