Projectivization

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 $S$.

Properties

 * 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


 * $$ f: V\to W $$


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


 * $$ \mathbf{P}(f): \mathbf{P}(V)\to \mathbf{P}(W).$$


 * In particular, the general linear group GL(V) acts on the projective space $P(V)$ by automorphisms.

Projective completion
A related procedure embeds a vector space $V$ over a field $K$ into the projective space $P(V &oplus; K)$ of the same dimension. To every vector $v$ of $V$, it associates the line spanned by the vector $(v, 1)$ of $V &oplus; K$.

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 $P(S)$. 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.