Projectivization


In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space, whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalar multiplication defines a subset of formed by the lines contained in S and is called the projectivization of S.

Properties

A related procedure embeds a vector space V over a field K into the projective space of the same dimension. To every vector v of V, it associates the line spanned by the vector of.

Generalization

In algebraic geometry, there is a procedure that associates a projective variety Proj S with a graded commutative algebra S. If S is the algebra of polynomials on a vector space V then Proj S is This Proj construction gives rise to a contravariant functor from the category of graded commutative rings and surjective graded maps to the category of projective schemes.