Let $k$ be a field and $X$ be a smooth, complete, geometrically irreducible curve over $k$. For a close point $P$ of $X$ (for the Zariski topology) of degree $d$, let $A$ be the subring of $k(X)$ (the function field of $X$) consisting in elements that have no poles away from $P$. For $a\in A$, let $\deg_P a=-d v_P(a)$ be the degree map at $P$.
Starting from $A$, I want to recover $X$. To do so, I define $$A_{\operatorname{Gr}}=\prod_{n\geq 0}{\{a\in A| \deg_P a\leq n\}}$$ and I give to it the structure of a graded $k$-algebra, where the $n^{th}$ grading is given by the $n^{th}$ factor.
Is that true that $X=\operatorname{Proj}(A_{\operatorname{Gr}})$ ?
Similary, for $\mathfrak{a}$, ideal of $A$, we define $$\mathfrak{a}_{\operatorname{Gr}}=\prod_{n\geq 0}{\{a\in \mathfrak{a}| \deg_P a\leq n\}}.$$
Is that true that $\tilde{\mathfrak{a}}_{\operatorname{Gr}}$ is a rank one coherent sheaf of $\mathcal{O}_X$-modules? What should be the Cartier divisor $V_{\mathfrak{a}}$ corresponding to it?
A reference will be enough I think. Many thanks for your clarifications!