nLab projective scheme

Idea

A projective $k$-variety (Zariski closed subvarieties of a projective space over $k$) is determined up to an isomorphism by its homogeneous coordinate ring which is an $\mathbf{N}$-graded commutative $k$-algebra; this coordinate ring depends slightly on the embedding of the variety into a projective space (see ample line bundle). More generally, to any $\mathbf{N}$-graded commutative ring $A$ one can define a locally ringed space, in fact an algebraic scheme, $Proj(A)$, the “Proj” or projective spectrum of $A$.

A projective scheme is an algebraic scheme which is isomorphic as a locally ringed space to a Proj of some finitely generated graded commutative ring. Alternatively a projective scheme is a closed subscheme of a projective space.

More generally Grothendieck defines projective morphisms of schemes or relative projective schemes, which can be presented as generalized version of Proj for projective fiber bundles associated to locally free modules over a target.

Construction of Proj

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Properties

(Serre's theorem on Proj) The category of quasicoherent sheaves of $\mathcal{O}_X$-modules where $X = Proj(A)$ is equivalent to the localization of the category of graded $A$-modules on the torsion subcategory of graded $A$-modules of finite length.

Generalizations

There are also categorical versions of Proj for noncommutative $\mathbf{N}$-graded rings, which serve as a basis for noncommutative algebraic geometry.

References

• The Stacks project: Proj of a graded ring, tag/00JM
• Robin Hartshorne, Algebraic geometry, Springer Graduate Texts in Mathematics
• eom: projective scheme
• Ravi Vakil, Foundations of algebraic geometry, lecture 14, course notes pdf

category: algebraic geometry