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Let be an abelian Lawvere theory (one containing the theory of abelian groups). Write for its canonical line object and for the corresponding multiplicative group-object.
The projective space of is the quotient
of the complement of the origin inside the -fold Cartesian product of the line with itself by the canonical action of .
Any point gives homogeneous coordinates for its image under the quotient map. When considered in this fashion, one often writes . Homogeneous coordinates were introduced in Möbius 27
More generally, for a pointed topological space with (pointed) -action, the quotient
is the corresponding projective space.
If instead of forming the quotient one forms the quotient stack/action groupoid, one speaks of the projective stack
For the theory of commutative rings or more generally commutative associative algebras over a ring , is the standard affine line over . In this case is (…) A closed subscheme of is a projective scheme.
For a commutative -algebra, there is a natural isomorphism between
The proof is spelled out at affine line.
In particular, as ranges over the four finite-dimensional real normed division algebras, we have
real numbers: real projective space
including the circle
complex numbers: complex projective space
including the Riemann sphere
quaternions: quaternionic projective space
including the 4-sphere
octonions: octonionic projective space
(for )
including the 8-sphere
(topological projective space)
Let . Consider the Euclidean space equipped with its metric topology, let be the topological subspace which is the complement of the origin, and consider on its underlying set the equivalence relation which identifies two points if they differ by multiplication with some (necessarily non-zero):
The equivalence class is traditionally denoted
Then the projective space is the corresponding quotient topological space
(canonical inclusion of projective spaces)
For the function between topological projective spaces from def. given by
is a continuous function.
There is a commuting square of functions of underlying sets of the form
where the two vertical functions are the defining quotient co-projections, which are continuous functions by nature of quotient spaces. The top function is clearly continuous (polynomials are continuous) and hence so is its composite with the right co-projection, inducated by the diagonal arrow in the above diagram.
This implies that the bottom function is continuous by the nature (the universal property) of the quotient space topology.
(projective space as quotient space of an -sphere)
For there are homeomorphisms
between
the quotient space of the Euclidean n-sphere canonically regarded as a subspace of the Euclidean space by the equivalence relation which identifies two points if they differ by multiplication by
real projective space (def. )
between
the quotient space of the Euclidean (2n+2)-sphere, canonically regarded as a subspace of the Euclidean space by the equivalence relation which identifies two points if they differ by multiplication by a complex number of unit norm
complex projective space (def. ).
It is clear that there is a bijection of underlying sets as claimed: Under the equivalence relation defining projective space, every element is equivalent to one of unit norm, namely , hence lying on the unit sphere. Representatives of this form are unique up to further multiplication by elements in of unit norm.
It remains to see that this bijection is a homeomorphism. For definiteness of notation, we discuss this for the case , the case works verbatim the same, with the evident substitutions.
So we have a commuting diagram of functions of underlying sets
where the top horizontal and the two vertical functions are continuous, and where the bottom function is is a bijection. Since the diagonal composite is also continuous, the nature of the quotient space topology implies that the bottom function is also continuous. To see that it is a homeomorphism it hence remains to see that it is an open map (by this prop.).
So let be an open set, which means that is an open set. We need to see that is open, hence that is open. Now by the nature of the Euclidean metric topology, the open subset is a union of open balls in intersected with . But then is their orbit under the multiplicative action by , hence is a cylinder . This is clearly open.
(cell structure of K-projective space)
There is a CW-complex structure on real projective space (def. ) for , given by induction, where arises from by attaching a single cell of dimension with attaching map the projection from prop. :
Similarly, there is a CW-complex structure on complex projective space (def. ) for , given by induction, where arises from by attaching a single cell of dimension with attaching map the projection from prop. :
we discuss the case . The case works verbatim the same, with the evident substitutions.
Given homogeneous coordinates for , let
be the phase of . Then under the equivalence relation defining these coordinates represent the same element as
where
is the absolute value of . Representatives of this form ( and ) parameterize the 2n+2-disk with boundary the -sphere at .
The resulting function is continuous: It may be factored as
and here the first map is the embedding of the disk as a hemisphere in , while the second is the defining quotient space projection. Both of these are continuous, and hence so is their composite.
The only remaining part of the action of which fixes the conditions and is acting on the elements with by phase shifts on the . The quotient of this remaining action on identifies its boundary -sphere with , by prop. .
This shows that the above square is a pushout diagram of underlying sets.
By the nature of colimits in Top (this prop.) it remains to see that the topology on is the final topology induced by the functions and , hence that a subset of is open precisely if its pre-images under these two functions are open.
We saw above that is continuous. Moreover, also the function is continuous (by this lemma).
This shows that if a subset of is open, then its pre-images under these functions are open. It remains to see that if is a subset with open and open, then is open.
Notice that contains with every point also its orbit under the action of , and that every open subset of is a unions of open balls. By the above factorization of this means that if is open, then is a union of open cyclinders, hence is open. By the nature of the quotient topology, this means that is open.
(standard open cover of topological projective space)
For the standard open cover of the projective space (def. ) is
with
To see that this is an open cover:
This is a cover because with the orgin removed in at every point at least one of the has to be non-vanishing.
These subsets are open in the quotient topology , since their pre-image under the quotient co-projection coincides with the pre-image under the projection onto the th coordinate in the product topological space (where we write ).
(n-sphere projecting to real projective space is covering space projection)
For , the continuous function from prop. is a covering space projection.
We need to produce an open cover such that the restrictions of the projection to this cover are homeomorphic over the base to a product topological space
Consider the standard open cover from def. . Hence and consists of those lines through the origin in which do not lie in the subspace defined by . The intersection of this subspace with the unit sphere is an equator of the -sphere, and so the complement of this equator is the disjoint union of the two open hemispheres . Hence
Moreover, each line in which corresponds to an element in intersects as well as exactly once. In particular therefore the -action on restricts over to the interchange of these two hemispheres, and hence prop. gives the required homeomorphism as above.
(standard open cover is atlas)
The charts of the standard open cover of def. are homeomorphic to Euclidean space .
If then
and the representatives of the form on the right are unique.
This means that
is a bijection of sets.
To see that this is a continuous function, notice that it is the composite
of the function
with the quotient projection. Now is a polynomial function and since polynomials are continuous, and since the projection to a quotient topological space is continuous, and since composites of continuous functions are continuous, it follows that is continuous.
It remains to see that also the inverse function is continuous. Since
is a rational function, and since rational functions are continuous, it follows, by nature of the quotient topology, that takes open subsets to open subsets, hence that is continuous.
(real/complex projective space is smooth manifold)
For the topological projective space (def. ) is a topological manifold.
Equipped with the standard open cover of def. regarded as an atlas by prop. , it is a differentiable manifold, in fact a smooth manifold.
By prop. is a locally Euclidean space. Moreover, admits the structure of a CW-complex (by prop. ) and therefore it is a paracompact Hausdorff space since CW-complexes are paracompact Hausdorff spaces. This means that it is a topological manifold.
It remains to see that the gluing functions of this atlas are differentiable functions and in fact smooth functions. But by prop. they are even rational functions.
Textbook account:
See also:
Miles Reid, Graded rings and varieties in weighted projective space (pdf)
Aurelio Carboni, Marco Grandis , Categories of projective spaces , JPAA 110 (1996) pp.241-258.
Discussion in synthetic algebraic geometry:
Last revised on May 24, 2024 at 05:23:38. See the history of this page for a list of all contributions to it.