Classifying Quasi-coherent Sheaves on Projective Schemes I know some references where I can find this, but they seem tedious. Both Hartshorne and Ueno cover this.
I am wondering if there is an elegant way to describe these. If this task is too difficult in general, how about just $\mathbb{P}^n$? 
Thanks!
 A: Quasi-coherent sheaves on affine schemes (say $Spec(A)$) are obtained by taking an $A$-module $M$ and the associated sheaf (by localizing $M$).  This gives an equivalence of categories between $A$-modules and q-c sheaves on $Spec(A)$.
Let $R$ be a graded ring, $R = R_0 + R_1 + \dots$ (direct sum).  Then we can, given a graded $R$-module $M$, consider its associated sheaf $\tilde{M}$.  The stalk of this at  a homogeneous prime ideal $P$ is defined to be the localization $M_{(P)}$, which is defined as generated by quotients $m/s$ for $s$ homogeneous of the same degree as $m$ and not in $P$.  
In short, we get sheaves of modules on the affine scheme just as we get the normal  sheaves of rings.  We get sheaves of modules on the projective scheme in the same homogeneous localization way as we get the sheaf of rings.
However, it's no longer an equivalence of categories. Why? Say you had a graded module $M= M_0 + M_1 + \dots$ (in general, we allow negative gradings as well).  Then it is easy to check that the sheaves associated to $M$ and $M' = M_1 + M_2 + \dots$ are exactly the same.
Nevertheless, it is possible to get every sheaf on $Proj(R)$ for $R$ a graded ring in this way.  See Proposition II.5.15 in Hartshorne.
