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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.

Somatic Custard
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    How do you define multiplication on $\oplus \mathcal{E}(n)$? – Sasha Nov 17 '21 at 18:36
  • @Sasha Argh, yes good point. It looks like I have some more confusion to sort out, before I should even ask the question. At least for line bundles, one description of the multiplication is given here: https://math.stackexchange.com/questions/4253096/section-rings-map – Somatic Custard Nov 17 '21 at 19:42
  • I believe $\mathscr{A}_{\mathscr{E}}$ is the tensor algebra of $\mathscr{E}$ (see Hartshorne exercise II.5.16). It is not necessarily a sheaf of algebras since multiplication need not be commutative, whereas it is when we take the symmetric algebra. As such, taking Spec might not make sense unless you define Spec for noncommutative rings. – Daniel Nov 17 '21 at 20:21

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