= Toric irrelevant ideal
{title2=$B_\Sigma$}
For the polynomial <Cox ring> $S=k[z_\rho]$ of a <normal variety>[normal] split <toric variety> whose <toric rays> span the ambient real <toric lattice> space, the <toric irrelevant ideal> is
$$
B_\Sigma=\left(\prod_{\rho\notin\sigma(1)}z_\rho:\sigma\text{ maximal in }\Sigma\right).
$$
The <Cox construction> uses the open <set> $\operatorname{Spec}S\setminus V(B_\Sigma)$. For the <projective line>, the two maximal <toric cones> give generators $z_0,z_1$, so the excluded locus is the origin. For <affine space>, one maximal <toric cone> contains every <toric ray>; the empty product is one, hence $B=(1)$ and nothing is excluded.
The ray-span/no-torus-factor hypothesis belongs to this polynomial presentation. A pure <algebraic torus> has no <toric rays> and requires its <algebraic torus> coordinates in addition to the displayed polynomial data.<The Homogeneous Coordinate Ring of a Toric Variety by David A. Cox>[section1, irrelevant monomial ideal and ray-span hypothesis]{ref}
// Target: geometry-and-topology.bigb
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