Projective spectrum Proj assigns a locally ringed space $Proj(A)$ to a graded ring $A$. This construction has been introduced by Serre.
Serre proved two elementary theorems concerning the structure of sheaves of $\mathcal{O}_X$-modules on $X = Proj(A)$ relating graded modules over $A$ and sheaves of $\mathcal{O}_X$-modules: one on coherent sheaves and another on quasicoherent sheaves. In both cases, the sheaves correspond to objects in the localization of a category of graded modules by a torsion (Serre’s) subcategory.
There is an analogue for affine schemes, (affine) Serre's theorem on quasicoherent sheaves over affine schemes.
There is a functor $M\mapsto \tilde{M}$ (to do: should be described here) from the category of graded modules over a $\mathbf{Z}_{\geq 0}$-graded ring $A$ to the category of quasicoherent sheaves of $\mathcal{O}_X$-modules on $X = Proj(A)$ which has a fully faithful right adjoint, hence inducing an equivalence of the category of quasicoherent modules $qcoh(X)$ with a localization of the category of graded $A$-modules. The kernel of that localization is the subcategory of modules of finite length.
If $A$ is Noetherian, the functor $M\mapsto \tilde{M}$ restricted to the category of finitely generated projective modules takes values in the category of coherent sheaves and the induced functor also has a fully faithful right adjoint, which moreover induces an equivalence between the localization of the category of finitely generated projective $A$-modules by the Serre’s subcategory of modules of finite length and the category of coherent sheaves over $Proj(A)$.
The subject of noncommutative projective algebraic geometry by Artin and Zhang, as well as independent (and different mainly in the level of generality) attempts to noncommutative geometry by other authors take the Serre’s theorem to the status of definition of the category of quasicoherent sheaves when the graded ring is generalized to a noncommutative one.
This is Section 59, Proposition. 7.8, p. 252 in
See also 3.3.5 in
Hartshorne proves as Proposition 5.15 that the counit of the corresponding adjunction is an isomorphism (hence we have the reflective subcategory that is localization functor)
See also
Serre’s theorem on Proj, mathOverflow
We prove that an abelian category equipped with an ample sequence of objects is equivalent to the quotient of the category of coherent modules over the corresponding algebra by the subcategory of finite-dimensional modules. In the Noetherian case a similar result was proved by Artin and Zhang.
Last revised on April 28, 2024 at 19:40:54. See the history of this page for a list of all contributions to it.