# Serre's Theorem for Coherent Sheaves

I recently heard a discussion about a certain of Serre which reconstructs the category of coherent sheaves of a variety $V$ as the category of modules over the homogeneous space of $V$ modulo modules with torsion . . . or something quite close to this. Googling for a result of Serre is difficult, given his fantastic output, so I was unable to find a presentation of the result. Can anyone point me in the direction of a readable presentation of this result? Moreover, what is the result known as?

It might be the general fact about the proj construction, which can be found in any basic textbook (eg hartshorne), if by variety you mean projective variety. Otherwise I'm not sure.

• yes, it's a projective variety. – Asten Matshink Mar 3 '15 at 19:58

The following is apparently a theorem of Serre (which can be found in his FAC paper, although I haven't spotted it http://www1.mat.uniroma1.it/people/arbarello/FAC.pdf).

Let's say that a graded module is torsion if it's zero for all degrees high enough.

Let R be a commutative graded ring generated in dimension one. Denote Gr the category of graded R-modules and Tors the subcategory of torsion modules (which is a Serre subcategory). Then the quotient category Gr/Tors is equivalent (through the ~ construction) to the category of quasi-coherent modules on Proj R.

I believe that when R is noetherian then this equivalence restricts to one between finitely generated graded modules and coherent modules.

(presumably there is also a relative version of this -- but I am quite allergic to Proj so I'm not going to go any further on this)