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I would like to know which kind of adjunction in algebraic geometry can be prove by the adjoint functor theorem ?

thanks in advance !

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 !

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 !

I would like to know which kind of adjunction in algebraic geometry can be prove by the adjoint functor theorem ?

thanks in advance !

I dont have so much expercience in category theory so my questionn may be stupid and non-sense.

1.there is a classical adjonction 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 !

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 !