§1.1. Introduction to Affine Toric Varieties 17 of S. To make this precise, we think of M as the character lattice of a torus TN, so that m ∈ M gives the character χm. Then C[S] = m∈S cmχm | cm ∈ C and cm = 0 for all but finitely many m , with multiplication induced by χm · χm = χm+m . If S = NA for A = {m1,...,ms}, then C[S] = C[χm1,...,χms]. Here are two basic examples. Example 1.1.12. The affine semigroup Nn ⊆ Zn gives the polynomial ring C[Nn] = C[x1,...,xn], where xi = χei and e1,...,en is the standard basis of Zn. ♦ Example 1.1.13. If e1,...,en is a basis of a lattice M, then M is generated by A = {±e1,...,±en} as an affine semigroup. Setting ti = χei gives the Laurent polynomial ring C[M] = C[t±1,...,t±1]. 1 n Using Example 1.0.2, one sees that C[M] is the coordinate ring of the torus TN. ♦ Affine semigroup rings give rise to affine toric varieties as follows. Proposition 1.1.14. Let S ⊆ M be an affine semigroup. Then: (a) C[S] is an integral domain and finitely generated as a C-algebra. (b) Spec(C[S]) is an affine toric variety whose torus has character lattice ZS, and if S = NA for a finite set A ⊆ M, then Spec(C[S]) = YA . Proof. As noted above, A = {m1,...,ms} implies C[S] = C[χm1,...,χms], so C[S] is finitely generated. Since C[S] ⊆ C[M] follows from S ⊆ M, we see that C[S] is an integral domain by Example 1.1.13. Using A = {m1,...,ms}, we get the C-algebra homomorphism π : C[x1,...,xs] −→ C[M] where xi → χmi ∈ C[M]. This corresponds to the morphism ΦA : TN −→ Cs from (1.1.4), i.e., π = (ΦA )∗ in the notation of §1.0. One checks that the kernel of π is the toric ideal I(YA ) (Exercise 1.1.4). The image of π is C[χm1,...,χms] = C[S], and then the coordinate ring of YA is (1.1.5) C[YA ] = C[x1,...,xs]/I(YA ) = C[x1,...,xs]/Ker(π) Im(π) = C[S].

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