10 Chapter 1. Affine Toric Varieties 1.0.10. Let V be irreducible and suppose that p V is smooth. The goal of this exercise is to prove that OV,p is normal using standard results from commutative algebra. Set n = dim V and consider the ring of formal power series C[[x1,...,xn]]. This is a local ring with maximal ideal m = x1,...,xn . We will use three facts: C[[x1,...,xn]] is a UFD by [281, p. 148] and hence normal by Exercise 1.0.5. Since p V is smooth, [208, §1C] proves the existence of a C-algebra homomorphism OV,p C[[x1,...,xn]] that induces isomorphisms OV,p/mV,p C[[x1,...,xn]]/m for all 0. This implies that the completion (see [10, Ch. 10]) OV,p = limOV,p/mV,p ←− is isomorphic to a formal power series ring, i.e., OV,p C[[x1,...,xn]]. Such an iso- morphism captures the intuitive idea that at a smooth point, functions should have power series expansions in “local coordinates” x1,...,xn. If I OV,p is an ideal, then I = =1 (I + m V ,p ). This theorem of Krull holds for any ideal I in a Noetherian local ring A and follows from [10, Cor. 10.19] with M = A/I. Now assume that p V is smooth. (a) Use the third bullet to show that OV,p C[[x1,...,xn]] is injective. (b) Suppose that a,b OV,p satisfy b|a in C[[x1,...,xn]]. Prove that b|a in OV,p. Hint: Use the second bullet to show a bOV,p + m V ,p and then use the third bullet. (c) Prove that OV,p is normal. Hint: Use part (b) and the first bullet. This argument can be continued to show that OV,p is a UFD. See [208, (1.28)] 1.0.11. Let V and W be affine varieties and let S ⊆V be a subset. Prove that S×W = S ×W. 1.0.12. Let V and W be irreducible affine varieties. Prove that V ×W is irreducible. Hint: Suppose V ×W = Z1 ∪Z2, where Z1,Z2 are closed. Let Vi = {v V | {v}×W Zi}. Prove that V = V1 ∪V2 and that Vi is closed in V. Exercise 1.0.11 will be useful. §1.1. Introduction to Affine Toric Varieties We first discuss what we mean by “torus” and then explore various constructions of affine toric varieties. The Torus. The affine variety (C∗)n is a group under componentwise multiplica- tion. A torus T is an affine variety isomorphic to (C∗)n, where T inherits a group structure from the isomorphism. The term “torus” is taken from the language of linear algebraic groups. We will use (without proof) basic results about tori that can be found in standard texts on algebraic groups such as [37], [152], and [257]. See also [36, Ch. 3] for a self-contained treatment of tori.
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