Exercises in Commutative Ring Theory The Enhanced Program for Graduate Study, BICMR Wen-Wei Li, Chunlin Shao Peking University, Spring 2019 Preface The present document comprises the homework for the course Commutative Ring Theory (Spring 2019) for the 11th Enhanced Program for Graduate Study organized by the Beijing International Cen- ter for Mathematical Research. They are divided into 11 problem sets, corresponding roughly to the materials in the lecture notes of the first-named author. Due to time constraints, they cover onlya very limited terrain of Commutative Algebra. Nevertheless, we hope this will be of some use to the readers. Most of the exercises herein are collected from existing resources such as the textbooks by Atiyah– Macdonald, Eisenbud, Matsumura, etc., or the websites MathOverflow and Stacks Project. The au- thors claim no originality of the exercises. We are deeply grateful to the students of this course, coming from all over mainland China, for their active participation and valuable feedback. Last update: May 25, 2019 General conventions • Unless otherwise specified, the rings are assumed to be commutative with unit 1. The ring of integers, rational, real and complex numbers are denoted by the usual symbols ℤ, ℚ, ℝ and ℂ respectively. • An expression 퐴 ∶= 퐵 means that 퐴 is defined to be 퐵. Injections and surjections are indicated by ↪ and ↠, respectively. The difference of sets is denoted as 퐴 ∖ 퐵, etc. • The group of invertible elements in a ring 푅 is denoted by 푅×. The localization of 푅 with respect to a multiplicative subset 푆 is denoted by 푅[푆 −1], and the localization with respect to a prime ideal 픭 by 푅픭. The space of prime ideals (resp. maximal ideals) of 푅 is denoted by Spec(푅) (resp. MaxSpec(푅)) and we let 푉 (픞) ∶= {픭 ∈ Spec(푅) ∶ 픭 ⊃ 픞} for any ideal 픞. The ideal generated by elements 푎, 푏, … is denoted by (푎, 푏, …). • The field of fractions of an integral domain 푅 is denoted by Frac(푅). • The support of an 푅-module 푀 is denoted by Supp(푀), and the set of associated prime by Ass(푀). The annihilator ideals are denoted by ann(⋯). • For a ring 푅, the 푅-algebra of polynomials (resp. formal power series) is denoted by 푅[푋, 푌 , …] (resp. 푅 푋, 푌 ) where 푋, 푌 , … stand for the indeterminates. J K 1 Problem set 1 28 February You may select 5 of the following problems as your homework. Ring Theory Revisited 1. Let 푥 be a nilpotent element of ring A. Show that 1 + 푥 is a unit of 퐴 . Deduce that the sum of a nilpotent element and a unit is a unit. 2. Let 퐴 be a ring and let 퐴[푥] be the ring of polynomials in an an indeterminate 푥, with coeffi- 푛 cients in 퐴 . Let 푓 = 푎0 + 푎1푥 + ⋯ + 푎푛푥 ∈ 퐴[푥]. Prove that: (1) 푓 is a unit in 퐴[푥] ⟺ 푎0 is a unit in 퐴 and 푎1, … , 푎푛 are nilpotent. 푚 푟 +1 (Hint: If 푏0 + 푏1푥 + ⋯ + 푏푚푥 is the inverse of 푓 , prove by induction on 푟 that 푎푛 푏푚−푟 = 0. Hence show that 푎푛 is nilpotent, and then use Ex.1.) (2) 푓 is nilpotent ⟺ 푎0, 푎1, … , 푎푛 are nilpotent. (3) 푓 is a zero-divisor ⟺ there exists 푎 ≠ 0 in 퐴 such that 푎푓 = 0. 푚 (Hint: Choose a polynomial 푔 = 푏0 + 푏1푥 + ⋯ + 푏푚푥 of least degree 푚 such that 푓 푔 = 0. Then 푎푛푏푚 = 0, hence 푎푛푔 = 0.) (4) 푓 is said to be primitive if (푎0, 푎1, … , 푎푛) = (1). Prove that if 푓 , 푔 ∈ 퐴[푥], then 푓 푔 is primitive ⟺ 푓 and 푔 are primitive. 3. Prove that a local ring contains no element 푒 ≠ 0, 1, such that 푒2 = 푒. Zariski Topology 4. For each 푓 ∈ 퐴, let 푋푓 denote the complement of 푉 (푓 ) in 푋 = Spec(퐴). The sets 푋푓 are open. Show that they form a basis of open sets for the Zariski topology, and that (1) 푋푓 ∩ 푋푔 = 푋푓 푔 (2) 푋푓 = ∅ ⟺ 푓 is nilpotent. (3) 푋푓 = 푋 ⟺ 푓 is a unit. Prime Avoidance 2 5. Here are two examples that show how the prime avoidance cannot be improved. (1) Show that if 푘 = ℤ/(2) then the ideal (푥, 푦) ⊂ 푘[푥, 푦]/(푥, 푦)2 is the union of 3 properly smaller ideals. 2 (2) Let 푘 be any field. In the ring 푘[푥, 푦]/ (푥푦, 푦 ), consider the ideals 퐼1 = (푥), 퐼2 = (푦), 2 and 퐽 = (푥 , 푦) . Show that the homogeneous elements of 퐽 are contained in 퐼1 ∪ 퐼2, but that 퐽 ⊈ 퐼1 and 퐽 ⊈ 퐼2 . Note that one of the 퐼푗 is prime. Localization of rings and modules 6. Let 푓 ∶ 퐴 → 퐵 be a homomorphism of rings and let 푆 be a multiplicatively closed subset of 퐴. Let 푇 = 푓 (푆). Show that 퐵[푆 −1] and 퐵[푇 −1] are isomorphic as 퐴[푆 −1] -modules. 7. Let 푀 be an 퐴 -module. Then the following are equivalent: (1) 푀 = 0. (2) 푀픭 = 0 for all prime ideals 픭 of 퐴. (3) 푀픪 = 0 for all maximal ideals 픪 of 퐴. Nakayama’s lemma 8. Let 퐴 be a ring, 픞 an ideal contained in the Jacobson radical ℜ of 퐴, let 푀 be an 퐴 -module and 푁 a finitely generated 퐴 -module, and let 푢 ∶ 푀 → 푁 be a homomorphism. If the induced homomorphism 푀/픞푀 → 푁 /픞푁 is surjective, then 푢 is surjective. 3 Problem set 2 7 March You may select 3 of the following problems as your homework. Radicals 1. A ring 퐴 is such that every ideal not contained in the nilradical (i.e. √0) contains a non-zero idempotent (that is, an element 푒 such that 푒2 = 푒 ≠ 0). Prove that the nilradical and Jacobson radical of 퐴 are equal. Noetherian and Artinian rings 2. In a Noetherian ring 퐴, show that every ideal a contains a power of its radical. 3. Let 푀 be an 퐴-module and let 푁1, 푁2 be submodules of 푀. If 푀/푁1 and 푀/푁2 are Noetherian, so is 푀/ (푁1 ∩ 푁2). Similarly with Artinian in place of Noetherian. Support of a module 4. (1) Let 퐴 be a local ring, 푀 and 푁 finitely generated 퐴 -modules. Prove that if 푀 ⊗퐴 푁 = 0, then 푀 = 0 or 푁 = 0. (Hint: Let 픪 be the maximal ideal, 푘 = 퐴/픪 the residue field. Let 푀푘 = 푘⊗퐴푀 ≅ 푀/픪푀, and similarly definition for 푁푘. Then use Nakayama’s lemma, and note that 푀푘, 푁푘 are vector spaces.) (2) Let 퐴 be a ring, 푀, 푁 are finitely generated 퐴-module. Prove that Supp (푀 ⊗퐴 푁 ) = Supp(푀) ∩ Supp(푁 ) 4 Problem set 3 14 March You may select 4 of the following problems as your homework. Support, Associated primes and Primary decompositions 1. Show that the support of the ℤ-module 푀 ∶= ⨁푛≥1 ℤ/푛ℤ is not a closed subset of Spec(ℤ). Show that its closure equals 푉 (annℤ(푀)). ∞ 푎 2. Let 푝 be a prime number. Show that the support of the ℤ-module ∏푎=1 ℤ/푝 ℤ is not equal ∞ 푎 to the closure of ⋃푎=1 Supp (ℤ/푝 ℤ). 3. Let 푀 be an 푅-module, which is “graded” in the following sense: we are given decompositions ∞ ∞ of abelian groups 푅 = ⨁푎=0 푅푎 and 푀 = ⨁푏=0 푀푏 , such that 1 ∈ 푅0 and for all 푎, 푏 ∈ ℤ≥0, 푅푎 ⋅ 푅푏 ⊂ 푅푎+푏 , 푅푎 ⋅ 푀푏 ⊂ 푀푎+푏 . Suppose that 푚 ∈ 푀 and 픭 ∶= ann푅(푚) is a prime ideal of 푅. (1) Show that 픭 = ⨁푎≥0(픭 ∩ 푅푎) .We say 픭 is a “homogeneous ideal”. ′ ′ (2) Show that there exist some 푏 ≥ 0 and 푚 ∈ 푀푏 such that 픭 = ann푅(푚 ) .Thus 픭 is actually the annihilator of some “homogeneous element”. 4. Let 1 1 0 퐴 = ( 0 1 0 ) 0 0 2 3 푚 푗 푚 푗 Regard the ℂ-vector space ℂ as a ℂ[푇 ] -module via (∑푗 =0 푎푗 푇 ) 푣 ∶= ∑푗 =0 푎푗 (퐴 푣). What are the associated prime ideals of this ℂ[푇 ] -module? 5. Try to give a primary decomposition of the ideal 퐼 = (푥3푦, 푥푦 4) of the ring 푅 = 푘[푥, 푦]. (Hint:If 퐼 is a monomial ideal with a generator 푎푏, where 푎 and 푏 are coprime, say 퐼 = (푎푏)+ 퐽 , then 퐼 = ((푎) + 퐽 ) ∩ ((푏) + 퐽 ). Applying this recursively gives a primary decomposition.) 5 Problem set 4 21 March You may select 4 of the following problems as your homework. Integral dependence, Nullstellensatz 1. Prove that the integral closure of 푅 ∶= ℂ[푋, 푌 ]/(푌 2 −푋 2 −푋 3) in Frac(푅) equals ℂ[푡] with 푡 ∶= 푌̄ /푋̄ , where 푋,̄ 푌̄ denote the images of 푋, 푌 in 푅. (Hint: First, show that the homomorphism 휉 ∶ ℂ[푋, 푌 ] → ℂ[푇 ] given by 푋 ↦ 푇 2 − 1 and 푌 ↦ 푇 3 − 푇 has kernel equal to (푌 2 − 푋 2 − 푋 3). Secondly, show that 푇 is in the field of fractions of 푅 ≃ image(휉 ) ⊂ ℂ(푇 ), integral over 푅, and argue that ℂ[푇 ] is the required integral closure. Try to motivate the choice of 휉 .) 2. Consider a Noetherian ring 푅 with 퐾 ∶= Frac(푅). Show that 푥 ∈ 퐾 is integral over 푅 if and only if there exists 푢 ∈ 푅 such that 푢 ≠ 0 and 푢푥푛 ∈ 푅 for all 푛. (Hint: When this condition holds, 푅[푥] is an 푅-submodule of 푢−1푅.) 3. Let 푅 = ℚ[푋1, 푋2, …] (finite or infinitely many variables).