Contents

Definition

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

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

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

of the $(n+1)$-fold product of the line with itself by the canonical action of $\mathbb{G}_m$. Any point $(x_0,x_1,\ldots,x_n)\in \mathbb{A}^{n+1} - \{0\}$ gives homogeneous coordinates for its image under the quotient map. When considered in this fashion, one often writes $[x_0:x_1:\ldots:x_n]$. Homogeneous coordinates were introduced by August Ferdinand Möbius? in his 1827 work Der barycentrische Calcül.

More generally, for $(X,0)$ a pointed space with (pointed) $\mathbb{G}_m$-action, the quotient

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

is the corresponding projective space.

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

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

Examples

For commutative rings and algebras

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

Proposition

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

• $\mathbb{Z}$-gradings on $R$;

• $\mathbb{G}_m$-actions on $Spec R$.

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)

• 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 (58.179.228.38)