In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space P(V), whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalar multiplication defines a subset of P(V) formed by the lines contained in S and is called the projectivization of S.[1][2] A related procedure embeds a vector space V over a field K into the projective space P(V ⊕ K) of the same dimension.
To every vector v of V, it associates the line spanned by the vector (v, 1) of V ⊕ K. 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 P(V).
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.