Projectivization
Template:Short description Template:Refimprove In mathematics, projectivization is a procedure which associates with a non-zero vector space Template:Math a projective space Template:Math, whose elements are one-dimensional subspaces of Template:Math. More generally, any subset Template:Math of Template:Math closed under scalar multiplication defines a subset of Template:Math formed by the lines contained in Template:Math and is called the projectivization of Template:Math.[1][2]
Properties
- Projectivization is a special case of the factorization by a group action: the projective space Template:Math is the quotient of the open set Template:Math of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The dimension of Template:Math in the sense of algebraic geometry is one less than the dimension of the vector space Template:Math.
- Projectivization is functorial with respect to injective linear maps: if
- is a linear map with trivial kernel then Template:Math defines an algebraic map of the corresponding projective spaces,
- In particular, the general linear group GL(V) acts on the projective space Template:Math by automorphisms.
Projective completion
A related procedure embeds a vector space Template:Math over a field Template:Math into the projective space Template:Math of the same dimension. To every vector Template:Math of Template:Math, it associates the line spanned by the vector Template:Math of Template:Math.
Generalization
Template:Main In algebraic geometry, there is a procedure that associates a projective variety Template:Math with a graded commutative algebra Template:Math (under some technical restrictions on Template:Math). If Template:Math is the algebra of polynomials on a vector space Template:Math then Template:Math is Template:Math. 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.