Projectivization

• Projectivization is a special case of the factorization by a group action: the projective space is the quotient of the open set of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The dimension of in the sense of algebraic geometry is one less than the dimension of the vector space . • Projectivization is functorial with respect to injective linear maps: if :: f: V\to W : is a linear map with trivial kernel then 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 by automorphisms. == Projective completion ==

A related procedure embeds a vector space over a field into the projective space of the same dimension. To every vector of , it associates the line spanned by the vector of . == Generalization ==

In algebraic geometry, there is a procedure that associates a projective variety with a graded commutative algebra (under some technical restrictions on ). If is the algebra of polynomials on a vector space then is . 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. == References ==