nLab
projective space

Definition

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

The projective space ${\mathbb{P}}_n$ of $T$ is the quotient

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

of the $(n+1)$-fold product of the line with itself by the canonical action of ${𝔾}_m$.

More generally, for $X$ a pointed space with ${𝔾}_m$-action, the quotient

$$\mathbb{P}(X):=(X-\{0\})/{𝔾}_m$$\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

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

Examples

For commutative rings and algebras

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

Proposition

For $R$ a commutative $k$-algebra, there is a natural isomorphism between

• $\mathbb{Z}$-gradings on $R$;

• ${𝔾}_m$-actions on $\mathrm{Spec}R$.

The proof is spelled out at affine line.

Over the real and complex numbers

(…)

Related concepts

• projective variety

• Grassmannian

References

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)