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 -fold product of the line with itself by the canonical action of .
More generally, for a pointed space with (pointed) -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 commutative rings and algebras
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?.
The proof is spelled out at affine line.
Over the real and complex numbers
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.
Revised on May 3, 2016 09:05:11
by Urs Schreiber