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18 Chapter 1. Affine Toric Varieties This proves that Spec(C[S]) = YA . Since S = NA implies ZS = ZA , the torus of YA = Spec(C[S]) has the desired character lattice by Proposition 1.1.8. Here is an example of this proposition. Example 1.1.15. Consider the affine semigroup S ⊆ Z generated by 2 and 3, so that S = {0,2,3,...}. To study the semigroup algebra C[S], we use (1.1.5). If we set A = {2,3}, then ΦA (t) = (t2,t3) and the toric ideal is I(YA ) = x3 − y2 by Example 1.1.4. Hence C[S] = C[t2,t3] C[x,y]/ x3 − y2 and the affine toric variety YA is the curve x3 = y2 from Example 1.1.4. ♦ Equivalence of Constructions. Before stating our main result, we need to study the action of the torus TN on the semigroup algebra C[M]. The action of TN on itself given by multiplication induces an action on C[M] as follows: if t ∈ TN and f ∈ C[M], then t · f ∈ C[M] is defined by p → f (t−1 · p) for p ∈ TN. The minus sign will be explained in §5.0. The following lemma will be used several times in the text. Lemma 1.1.16. Let A ⊆ C[M] be a subspace stable under the action of TN. Then A = χm∈A C · χm. Proof. Let A = χm∈A C · χm and note that A ⊆ A. For the opposite inclusion, pick f = 0 in A. Since A ⊆ C[M], we can write f = m∈B cmχm, where B ⊆ M is finite and cm = 0 for all m ∈ B. Then f ∈ B ∩ A, where B = Span(χm | m ∈ B) ⊆ C[M]. An easy computation shows that t · χm = χm(t−1)χm. It follows that B and hence B ∩ A are stable under the action of TN. Since B ∩ A is finite-dimensional, Proposition 1.1.2 implies that B∩A is spanned by simultaneous eigenvectors of TN. This is taking place in C[M], where simultaneous eigenvectors are characters. It follows that B∩A is spanned by characters. Then the above expression for f ∈ B∩A implies that χm ∈ A for m ∈ B. Hence f ∈ A , as desired. We can now state the main result of this section, which asserts that our various approaches to affine toric varieties all give the same class of objects. Theorem 1.1.17. Let V be an affine variety. The following are equivalent: (a) V is an affine toric variety according to Definition 1.1.3. (b) V = YA for a finite set A in a lattice.