higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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
of the $(n+1)$-fold 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 by August Ferdinand Möbius? in his 1827 work Der barycentrische Calcül.
More generally, for $(X,0)$ a pointed space with (pointed) $\mathbb{G}_m$-action, the quotient
is the corresponding projective space.
If instead of forming the quotient one forms the weak quotient/action groupoid, one speaks of the projective stack
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?.
For $R$ a commutative $k$-algebra, there is a natural isomorphism between
The proof is spelled out at affine line.
$\mathbb{C}P^1$ is the Riemann sphere
for $\mathbb{C}P^n$ see at complex projective space
for $\mathbb{R}P^n$ see at real projective space
An introduction to projective spaces over the theory of ordinary commutative rings is in
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.