Projectivization

In mathematics, projectivization is a procedure which associates to a vector space V its projective space ${\mathbb {P}}V$, which is a space of lines $l\subset V$ passing through 0. If $S\subset V$ is a subset which contains with each point a 1-dimensional subspace $l\subset V$ passing through this point, the corresponding subset of ${\mathcal {P}}V$ is called a projectivization of S.

Projectivization is functorial, and defines a functor from the category of vector spaces to the category of algebraic varieties.

In algebraic geometry, a similar procedure is used to associate with a graded commutative ring A the corresponding projective variety, which is a space of all graded prime ideals in A, equipped with Zariski topology. This procedure is also called a projectivization, and gives a contravariant functor from the category of graded commutative rings to the category of projective varieties.