§1.0. Background: Affine Varieties 9 in C2. This makes it easy to see that Zariski closed sets in C2 such as V(y − x2) cannot be closed in the product topology. ♦ Exercises for §1.0. 1.0.1. Prove Lemma 1.0.1. Hint: You will need the Nullstellensatz. 1.0.2. Let R be a commutative C-algebra. A subset S ⊆ R is a multipliciative subset pro- vided 1 ∈ S, 0 / S, and S is closed under multiplication. The localization RS consists of all formal expressions g/s, g ∈ R, s ∈ S, modulo the equivalence relation g/s ∼ h/t ⇐⇒ u(tg − sh) = 0 for some u ∈ S. (a) Show that the usual formulas for adding and multiplying fractions induce well-defined binary operations that make RS into C-algebra. (b) If R has no nonzero nilpotents, then prove that the same is true for RS. For more on localization, see [10, Ch. 3] or [89, Ch. 2]. 1.0.3. Let R be a finitely generated C-algebra without nilpotents as in Lemma 1.0.1 and let f ∈ R be nonzero. Then S = {1, f , f 2 ,...} is a multiplicative set. The localization RS is denoted R f and is called the localization of R at f . (a) Show that R f is a finitely generated C-algebra without nilpotents. (b) Show that R f satisfies Spec(R f ) = Spec(R) f . (c) Show that R f is given by (1.0.2) when R is an integral domain. 1.0.4. Let V be an affine variety with coordinate ring C[V]. Given a point p ∈ V , let S = {g ∈ C[V] | g(p) = 0}. (a) Show that S is a multiplicative set. The localization C[V ]S is denoted OV,p and is called the local ring of V at p. (b) Show that every φ ∈ OV,p has a well-defined value φ(p) and that mV,p = {φ ∈ OV,p | φ(p) = 0} is the unique maximal ideal of OV,p. (c) When V is irreducible, show that OV,p agrees with the definition given in the text. 1.0.5. Prove that a UFD is normal. 1.0.6. In the setting of Example 1.0.5, show that C[¯/¯] ⊆ C(C) is the integral closure of C[C] and that the normalization C → C is defined by t → (t2,t3). 1.0.7. In this exercise, you will prove some properties of normal domains needed for §1.3. (a) Let R be a normal domain with field of fractions K and let S ⊆ R be a multiplicative subset. Prove that the localization RS is normal. (b) Let Rα, α ∈ A, be normal domains with the same field of fractions K. Prove that the intersection α∈A Rα is normal. 1.0.8. Prove that dim Tp(Cn) = n for all p ∈ Cn. 1.0.9. Use Lemma 1.0.6 to prove the claim made in the text that smoothness is determined by the rank of the Jacobian matrix (1.0.3).

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