projective space



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

The projective space n\mathbb{P}_n of TT is the quotient

n:=(𝔸 n+1{0})/𝔾 m \mathbb{P}_n := (\mathbb{A}^{n+1} - \{0\})/\mathbb{G}_m

of the (n+1)(n+1)-fold product of the line with itself by the canonical action of 𝔾 m\mathbb{G}_m. Any point (x 0,x 1,,x n)𝔸 n+1{0}(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::x n][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)(X,0) a pointed space with (pointed) 𝔾 m\mathbb{G}_m-action, the quotient

(X):=(X{0})/𝔾 m \mathbb{P}(X) := (X-\{0\})/\mathbb{G}_m

is the corresponding projective space.

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

^(X):=(X{0})//𝔾 m. \hat \mathbb{P}(X) := (X-\{0\})//\mathbb{G}_m \,.


For commutative rings and algebras

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


For RR a commutative kk-algebra, there is a natural isomorphism between

  • \mathbb{Z}-gradings on RR;

  • 𝔾 m\mathbb{G}_m-actions on SpecRSpec R.

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 July 29, 2016 03:31:14 by David Roberts (