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.
Projectivization is a special case of the factorization by a group action: the projective space
P
(
V
)
is the quotient of the open set V\{0} of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The dimension of
P
(
V
)
in the sense of algebraic geometry is one less than the dimension of the vector space V.
Projectivization is functorial with respect to injective linear maps: if
is a linear map with trivial kernel then
f defines an algebraic map of the corresponding projective spaces,
In particular, the general linear group GL(
V) acts on the projective space
P
(
V
)
by automorphisms.
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.
