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Home , Primary decomposition

Exercises in 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 . 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 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 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 of fractions of an 푅 is denoted by Frac(푅). • The support of an 푅- 푀 is denoted by Supp(푀), and the set of 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 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.

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 . 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 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 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). Show that nil(푅) = rad(푅) = {0}.

(Hint: To show rad(푅) = {0}, produce a surjective homomorphism 휑푓 ∶ 푅 → ℚ such that 휑푓 (푓 ) ≠ 0, for any given nonzero 푓 ∈ 푅.)

4. Let 푅 = ℚ[푋1, 푋2, …] (infinitely many variables). Show that 푅 is not a Jacobson ring. (Hint: Exhibit a surjective homomorphism 휓 ∶ 푅 → 푅′ such that 푅′ is an integral domain with unique maximal ideal ≠ {0}. Such 푅′ can be obtained by localizing ℚ[푋] at some prime ideal.) 5. (1) Let 퐴 be a subring of an integral domain 퐵 , and let 퐶 be the integral closure of 퐴 in 퐵. Let 푓 , 푔 be monic polynomials in 퐵[푥] such that 푓 푔 ∈ 퐶[푥] . Then 푓 , 푔 are in 퐶[푥]. (Hint: Take a field containing 퐵 in which the polynomials 푓 , 푔 split into linear factors, prove that the coefficients of 푓 and 푔 are integral over 퐶.) (2) Prove the same result without assuming that 퐵 (or 퐴) is an integral domain.

6 6. Let 푓 ∶ 퐴 → 퐵 be an injective map, and 퐵 integral over 퐴. Assume that neither 퐴 nor 퐵 have zero divisors. (1) Show that if 퐴 is a field, then so is 퐵 . (2) Deduce that a field 푘 is algebraically closed (i.e., every polynomial has a root) if and only if for every finite field extension 푘 ⊂ 푘′ i.e., 푘′ is f.d. as a 푘-vector space, we have 푘 = 푘′. (3) Show that if 퐵 is a field, then so is 퐴 .

7 Problem set 5

28 March

Flatness 1. For a field 푘 , show that 푘 푡 [푌 , 푍]/(푌 푍 − 푡) is flat over 푘 푡 . 2. Let 푁 ′, 푁 , 푁 ′′ be 퐴-modules, and 0→ 푁 ′ → 푁 → 푁 ′′ → 0 be an exact sequence, with 푁 ′′ flat. Prove that 푁 ′ is flatJ K ⟺ 푁 is flat. J K ′′ (Hint: Use the Tor exact sequence, and note that for any 퐴-module 푀, Tor푛(푀, 푁 ) = ′′ Tor푛(푁 , 푀) = 0. Then use 5-lemma.) 3. A ring 퐴 is absolutely flat if every 퐴 -module is flat. Prove that the following are equivalent: (1) 퐴 is absolutely flat. (2) Every is idempotent. (3) Every finitely generated ideal is a direct summand of 퐴. (Hint: For (1) ⟹ (2), let 푥 ∈ 퐴, consider the following diagram:

(푥) ⊗ 퐴 퐴/(푥)

퐴 퐴/(푥)

For (2) ⟹ (3), prove that every finitely generated ideal is principal.) 4. Prove the following properties of absolutely flat: (1) Every homomorphic image of an absolutely flat ring is absolutely flat. (2) If a local ring is absolutely flat, then it is a field. (3) If a ring 퐴 is absolutely flat, then every non-unit in 퐴 is a zero-divisor.

8 Problem set 6

4 April

Going-up and Going-down 1. Let 푓 ∶ 퐴 → 퐵 be an integral homomorphism of rings, i.e. 퐵 is integral over its subring 푓 (퐴). Show that 푓 # ∶ Spec(퐵) → Spec(퐴) is a closed mapping, i.e. that it maps closed sets to closed sets. (Hint: This is a geometrical equivalent of the first part of Krull–Cohen–Seidenberg theorem. ) 2. Let 퐴 ⊂ 퐵 be an extension of rings, making 퐵 integral over 퐴, and let 픭 be a prime ideal of 퐴. Suppose there is a unique prime ideal 픮 of 퐵 with 픮 ∩ 퐴 = 픭. Show that −1 (a) 픮퐵픭 is the unique maximal ideal of 퐵픭 ∶= 퐵[(퐴 ∖ 픭) ];

(b) 퐵픮 = 퐵픭;

(c) 퐵픮 is integral over 퐴픭.

(Hint: Observe that 퐵픭 is integral over 퐴픭. For (a), show that 픮퐵픭 is the unique prime of 퐵픭 over 픭퐴픭, then apply the local case of Krull–Cohen–Seidenberg Theorem. For (b) and (c), show that every 푦 ∈ 퐵 ∖ 픮 becomes invertible in 퐵픭 by using the integrality over 퐴.) 3. Let the integral extension 퐴 ⊂ 퐵 and the prime ideal 픭 be as above. Suppose that 퐴 is a domain and 픮, 픮′ are distinct prime ideals of 퐵, both mapping to 픭 under Spec(퐵) → Spec(퐴). Show that 퐵픮 is not integral over 퐴픭. ′ −1 (Hint: Take 푦 ∈ 픮 ∖ 픮. Claim: 푦 ∈ 퐵픮 is not integral over 퐴픭. Otherwise we would have

푛 푛−1 푎0푦 + 푎1푦 + ⋯ + 푎푛−1푦 = −1

for some 푎0, … 푎푛−1 ∈ 퐴픭 with 푎0 ≠ 0. Clean denominators of 푎0, … , 푎푛−1 to derive a contra- diction.)

9 Problem set 7

11 April

Graded rings and modules, Filtrations 1. Many basic operations on ideals, when applied to homogeneous ideals in ℤ-graded rings, lead to homogeneous ideals. Let 퐼 be a homogeneous ideal in a ℤ-graded ring 푅. Show that: (1) The radical of 퐼 is homogeneous, that is, the radical of 퐼 is generated by all the homogeneous elements 푓 such that 푓 푛 ∈ 퐼 for some 푛. (2) If 퐼 and 퐽 are homogeneous ideals of 푅, then (퐼 ∶ 퐽 ) ∶= {푓 ∈ 푅|푓 퐽 ⊂ 퐼 } is a homoge- neous ideal. (3) Suppose that for all 푓 , 푔 homogeneous elements of 푅 such that 푓 푔 ∈ 퐼 , one of 푓 and 푔 is in 퐼 . Show that 퐼 is prime.

2. Suppose 푅 is a ℤ-graded ring and 0 ≠ 푓 ∈ 푅1. (1) Show that 푅[푓 −1] is again a ℤ-graded ring. −1 −1 −1 (2) Let 푆 = 푅[푓 ]0, show that 푆 ≅ 푅/(푓 − 1), and 푅[푓 ] ≅ 푆[푥, 푥 ], where 푥 is a new variable.

3. Show that if 푅 is a graded ring with no nonzero homogeneous prime ideals, then 푅0 is a field −1 and either 푅 = 푅0 or 푅 = 푅0[푥, 푥 ]. 4. Taking the associated graded ring can also simplify some features of the structure of 푅. For example, let 푘 be a field, and let 푅 = 푘[푥1, … , 푥푟 ] ⊂ 푅1 = 푘 푥1, … , 푥푟 be the rings of poly- nomials in 푟 variables and formal power series in 푟 variables over 푘, and write 퐼 = (푥1, … , 푥푟 ) for the ideal generated by the variables in either ring. Show thatJ gr퐼 푅 =Kgr퐼 푅1.

10 Problem set 8

18 April

You may select 3 of the following problems as your homework.

Completions 1. Let 퐴 be a local ring, m its maximal ideal. Assume that 퐴 is m -adically complete. For any polynomial 푓 (푥) ∈ 퐴[푥], let 푓 (푥) ∈ (퐴/m)[푥] denote its reduction mod. m. Prove Hensel’s lemma: if 푓 (푥) is monic of degree 푛 and if there exist coprime monic polynomials 푔(푥), ℎ(푥) ∈ (퐴/m)[푥] of degrees 푟 , 푛−푟 with 푓 (푥) = 푔(푥)ℎ(푥), then we can lift 푔(푥), ℎ(푥) back to monic polynomials 푔(푥), ℎ(푥) ∈ 퐴[푥]such that 푓 (푥) = 푔(푥)ℎ(푥).

(Hint:Assume inductively that we have constructed 푔푘(푥), ℎ푘(푥) ∈ 퐴[푥] such that

푘 푔푘(푥)ℎ푘(푥) − 푓 (푥) ∈ 픪 퐴[푥].

Then use the fact that since 푔(푥) and ℎ(푥) are coprime we can find 푎푝 (푥), 푏 푝 (푥), of degrees ⩽ 푝 푛−푟 , 푟 respectively, such that 푥 = 푎푝 (푥)푔 푘(푥)+푏 푝 (푥)ℎ 푘(푥), where 푝 is any integer such that 1⩽ 푝 ⩽ 푛. Finally, use the completeness of 퐴 to show that the sequences 푔푘(푥), ℎ푘(푥)converge to the required 푔(푥), ℎ(푥).) 2. (1) With the notation of Exercise 1, deduce from Hensel’s lemma that if 푓 (푥) has a simple root 훼 ∈ 퐴/m, then 푓 (푥) has a simple root 푎 ∈ 퐴 such that 훼 = 푎 mod m. (2) Show that 2 is a square in the ring of 7 -adic integers.

(3) Let 푓 (푥, 푦) ∈ 푘[푥, 푦], where 푘 is a field, and assume that 푓 (0, 푦) has 푦 = 푎0 as a simple root. ∞ 푛 Prove that there exists a formal power series 푦(푥) = ∑푛=0 푎푛푥 such that 푓 (푥, 푦(푥)) = 0. This gives the ”analytic branch” of the curve 푓 = 0 through the point (0, 푎0). 3. Let 퐴 be a Noetherian ring, 픞 an ideal in 퐴, and 퐴̂ the 픞-adic completion. For any 푥 ∈ 퐴, let ̂푥 be the image of 푥 in 퐴̂ . Show that 푥 not a zero-divisor in 퐴 implies ̂푥 not a zero-divisor in 퐴̂ . Does this imply that if 퐴 is an integral domain then 퐴̂ is an integral domain? (Hint: For a counter-example, consider 푅 = 푘[푥, 푦] and 픪 = (푥, 푦), 퐴 = 푘[푥, 푦]/(푦 2 − 푥2 − 푥3).

11 2 1 Let 푓 ∈ 푘 푥, 푦 be such that 푓 = 1 + 푥, such a power series can be obtained푓 = 1 + 2 푥 − 1 2 8 푥 + ⋯ ∈ 푘 푥 ⊂ 푘 푥, 푦 )) 4. Let 한 be aJ fieldK and consider the quotient of infinite J K J K 한[푋, 푍, 푌 , 푌 , 푌 , …] 푅 ∶= 1 2 3 . 2 3 (푋 − 푍푌1, 푋 − 푍 푌2, 푋 − 푍 푌3, …)

푛 Denote by 푍 the image of 푍 in 푅. Show that the ideal 퐼 ∶= (푍) of 푅 satisfies ⋂푛≥1 퐼 ≠ {0}. Why is this consistent with Krull’s intersection theorem?

12 Problem set 9

25 April

The materials below are taken from the Stacks Project.

Mittag-Leffler systems and Completions for non-Noetherian rings

휑푛+1 1. Consider an inverse system of sets ⋯ ← 퐴푛 ← 퐴푛+1 ← ⋯ (where 푛 = 1, 2, …). For each 푗 > 푖, let 휑푗 ,푖 ∶ 퐴푗 → 퐴푖 be the composition of 휑푗 , … , 휑푖 . We say that Mittag-Leffler conditions holds for (퐴푛, 휑푛)푛≥1 if for each 푖, we have

휑푘,푖 (퐴푘) = 휑푗 ,푖 (퐴푗 ) whenever 푗 , 푘 ≫ 푖.

Show that if (퐴푛, 휑푛)푛 is Mittag-Leffler and 퐴푛 ≠ ∅ for each 푛, then the limit

lim 퐴푛 ∶= {(푎푛)푛 ∈ ∏ 퐴푛 ∶ ∀푛, 휑푛+1(푎푛+1) = 푎푛} ←−푛 푛 is nonempty as well. 2. Suppose we are given an inverse system of short exact sequences of abelian groups, i.e. a com- mutative diagram ⋮ ⋮ ⋮

푓푛 푔푛 0 퐴푛 퐵푛 퐶푛 0

휑푛+1 휓푛+1 휉푛+1

0 퐴푛+1 퐵푛+1 푔 퐶푛+1 0 푓푛+1 푛+1

⋮ ⋮ ⋮

with exact rows, where 푛 = 1, 2, …. Show that if (퐴푛, 휑푛)푛 is Mittag-Leffler, then

lim 푓 lim 푔 0 → 퐴 −−−→푛 퐵 −−−→푛 퐶 → 0 ←−lim 푛 ←−lim 푛 ←−lim 푛

is exact. You only have to show the surjectivity of lim 푔푛.

13 3. Let 푅 be a ring (not necessarily Noetherian), 퐼 be a proper ideal, and 휑 ∶ 푀 → 푁 be a homomorphism of 푅-modules. Prove the following statements. 푀/퐼 푀 → 푁 /퐼 푁 휑̂ ∶ 푀̂ → 푁̂ 푀̂ = 푀/퐼 푛푀 (a) If is surjective, then so is . Here ←−lim 푛≥1 stands for the 퐼 -adic completion. 푛 푛 (Hint: First, show 푀/퐼 푀 → 푁 /퐼 푁 is surjective. Next, set 퐾푛 = Ker[푀 → 푁 /퐼 푛푁 ] to get exact sequences

푛 푛 푛 0 → 퐾푛/퐼 푀 → 푀/퐼 푀 → 푁 /퐼 푁 → 0,

푛+1 푛 then try to establish the surjectivity of 퐾푛+1/퐼 푀 → 퐾푛/퐼 푀 in order to apply Mittag-Leffler.) (b) If 0 → 퐾 → 푀 → 푁 → 0 is an exact sequence of 푅-modules and 푁 is flat, then 0 → 퐾̂ → 푀̂ → 푁̂ → 0 is exact. (Hint: Tensor by 푅/퐼 푛 to obtain an inverse system of short exact sequences, and apply Mittag-Leffler.) ̂ ̂ (c) If 푀 is finitely generated, then the natural homomorphism 푀 ⊗푅 푅 → 푀 given by ∞ ∞ 푚 ⊗ (푟푛)푛=1 ↦ (푟푛푚)푛=1 is surjective. (Hint: Show that if 푅⊕푁 → 푀 is a surjective homomorphism, then so is 푅̂⊕푁 → 푀̂ .) 4. Suppose 퐼 is finitely generated. Let 푀 be an 푅-module. Prove that

퐼 푛푀̂ = Ker[푀̂ → 푀/퐼 푛푀] = 퐼̂푛푀 ̂ for all 푛 ∈ ℤ≥1, and 푀 is 퐼 -adically complete as an 푅-module. 푛 (Hint: Fix 푛 and take generators 푓1, … , 푓푟 of 퐼 . This yields a surjective homomorphism of ⊕푟 푛 푅-modules (푓1, … , 푓푟 ) ∶ 푀 → 퐼 푀 ⊂ 푀. Pass to completions and show that

푛 ̂ ⊕푟 ̂푛 퐼 푀 ̂ 푛 ̂ (푓1, … , 푓푟 ) ∶ 푀 → 퐼 푀 = lim 푚 ≃ Ker[푀 → 푀/퐼 푀] ⊂ 푀, 푚≥푛←− 퐼 푀

̂ ⊕푟 ̂ which is surjective by the previous exercise. Identify the image of (푓1, … , 푓푟 ) ∶ 푀 → 푀 to infer that 푀/퐼̂ 푛푀̂ ≃ 푀/퐼 푛푀.)

14 Problem set 10

9 May

Hilbert series

1. Let 한 be a field and 푅 = 한[푋0, … , 푋푛], graded by total degree. Consider the graded 푅-module 푆 = 푅/(푓 ) where 푓 is a of total degree 푑 ≥ 1. Show that when 푚 ≥ 푑, 푚 + 푛 푚 + 푛 − 푑 휒(푆, 푚) ∶= dim 푆 = ( ) − ( ). 한 푚 푛 푛

2. Let 푅 = ⨁푛 푅푛 be a ℤ≥0-graded ring, finitely generated over 푅0. Assume 푅0 is Artinian (for example, a field) and let 푀 = ⨁푛 푀푛 be a finitely generated ℤ≥0-graded 푅-module. Define the Hilbert series in the variable 푇 as

푚 퐻푀 (푇 ) ∶= ∑ 휒(푀, 푚)푇 ∈ ℤ[[푇]] 푚≥0

where 휒(푀, 푚) denotes the length of the 푅0-module 푀푚, as usual. In what follows, graded means graded by ℤ≥0. (a) Show that if 0 → 푀 ′ → 푀 → 푀 ″ → 0 is a short exact sequence of graded 푅- modules, then 퐻푀 (푇 ) = 퐻푀 ′ (푇 ) + 퐻푀 ″ (푇 ).

(b) Relate 퐻푀 (푇 ) and 퐻푀(푘)(푇 ) for arbitrary 푘 ∈ ℤ, where 푀(푘)푑 ∶= 푀푑+푘.

(c) Suppose that 푅 is generated as an 푅0-algebra by homogeneous elements 푥1, … , 푥푛 with 푑푖 ∶= deg 푥푖 ≥ 1. Show that there exists 푄 ∈ ℚ[푇 ] such that 푄(푇 ) 퐻푀 (푇 ) = (1 − 푡 푑1 ) ⋯ (1 − 푡 푎푛 )

as elements of ℚ[[푇]]. (Hint: you may imitate the arguments for the quasi-polynomiality of 휒(푀, 푛).) 푚 (d) What can be said about the ℤ≥0-graded case, for general 푚?

15 Problem set 11

16 May

Dimension theory

1. Let ℤ3 be the 3-adic completion of the ring ℤ, so that ℤ ↪ ℤ3 naturally. Evaluate 1 + 3 + 2 3 3 + 3 + ⋯ in ℤ3. 2. Let 푅 be a Noetherian local ring. Suppose that there exists a principal prime ideal 픭 in 푅 such that ht(픭) ≥ 1. Prove that 푅 is an integral domain. (Hint: Below is one possible approach. Suppose 픭 = (푥) for some 푥 ∈ 푅. Let 픮 ⊂ 픭 be a minimal prime in 푅. Argue that (i) 푥 ∉ 픮, (ii) 픮 = 푥픮, and finally (iii) 픮 = {0}.) 3. Let 한 be a field and 푅 = 한 푋 × 한 푋 . Prove that 푅 is a Noetherian semi-local ring, 푅 contains a principal prime ideal of height 1, but 푅 is not an integral domain. (Hint: It is known that 한 푋 Jis NoetherianK J K local with maximal ideal (푋). Argue that the ideals in 한 푋 × 한 푋 take the form 퐼 × 퐽 where 퐼 , 퐽 are ideals in 한 푋 . Show that (푋) × 한 푋 and 한 푋 × (푋) are the onlyJ K maximal ideals, and both are of height 1.) J K J K J K J K J K

16 Final exam of Commutative Ring Theory The Enhanced Program for Graduate Study, BICMR / PKU June 14, 2019

The rings are assumed to be commutative and nonzero, but not necessarily Noetherian. We say that a ring homomorphism 휌 ∶ 퐴 → 퐵 is faithful if for all nonzero 퐴-module 푀, we have 푀 ⊗퐵 ≠ {0}. 퐴

1. (20 points) Show that the polynomial algebra ℚ[푋1, 푋2, 푋3, …] in infinitely many variables has infinite .

Solution: Consider the prime chain (푋1) ⊊ (푋1, 푋2) ⊊ ⋯. 2. (15 points) Let 휌 ∶ 퐴 → 퐵 be a faithful homomorphism of rings and let 푢 ∶ 푀 → 푁 be a homomorphism of 퐴-modules. Prove that if 푢 ⊗ id퐵 ∶ 푀 ⊗ 퐵 → 푁 ⊗ 퐵 is surjective, then 퐴 퐴 푢 ∶ 푀 → 푁 is surjective as well.

Solution: Recall that coker(푢 ⊗ id퐵 ) = coker(푢) ⊗ 퐵. 퐴 In what follows, we fix a ring 퐴 and an element 푓 ∈ 퐴 which is not a zero-divisor. Denote by 퐴̂the ̂ −1 ̂ ̂ −1 (푓 )-adic completion of 퐴. Weidentify 푓 with its image in 퐴 and write 퐴푓 ∶= 퐴[푓 ], 퐴푓 ∶= 퐴[푓 ] for the localizations with respect to the multiplicative subsets generated by 푓 . More generally, we −1 ̂ ∼ write 푀푓 ∶= 푀[푓 ] for any module 푀 over 퐴 or 퐴. Note that 푀 → 푀푓 if 푓 ∶ 푀 → 푀 is an automorphism. 3. (20 points) Show that 푓 is not a for 퐴̂. Solution: Let ̂푎= (푎 ) ∈ lim 퐴/(푓 푛). Then 푓 ̂푎= 0 is equivalent to 푎 ∈ (푓 푛) for 푛 푛≥1 푛≥1 푛+1 ←− 푛 all 푛 ≥ 1. This implies 푎푛 = 0 for all 푛 ≥ 1, since 푎푛 = 푎푛+1 mod 푓 . 4. (15 points) Let 푀 be an 퐴-module such that every element 푥 ∈ 푀 is annihilated by some power of 푓 . Show that (i) when 푓 푛푀 = {0} for some 푛 ≥ 1, the canonical homomorphism 푀 → 푀 ⊗ 퐴̂given 퐴 by 푥 ↦ 푥 ⊗ 1 is bijective; (ii) same as (i), but without assuming 푓 푛푀 = {0}; ̂ (iii) the “diagonal” homomorphism 휌 ∶ 퐴 → 퐴푓 × 퐴 is faithful.

You may use the easy fact that 퐴/푓̂ 푛퐴̂ ≃ 퐴/(푓 푛) (an earlier homework). Solution: Consider (i). We have 퐴 퐴̂ 푀 ≃ 푀 ⊗ →∼ 푀 ⊗ ≃ 푀 ⊗ 퐴,̂ 퐴 (푓 푛) 퐴 푓 푛퐴̂ 퐴 whose composite is exactly 푀 → 푀 ⊗ 퐴̂. For (ii), write 퐴 푀 = ⋃ ker [푓 푛 ∶ 푀 → 푀] 푛≥1

1 ⊗ 푀 = {0} and commute lim−→ past . As for (iii), note that our condition amounts to 푓 . 5. (20 points) Let 퐿 be an 퐴̂-module for which 푓 is not a zero-divisor, thus 퐿 can be viewed as a submodule of 퐿푓 . Prove that 퐴 ̂ 푛 ̂ (i) Tor1 (퐴, 퐿/푓 퐿) is zero for all 푛 ≥ 1, in particular 푓 is not a zero divisor for 퐿 ⊗ 퐴; 퐴 퐴 ̂ (ii) Tor1 (퐴, 퐿푓 /퐿) is zero. You may make free use of the long exact sequence for Tor:

퐴 ″ 퐴 ′ 퐴 퐴 ″ ⋯ → Tor푖+1(푋, 푌 ) → Tor푖 (푋, 푌 ) → Tor푖 (푋, 푌 ) → Tor푖 (푋, 푌 ) → ⋯

where 0 → 푌 ′ → 푌 → 푌 ″ → 0 is exact, as well as the fact that Tor commutes with direct limits and direct sums in each variable. Solution: In view of 1 1 퐿 /퐿 ≃ 퐿/퐿, 퐿/퐿 ≃ 퐿/푓 푛퐿, 푓 −→lim 푛 푛 푛≥1 푓 푓

it suffices to prove (i). The case 퐿 = 퐴̂amounts to the result that 푓 is not a zero divisor in 퐴̂, since 퐴/푓̂ 푛퐴̂ ≃ 퐴/(푓 푛). For general 퐿, one chooses a presentation of the 퐴/(푓 푛)-module 퐿/푓 푛퐿: there exists a set 퐼 and an exact sequence of 퐴-modules

⊕퐼 퐿 0 → 푀 → (퐴/(푓 푛)) → → 0. 푓 푛퐿

Then, by the previous result,

퐴 ̂ 푛 ̂ 푛 ⊕퐼 ̂ Tor1 (퐴, 퐿/푓 퐿) ≃ ker [푀 ⊗ 퐴 → (퐴/(푓 )) ⊗ 퐴 ] 퐴 퐴 ⊕퐼 ≃ ker [푀 → (퐴/(푓 푛)) ] = {0}.

6. (10 points) Below is a weak version of the celebrated Beauville–Laszlo theorem. Consider the following data

• an 퐴푓 -module 퐾, • an 퐴̂-module 퐿 for which 푓 is not a zero-divisor, ̂ ∼ ̂ • an isomorphism 휑 ∶ 퐾 ⊗퐴 퐴 → 퐿푓 of 퐴푓 -modules. Show that there exists an 퐴-module 푀, for which 푓 is not a zero-divisor, together with iso- morphisms ∼ ̂ ∼ 훼 ∶ 푀푓 → 퐾, 훽 ∶ 푀 ⊗ 퐴 → 퐿, 퐴 such that 휑 equals the composite of

−1 훽[푓 −1] ̂ 훼 ⊗id ̂ 퐾 ⊗ 퐴 푀푓 ⊗ 퐴 퐿푓 . 퐴 퐴

2 Solution: Given (퐾, 퐿, 휑), let us begin by showing the surjectivity of the composed homo- morphism 휑 ̂ 휑 ∶ 퐾 → 퐾 ⊗ 퐴 퐿푓 → 퐿푓 /퐿. 퐴 ̂ ̂ Since 퐴 → 퐴푓 ×퐴 is faithful and (퐿푓 /퐿)푓 = 0, this reduces to the surjectivity of the 퐴-linear map ̂ ̂ 퐾 ⊗ 퐴 → (퐿푓 /퐿) ⊗ 퐴 ≃ 퐿푓 /퐿, 퐴 퐴

휑 ̂ which agrees with 휑 when pulled-back to 퐾, thus it equals 퐾 ⊗ 퐴 ∼ 퐿푓 ↠ 퐿푓 /퐿. Hence 퐴 휑 is surjective. Next, put 푀 ∶= ker(휑) into the short exact sequence

푖 휑 0 → 푀 퐾 퐿푓 /퐿 → 0.

−1 Thus 훼 ∶= 푖[푓 ] ∶ 푀푓 → 퐾푓 ≃ 퐾 is an isomorphism since 퐾 is already an 퐴푓 -module. 퐴 ̂ Also, 푓 is not a zero-divisor for 푀. Using Tor1 (퐴, 퐿푓 /퐿) = 0, we conclude that the com- mutative diagram

̂ 푖⊗id ̂ 0 푀 ⊗ 퐴 퐾 ⊗ 퐴 퐿푓 /퐿 0 퐴 퐴

∼ 휑

0 퐿 퐿푓 퐿푓 /퐿 0

̂ ∼ has exact rows. We obtain 훽 ∶= 휑 ∘(푖 ⊗id) ∶ 푀 ⊗퐴 → 퐿. Since (퐿푓 /퐿)푓 = 0, by localizing 퐴 the diagram above with respect to 푓 we obtain

̂ 훼⊗id ̂ 푀푓 ⊗ 퐴 퐾 ⊗ 퐴 퐴 퐴

̂ ̂ (푀 ⊗ 퐴)푓 (퐾 ⊗ 퐴)푓 퐴 퐴

훽[푓 −1] 휑 퐿 퐿 푓 id 푓

in which all arrows are invertible. It follows that 훽[푓 −1] ∘ (훼−1 ⊗ id) = 휑. Note. Our approach to Beauville–Laszlo theorem follows the original paper A. Beauville, Y. Laszlo, Un lemme de descente. (French. Abridged English version). C. R. Acad. Sci., Paris, Sér. I 320, No. 3, 335-340 (1995). A more general version can be found on Stacks Project, 15.81.

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