Quasi-coherent sheaves of algebras over affine scheme

Question

Hi there I am study algebraic geometry for the first time and I have a question. If we consider an affine scheme $\operatorname{Spec}(R)$, is the category of quasi-coherent sheaves of $\mathcal{O}_{R}$-algebras equivalent to the category of finitely generated $A$-algebras? Can you provide a reference?

Answer 1

What you write is not quite correct - the category of quasi-coherent $\mathcal{O}_R$-algebras is equivalent to the category of all $R$-algebras (with no finite-generation hypothesis).

This can be broken down in to two parts: first, the fact that categories of $R$-modules and quasi-coherent $\mathcal{O}_R$-modules are equivalent; second, the fact that an algebra is characterized by being a monoid object in the module category, and this property is preserved under equivalence. A reference for the first part could be Stacks 01IB or Hartshorne proposition II.5.5, and a reference for the second part could be nLab or any abstract algebra text that defines an algebra in terms of the multiplication map $m:A\otimes_R A\to A$, unit map $\eta:R\to A$, and various compatibilities between them. (You may also need something like Stacks 0073 to convince yourself that a sheaf of algebras is a sheaf of modules with an algebra structure, or this may be obvious to you.)