§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 = 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 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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