Contents

Ingredients

Concepts

Constructions

Examples

Theorems

# Contents

## Definition

Let $T$ be an abelian Lawvere theory (one containing the theory of abelian groups). Write $\mathbb{A}^1$ for its canonical line object and $\mathbb{G}_m$ for the corresponding multiplicative group-object.

The projective space $\mathbb{P}_n$ of $T$ is the quotient

$\mathbb{P}_n \coloneqq (\mathbb{A}^{n+1} \setminus \{0\})/\mathbb{G}_m$

of the complement of the origin inside the $(n+1)$-fold Cartesian product of the line with itself by the canonical action of $\mathbb{G}_m$.

Any point $(x_0,x_1,\ldots,x_n)\in \mathbb{A}^{n+1} - \{0\}$ gives homogeneous coordinates for its image under the quotient map. When considered in this fashion, one often writes $[x_0:x_1:\ldots:x_n]$. Homogeneous coordinates were introduced in Möbius 27

More generally, for $(X,0)$ a pointed topological space with (pointed) $\mathbb{G}_m$-action, the quotient

$\mathbb{P}(X) \;\coloneqq\; (X \setminus\{0\}) / \mathbb{G}_m$

is the corresponding projective space.

If instead of forming the quotient one forms the quotient stack/action groupoid, one speaks of the projective stack

$\hat \mathbb{P}(X) \coloneqq (X\setminus\{0\})//\mathbb{G}_m \,.$

## Examples

### For commutative rings and algebras

For $T$ the theory of commutative rings or more generally commutative associative algebras over a ring $k$, $\mathbb{A}_k^1$ is the standard affine line over $k$. In this case $\mathbb{P}^n_k$ is (…) A closed subscheme of $\mathbb{P}^n_k$ is a projective scheme?.

###### Proposition

For $R$ a commutative $k$-algebra, there is a natural isomorphism between

• $\mathbb{Z}$-gradings on $R$;

• $\mathbb{G}_m$-actions on $Spec R$.

The proof is spelled out at affine line.

### Real, complex, quaternionic, octonionic projective space

In particular, as $k$ ranges over the four finite-dimensional real normed division algebras, we have

• $k =$ real numbers: real projective space $\mathbb{R}P^n$

including the circle $\mathbb{R}P^1 \,\simeq\, S^1$

• $k =$ complex numbers: complex projective space $\mathbb{C}P^n$

including the Riemann sphere $\mathbb{C}P^1 \,\simeq\, S^2$

• $k =$ quaternions: quaternionic projective space $\mathbb{H}P^n$

including the 4-sphere $\mathbb{H}P^1 \,\simeq\, S^4$

• $k =$ octonions: octonionic projective space $\mathbb{O}P^n$

(for $n \leq 2$)

including the 8-sphere $\mathbb{O}P^1 \,\simeq\, S^8$

###### Definition

(topological projective space)

Let $n \in \mathbb{N}$. Consider the Euclidean space $k^{n+1}$ equipped with its metric topology, let $k^{n+1} \setminus \{0\} \subset k^{n+1}$ 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 $c \in k$ (necessarily non-zero):

$(\vec x_1 \sim \vec x_2) \;\Leftrightarrow\; \left( \underset{c \in k}{\exists} ( \vec x_2 = c \vec x_1 ) \right) \,.$

The equivalence class $[\vec x]$ is traditionally denoted

$[x_1 : x_2 : \cdots : x_{n+1}] \,.$

Then the projective space $k P^n$ is the corresponding quotient topological space

$k P^n \;\coloneqq\; \left(k^{n+1} \setminus \{0\}\right) / \sim \,.$
###### Lemma

(canonical inclusion of projective spaces)

For $n \in \mathbb{N}$ the function between topological projective spaces from def. given by

$\array{ k P^n &\overset{}{\longrightarrow}& k P^{n+1} \\ [x_1 : \cdots : x_{n+1}] &\mapsto& [ x_1 : \cdots : x_{n+1} : 0] }$

is a continuous function.

###### Proof

There is a commuting square of functions of underlying sets of the form

$\array{ (x_1, \cdots, x_{n+1}) &\mapsto& (x_1, \cdots, x_{n+1}, 0) \\ k^{n+1} & \overset{}{\longrightarrow} & k^{n+2} \\ \downarrow &\searrow& \downarrow \\ k P^n &\longrightarrow& k P^{n+1} \\ [x_1 : \cdots : x_{n+1}] &\mapsto& [ x_1 : \cdots : x_{n+1} : 0] } \,,$

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.

###### Proposition

(projective space as quotient space of an $n$-sphere)

For $n \in \mathbb{C}$ there are homeomorphisms

1. $S^{n}/(\mathbb{Z}/2) \simeq \mathbb{R}P^n$

between

1. the quotient space of the Euclidean n-sphere canonically regarded as a subspace of the Euclidean space $\mathbb{R}^{n+1}$ by the equivalence relation which identifies two points $\vec p \in \mathbb{R}^{n+1}$ if they differ by multiplication by $-1$

2. real projective space (def. )

2. $S^{2n+1}/S^1 \simeq \mathbb{C}P^{n}$

between

1. the quotient space of the Euclidean (2n+2)-sphere, canonically regarded as a subspace of the Euclidean space $\mathbb{R}^{2n+2} \simeq \mathbb{C}^{n+1}$ by the equivalence relation which identifies two points $\vec p \in \mathbb{C}^{n+1}$ if they differ by multiplication by a complex number of unit norm

2. complex projective space (def. ).

###### Proof

It is clear that there is a bijection of underlying sets as claimed: Under the equivalence relation defining projective space, every element $\vec x = (x_1, \cdots, x_{n+1}) \in k^{n+1}$ is equivalent to one of unit norm, namely $\frac{1}{\vert \vec x\vert} \vec x$, hence lying on the unit sphere. Representatives of this form are unique up to further multiplication by elements in $k \setminus \{0\}$ of unit norm.

It remains to see that this bijection is a homeomorphism. For definiteness of notation, we discuss this for the case $k = \mathbb{C}$, the case $k = \mathbb{R}$ works verbatim the same, with the evident substitutions.

So we have a commuting diagram of functions of underlying sets

$\array{ S^{2n+1} &\hookrightarrow& \mathbb{C}^{n+1} \setminus \{0\} \\ {}^{\mathllap{q_{S^{2n+1}}}}\downarrow &\searrow^{\mathrlap{f}}& \downarrow^{\mathrlap{q_{\mathbb{C}^{n+1}}}} \\ S^{2n+1}/S^1 &\longrightarrow& \mathbb{C}P^n }$

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 $U \subset S^{2n+1}/S^1$ be an open set, which means that $q_{S^{2n+1}}^{-1}(U) \subset S^{2n+1}$ is an open set. We need to see that $f(q_{S^{2n+1}}^{-1}(U)) \subset \mathbb{C}P^{n}$ is open, hence that $q_{\mathbb{C}^{n+1}}^{-1}(f(q_{S^{2n+1}}^{-1}(U))) \subset \mathbb{C}^{n+1}$ is open. Now by the nature of the Euclidean metric topology, the open subset $q_{S^{2n+1}}^{-1}(U)$ is a union of open balls $B^\circ_x(\epsilon)$ in $\mathbb{C}^{n+1}$ intersected with $S^{2n+1}$. But then $q_{\mathbb{C}^{n+1}}^{-1}(f(B^\circ_x(\epsilon)\vert_{S^{2n+1}}))$ is their orbit under the multiplicative action by $\mathbb{C} \setminus \{0\}$, hence is a cylinder $B^\circ_x(\epsilon)\vert_{S^{2n+1}} \times (\mathbb{C} \setminus \{0\})$. This is clearly open.

###### Proposition

(cell structure of K-projective space)

There is a CW-complex structure on real projective space $\mathbb{R}P^n$ (def. ) for $n \in \mathbb{N}$, given by induction, where $\mathbb{R}P^{n+1}$ arises from $\mathbb{R}P^n$ by attaching a single cell of dimension $n+1$ with attaching map the projection $S^{n} \longrightarrow \mathbb{C}P^n$ from prop. :

$\array{ S^{n} &\longrightarrow& S^{n}/(\mathbb{Z}/2) \simeq \mathbb{R}P^n \\ {}^{\mathllap{\iota_{n+1}}}\downarrow &(po)& \downarrow \\ D^{n+1} &\underset{q}{\longrightarrow}& \mathbb{R}P^{n+1} } \,.$

Similarly, there is a CW-complex structure on complex projective space $\mathbb{C}P^n$ (def. ) for $n \in \mathbb{N}$, given by induction, where $\mathbb{C}P^{n+1}$ arises from $\mathbb{C}P^n$ by attaching a single cell of dimension $2(n+1)$ with attaching map the projection $S^{2n+1} \longrightarrow \mathbb{C}P^n$ from prop. :

$\array{ S^{2n+1} &\longrightarrow& S^{2n+1}/S^1 \simeq \mathbb{C}P^n \\ {}^{\mathllap{\iota_{2n+2}}}\downarrow &(po)& \downarrow \\ D^{2n+2} &\underset{q}{\longrightarrow}& \mathbb{C}P^{n+1} } \,.$
###### Proof

we discuss the case $k = \mathbb{C}$. The case $k= \mathbb{R}$ works verbatim the same, with the evident substitutions.

Given homogeneous coordinates $(z_0 , z_1 , \cdots , z_n , z_{n+1} , z_{n+2}) \in \mathbb{C}^{n+2}$ for $\mathbb{C}P^{n+1}$, let

$\phi \coloneqq -arg(z_{n+2})$

be the phase of $z_{n+2}$. Then under the equivalence relation defining $\mathbb{C}P^{n+1}$ these coordinates represent the same element as

$\frac{1}{\vert \vec z\vert}(e^{i \phi} z_0, e^{i \phi}z_1,\cdots, e^{i \phi}z_{n+1}, r) \,,$

where

$r = {\vert z_{n+2}\vert}\in [0,1) \subset \mathbb{C}$

is the absolute value of $z_{n+2}$. Representatives $\vec z'$ of this form (${\vert \vec z' \vert = 1}$ and $z'_{n+2} \in [0,1]$) parameterize the 2n+2-disk $D^{2n+2}$ with boundary the $(2n+1)$-sphere at $r = 0$.

The resulting function $q \colon D^{2n+2} \to \mathbb{C}P^{n+1}$ is continuous: It may be factored as

$\array{ q_{D^{2n+2}} \colon & D^{2n+2} &\overset{\phantom{AAA}}{\hookrightarrow}& \mathbb{C}^{n+2} \setminus \{0\} &\overset{q_{\mathbb{C}^{n+2}}}{\longrightarrow}& \mathbb{C}P^{n+1} \\ & (Re(z_1), Im(z_1), \cdots, Re(z_{n+1}), Im(z_{n+1}), r) &\mapsto& (z_1, \cdots, z_{n+1}, r) &\mapsto& [ z_1 : \cdots : z_{n+1} : r ] }$

and here the first map is the embedding of the disk $D^{2n+2}$ as a hemisphere in $\mathbb{R}^{2n+1} \hookrightarrow \mathbb{R}^{2n+2} \simeq \mathbb{C}^{2n+2}$, 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 $\mathbb{C}-\{0\}$ which fixes the conditions ${\vert z'\vert} = 0$ and $z'_{n+2}$ is $S^1 \subset \mathbb{C} \setminus \{0\}$ acting on the elements with $r = \{z'_{n+2}\} = 0$ by phase shifts on the $z_0, \cdots, z_{n+1}$. The quotient of this remaining action on $D^{2(n+1)}$ identifies its boundary $S^{2n+1}$-sphere with $\mathbb{C}P^{n}$, 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 $\mathbb{C}P^{n+1}$ is the final topology induced by the functions $D^{2n+2} \to \mathbb{C}P^{n+1}$ and $\mathbb{C}P^n \to \mathbb{C}P^{n+1}$, hence that a subset of $\mathbb{C}P^{n+1}$ is open precisely if its pre-images under these two functions are open.

We saw above that $q_{D^{2n+2}}$ is continuous. Moreover, also the function $i_n \colon \mathbb{C}P^n \to \mathbb{C}P^{n+1}$ is continuous (by this lemma).

This shows that if a subset of $\mathbb{C}P^{n+1}$ is open, then its pre-images under these functions are open. It remains to see that if $S \subset \mathbb{C}P^{n+1}$ is a subset with $q_{S^{2n+2}}^{-1}(S) \subset D^{2n+2}$ open and $i_n^{-1}(S) \subset \mathbb{C}P^n$ open, then $S \subset \mathbb{C}P^{n+1}$ is open.

Notice that $q_{\mathbb{C}^{n+2}}^{-1}(S)$ contains with every point also its orbit under the action of $\mathbb{C} \setminus \{0\}$, and that every open subset of $D^{2n+2}$ is a unions of open balls. By the above factorization of $q_{D^{2n+2}}$ this means that if $q_{D^{2n+2}}^{-1}(S)$ is open, then $q_{\mathbb{C}^{n+2}}^{-1}(S)$ is a union of open cyclinders, hence is open. By the nature of the quotient topology, this means that $S \subset \mathbb{C}P^n$ is open.

###### Definition

(standard open cover of topological projective space)

For $n \in \mathbb{N}$ the standard open cover of the projective space $k P^n$ (def. ) is

$\left\{ U_i \subset k P^n \right\}_{i \in \{1, \cdots, n+1\}}$

with

$U_i \coloneqq \left\{ [x_1 : \cdots : x_{n+1}] \in k P^n \;\vert\; x_i \neq 0 \right\} \,.$

To see that this is an open cover:

1. This is a cover because with the orgin removed in $k^n \setminus \{0\}$ at every point $[x_1: \cdots : x_{n+1}]$ at least one of the $x_i$ has to be non-vanishing.

2. These subsets are open in the quotient topology $k P^n = (k^n \setminus \{0\})/\sim$, since their pre-image under the quotient co-projection $k^{n+1} \setminus \{0\} \to k P^n$ coincides with the pre-image $(pr_i\circ\iota)^{-1}( k \setminus \{0\} )$ under the projection onto the $i$th coordinate in the product topological space $k^{n+1} = \underset{i \in \{1,\cdots, n\}}{\prod} k$ (where we write $k^n \setminus \{0\} \overset{\iota}{\hookrightarrow} k^n \overset{pr_i}{\to} k$).

###### Proposition

(n-sphere projecting to real projective space is covering space projection)

For $n \in \mathbb{N}$, the continuous function $p \;\colon\; S^n \to \mathbb{R}P^n$ from prop. is a covering space projection.

###### Proof

We need to produce an open cover $\{U_i \subset \mathbb{R}P^n\}_{i \in I}$ such that the restrictions of the projection to this cover are homeomorphic over the base to a product topological space

$\array{ U_i \times Disc(\mathbb{Z}_2) && \overset{\simeq}{\longrightarrow} && S^n|_{U_i} \\ & \searrow && \swarrow \\ && U_i } \,.$

Consider the standard open cover from def. . Hence $i \in \{1, \cdots, n+1\}$ and $U_i$ consists of those lines through the origin in $\mathbb{R}^{n+1}$ which do not lie in the subspace defined by $x_i = 0$. The intersection of this subspace with the unit sphere $S^n \subset \mathbb{R}^{n+1}$ is an equator of the $n$-sphere, and so the complement of this equator is the disjoint union of the two open hemispheres $D_i^\pm \subset S^n$. Hence

$\array{ S^n\vert_{U_i} & \simeq D_i^+ \sqcup D_i^- } \,.$

Moreover, each line in $\mathbb{R}^{n+1}$ which corresponds to an element in $U_i$ intersects $D^+_i$ as well as $D^-_i$ exactly once. In particular therefore the $\mathbb{Z}_2$-action on $S^n$ restricts over $U_i$ to the interchange of these two hemispheres, and hence prop. gives the required homeomorphism as above.

###### Proposition

(standard open cover is atlas)

The charts of the standard open cover of def. are homeomorphic to Euclidean space $k^n$.

###### Proof

If $x_i \neq 0$ then

$[x_1 : \cdots : x_i : \cdots : x_{n+1}] = \left[ \frac{x_1}{x_i} : \cdots : 1 : \cdots \frac{x_{n+1}}{x_i} \right]$

and the representatives of the form on the right are unique.

This means that

$\array{ \mathbb{R}^n &\overset{\phi_i}{\longrightarrow}& U_i \\ (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) &\mapsto& [x_1: \cdots: 1: \cdots : x_n+1] }$

is a bijection of sets.

To see that this is a continuous function, notice that it is the composite

$\array{ && \mathrlap{\mathbb{R}^{n+1} \setminus \{x_i = 0\}} \\ & {}^{\mathllap{\hat \phi_i}}\nearrow & \downarrow \\ \mathbb{R}^n & \underset{\phi_i}{\longrightarrow} & U_i }$

of the function

$\array{ \mathbb{R}^n &\overset{\hat \phi_i}{\longrightarrow}& \mathbb{R}^{n+1} \setminus \{x_i = 0\} \\ (x_1, \cdots, x_{i-1}, x_{i+1}, \cdots, x_{n+1}) &\mapsto& (x_1, \cdots, 1, \cdots ,x_n+1) }$

with the quotient projection. Now $\hat \phi_i$ 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 $\phi_i$ is continuous.

It remains to see that also the inverse function $\phi_i^{-1}$ is continuous. Since

$\array{ \mathbb{R}^{n+1} \setminus \{x_i = 0\} &\overset{}{\longrightarrow}& U_i &\overset{\phi_i^{-1}}{\longrightarrow}& \mathbb{R}^n \\ (x_1, \cdots, x_{n+1}) && \mapsto && ( \frac{x_1}{x_i}, \cdots, \frac{x_{i-1}}{x_i}, \frac{x_{i+1}}{x_i}, \cdots , \frac{x_{n+1}}{x_i}) }$

is a rational function, and since rational functions are continuous, it follows, by nature of the quotient topology, that $\phi_i$ takes open subsets to open subsets, hence that $\phi_i^{-1}$ is continuous.

###### Proposition

(real/complex projective space is smooth manifold)

For $k \in \{\mathbb{R}, \mathbb{C}\}$ the topological projective space $k P^n$ (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.

###### Proof

By prop. $k P^n$ is a locally Euclidean space. Moreover, $kP^n$ 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.

## References

Textbook account: