mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, projectivization is a procedure which associates with a non-zero

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

a projective space , whose elements are one-dimensional subspaces of . More generally, any subset of closed under scalar multiplication defines a subset of formed by the lines contained in and is called the projectivization of .

Properties

* Projectivization is a special case of the factorization by a

group action In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under ...

: 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 In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

of in the sense of

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

is one less than the dimension of the vector space . * Projectivization is

functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

ial with respect to

injective In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...

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(f): \mathbf(V)\to \mathbf(W).

: In particular, the

general linear group In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...

GL(''V'') acts on the projective space by

automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...

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

, there is a procedure that associates a

projective variety In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...

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 In algebraic geometry, Proj is a construction analogous to the spectrum of a ring, spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective variety, projective varie ...

gives rise to a

contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

from the category of graded commutative rings and surjective graded maps to the category of projective schemes.

References

{{Reflist Projective geometry Linear algebra