Let $X$ be a scheme and $\mathscr{E}$ be a quasi-coherent $\mathcal{O}_X$-module. Take $X$ to be projective with a very ample bundle $\mathcal{O}_X(1)$ for a structure morphism $\pi: X \to S$, so that we can form the twists $\mathscr{E}(n) = \mathscr{E}\otimes\mathcal{O}_X(n)$.
We have two associated quasi-coherent $\mathcal{O}_X$-algebras
$$\mathscr{A}_\mathscr{E} := \bigoplus\limits_{n}\mathscr{E}^{\otimes n}$$ $$\mathscr{R}_\mathscr{E} := \bigoplus\limits_{n}\mathscr{E}(n)$$
I am trying to understand the roles of these two algebras and how to think about their similarities and differences, but I have had trouble getting a clear picture from the references I've viewed. I will permit $X$ and $\mathscr{E}$ to be sufficiently nice, if it clarifies the situation. I am most interested in the case when $\mathscr{E}$ is a line bundle, or at least locally free.
If $\mathscr{E} = \mathcal{O}_X(1)$, then the algebras are the same.
If $\mathscr{E}$ is a locally free sheaf, then I think that $\mathscr{A}_\mathscr{E} = \mathrm{Sym}^\bullet\mathscr{E}$ is the symmetric algebra, so that $\mathrm{Spec}(\mathscr{A}_\mathscr{E})$ is the total space of the bundle.
The functor $$\Gamma_*: \mathscr{E} \mapsto \bigoplus\limits_n \pi_*\mathscr{E}(n) = \pi_*\mathscr{R}_\mathscr{E}$$ takes quasi-coherent modules to graded modules over the graded coordinate ring of $X$, and establishes an (almost) equivalence between the categories.
Questions:
- What are the proper names and notations for these algebras?
- Should the index of the sum be $n\in\mathbb{N}$ or $n\in\mathbb{Z}$?
- How to think about these algebras, and what are their uses? In particular, how to think about their $\mathrm{Spec}$ and $\mathrm{Proj}$?
- When are the algebras isomorphic? If $\mathscr{E}$ is globally generated?
- What is the significance of $\mathscr{R}_\mathscr{E}$ depending on the choice of ample bundle, but $\mathscr{A}_\mathscr{E}$ not?
Thanks for helping me sort out this confusion.