Projectivization
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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.[1]
Properties
- Projectivization is a special case of the dimension of P(V) in the sense of algebraic geometryis one less than the dimension of the vector space V.
- Projectivization is injectivelinear maps: if
- 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 P(V) by automorphisms.
Projective completion
A related procedure embeds a vector space V over a field K into the projective space P(V ⊕ K) of the same dimension. To every vector v of V, it associates the line spanned by the vector (v, 1) of V ⊕ K.
Generalization
In
contravariant functor from the category of graded commutative rings and surjective graded maps to the category of projective schemes
.
References
- ^ Weisstein, Eric W. "Projectivization". mathworld.wolfram.com. Retrieved 2024-08-27.