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?
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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)
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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.
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$\begingroup$ yes, it's a projective variety. $\endgroup$ Commented Mar 3, 2015 at 19:58