Proj construction

In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. It is a fundamental tool in scheme theory.

In this article, all rings will be assumed to be commutative and with identity.

Let ${\displaystyle S}$ be a graded ring, where

is the direct sum decomposition associated with the gradation.

Define the set Proj S to be the set of all homogeneous prime ideals that do not contain the irrelevant ideal

For brevity we will sometimes write X for Proj S.

We may define a topology, called the Zariski topology, on Proj S by defining the closed sets to be those of the form

where a is a homogeneous ideal of S. As in the case of affine schemes it is quickly verified that the V(a) form the closed sets of a topology on X.

Indeed, if ${\displaystyle (a_{i})_{i\in I}}$ are a family of ideals, then we have ${\displaystyle \bigcap V(a_{i})=V(\Sigma a_{i})}$ and if the indexing set I is finite, then ${\displaystyle \bigcup V(a_{i})=V(\Pi a_{i})}$.

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