derived smooth geometry
The projective space of is the quotient
of the -fold 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 by August Ferdinand Möbius? in his 1827 work Der barycentrische Calcül.
More generally, for a pointed space with (pointed) -action, the quotient
is the corresponding projective space.
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.
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)