Projectivization

In mathematics, projectivization is a procedure which associates with a non-zero vector space "V" a projective space formula_1, whose elements are one-dimensional subspaces of "V". More generally, any subset "S" of "V" closed under scalar multiplication defines a subset of formula_1 formed by the lines contained in "S" and is called the projectivization of "S".
Projective completion.
A related procedure embeds a vector space "V" over a field "K" into the projective space formula_8 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" (under some technical restrictions on "S"). If "S" is the algebra of polynomials on a vector space "V" then Proj "S" is formula_9 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.