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 $\mathbb{G}_m$.
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.
projective linear group, [[]]
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.