Projectivization

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

Properties

• Projectivization is a special case of the factorization by a group action: the projective space ${\displaystyle {\mathbb {P} }(V)}$ 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 ${\displaystyle {\mathbb {P} }(V)}$ 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

${\displaystyle f:V\to W}$

is a linear map with trivial kernel then f defines an algebraic map of the corresponding projective spaces,

${\displaystyle \mathbb {P} (f):\mathbb {P} (V)\to \mathbb {P} (W).}$

In particular, the general linear group GL(V) acts on the projective space ${\displaystyle {\mathbb {P} }(V)}$ by automorphisms.

Projective completion

A related procedure embeds a vector space V over a field K into the projective space ${\displaystyle {\mathbb {P} }(V\oplus K)}$ of the same dimension. To every vector v of V, it associates the line spanned by the vector (v, 1) of V ⊕ K.

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 ${\displaystyle {\mathbb {P} }(V).}$ 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.