Contents

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 ${ℙ}_{n}$ of $T$ is the quotient

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

of the $\left(n+1\right)$-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

$ℙ\left(X\right):=\left(X-\left\{0\right\}\right)/{𝔾}_{m}$\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

$\stackrel{^}{ℙ}\left(X\right):=\left(X-\left\{0\right\}\right)//{𝔾}_{m}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${ℙ}_{k}^{n}$ is (…) A closed subscheme of ${ℙ}_{k}^{n}$ is a projective scheme?.

Proposition

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

• $ℤ$-gradings on $R$;

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

The proof is spelled out at affine line.

(…)

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)

Revised on January 9, 2013 22:47:33 by Urs Schreiber (89.204.137.52)