I don't have so much experience in category theory so my question may be stupid and non-sense.


1. There is a classical adjunction in algebraic geometry between the $M\rightarrow M^{\sim}$ and the global section in the affine case .
2. We know that if we deal with quasi-coherent sheaves is an equivalences of categories


can we prove 1 by the following way :


a) by the adjoint functor theorem $M\rightarrow M^{\sim}$ have an adjoint


b) since in a subcategory we know his adjoint (the global section) we can generalize this to the all category ( maybe by proving that there is an unique way to extend the global section functor...)


I know that is clearly not the simple way to do that but I want to improve my category theory skill.


thanks in advance !



prove classic proposition in algebraic geometry by category theory
I am not sure the example you choose is the most striking application of category theory in algebraic geometry
 However, there is a similar question where indeed category theory makes the heart of the phenomenon way clearer.


Let $ f : X \longrightarrow Y$ be a proper morphism of schemes. You get a functor $Rf\_\* : D(X) \longrightarrow D(Y)$ known as the derived push-forward (here $D(X)$ is the (unbounded) derived category of quasi-coherent sheaves on $X$). It is very natural to ask when this functor has a right adjoint.


If you make some serious hypothesis on the singularities of $X$ and $Y$, Grothendieck stated a general existence Theorem for this right-adjoint (this is Grothendieck-Serre duality). The proof of this duality statement as envisioned by the Master is very complicated and proceeds by identifying precisely who is the right adjoint. A complete proof has been written by Hartshorne in **Residues and Duality**. I personnally find it very hard to understand.


On the other hand, in the mid 90ies, Neemann found a beautiful and very simple categorical proof of the existence of such a right adjoint using Brown representability Theorem. This is beautifully written in **Grothendieck Duality via Bousfield's localization and Brown representability**.


Note however that there is minor drawback in Neeman approach : even if $X$ and $Y$ are smooth projective, you don't get a formula for the right adjoint. However, knowing that it exists allows you to find an explicit description of it quite quickly in the Gorenstein case.


