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| | In [[mathematics]], '''projectivization''' is a procedure which associates to a non-zero [[vector space]] ''V'' its associated [[projective space]] <math>{\Bbb P}(V)</math>, whose elements are one-dimensional [[Linear subspace|subspaces]] of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multiplication defines a subset of <math>{\Bbb P}(V)</math> formed by the lines contained in ''S'' and called the projectivization of ''S''. |
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| | == Properties == |
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| | * Projectivization is a special case of the [[quotient space|factorization]] by a [[group action]]: the projective space <math>{\Bbb P}(V)</math> 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 (mathematics)|dimension]] of <math>{\Bbb P}(V)</math> in the sense of [[algebraic geometry]] is one less than the dimension of the vector space ''V''. |
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| | * Projectivization is [[Functor|functorial]] with respect to [[injective]] linear maps: if |
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| | :: <math> f: V\to W </math> |
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| | : is a linear map with trivial [[kernel (linear algebra)|kernel]] then ''f'' defines an algebraic map of the corresponding projective spaces, |
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| | :: <math> \mathbb{P}(f): \mathbb{P}(V)\to \mathbb{P}(W).</math> |
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| | : In particular, the [[general linear group]] ''GL''(''V'') acts on the projective space <math>{\Bbb P}(V)</math> by [[automorphism]]s. |
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| | == Projective completion == |
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| | A related procedure embeds a vector space ''V'' over a [[field (mathematics)|field]] ''K'' into the projective space <math>{\Bbb P}(V\oplus K)</math> of the same dimension. To every vector ''v'' of ''V'', it associates the line spanned by the vector (''v'',1) of ''V''⊕''K''. |
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| | == Generalization == |
| | {{main|Proj construction}} |
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| | In [[algebraic geometry]], there is a procedure that associates a [[projective variety]] Proj ''S'' with a [[graded algebra|graded commutative algebra]] ''S'' (under some technical restrictions on ''S''). If ''S'' is the algebra of [[Symmetric algebra#Interpretation as polynomials|polynomials on a vector space]] ''V'' then Proj ''S'' is <math>{\Bbb P}(V).</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 [[scheme (mathematics)|schemes]]. |
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| | [[Category:Projective geometry]] |
| | [[Category:Linear algebra]] |
In mathematics, projectivization is a procedure which associates to a non-zero vector space V its associated projective space 
, whose elements are one-dimensional subspaces of V. More generally, any subset S of V closed under scalar multiplication defines a subset of formed by the lines contained in S and called the projectivization of S.
Properties
- is a linear map with trivial kernel then f defines an algebraic map of the corresponding projective spaces,
- In particular, the general linear group GL(V) acts on the projective space by automorphisms.
Projective completion
A related procedure embeds a vector space V over a field K into the projective space 
of the same dimension. To every vector v of V, it associates the line spanned by the vector (v,1) of V⊕K.
Generalization
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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 
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.