Commutative Algebra

Burt Totaro

Michaelmas 2011

Preface

Commutative algebra is about commutative rings such as ℤ, 푘[푡1, … , 푡푛], etc. The philosophy of commutative algebra is to think of every commutative ring as a ring of functions on some space. “Everything in algebra should have a geometric meaning.”

Scribe’s note

Some of the material has been reordered: for example, § 5.1 was lectured before § 4.1. The module theory appearing in these notes has been generalised to the non-commutative case (since doing so is virtually effortless), but the ring theory remains firmly rooted in the commutative context.

Recommended texts

• [A–M], Introduction to commutative algebra

• [Eisenbud, 1995], Commutative algebra with a view toward algebraic geometry

• [Lang, 2002], Algebra

• [Reid, 1995], Undergraduate commutative algebra

Contents

I. Introduction 1 1. Rings and homomorphisms ...... 1 2. Modules and linear maps ...... 4 3. Prime and maximal ideals ...... 5

II. Modules 13 1. Exactness ...... 13 2. Direct limits ...... 15 3. Tensor products ...... 17 4. Algebras ...... 22

III. Local algebra 25 1. Localisation of rings ...... 25 2. Localisation of modules ...... 28

IV. Homological algebra 33 1. Chain complexes ...... 33 2. Derived functors ...... 35

V. Finiteness conditions 41 1. Noetherian rings ...... 41 2. Integral extensions ...... 45 3. Noether normalisation and Nullstellensätze ...... 50

VI. Dimension theory 55 1. Artinian rings ...... 55

iii Contents

2. Discrete valuations and Dedekind domains ...... 57 3. Noetherian local rings ...... 60 4. Affine varieties ...... 62 5. Regular local rings ...... 64

Bibliography 69

iv I Introduction

1 Rings and homomorphisms

Definition 1.1.1. A ring is a set 푅 with two binary operations + and × such that 푅 is an abelian group under + with unit element 0 and 푅 is a monoid under × with unit element 1, and + distributes over ×, i.e. 푥 × (푦 + 푧) = 푥 × 푦 + 푥 × 푧 and (푥 + 푦) × 푧 = 푥 × 푧 + 푦 × 푧.A commutative ring is a ring 푅 which is also a commutative monoid under ×.

Examples 1.1.2.

• The ring of 푛 × 푛 matrices over a field 푘, denoted by M푛(푘), is a non- commutative ring.

• The group algebra 푘퐺 is non-commutative if 퐺 is a non-abelian group.

In this course, rings are assumed commutative unless otherwise stated.

Examples 1.1.3.

• Every field is a commutative ring: ℚ, ℝ, ℂ, 픽푝, etc.

• The ring of integers, denoted by ℤ, is a commutative ring.

• Given a field 푘, the polynomial ring 푘[푥1, … , 푥푛] is a commutative ring. • Given a topological space 푋, the set of all continuous functions 푋 → ℝ is naturally a commutative ring.

• Given a smooth manifold 푋, the set of all smooth functions 푋 → ℝ is naturally a commutative ring.

1 I. Introduction

Remark 1.1.4. We do not require 0 ≠ 1, so that we have the trivial ring 0. It has just one element and is characterised by the equation 0 = 1.

Exercise 1.1.5. Show the following equations hold in any ring:

• 0 × 푎 = 푎

• (−1) × 푎 = −푎

• etc.

Definition 1.1.6. Let 푅 be a commutative ring. An element 푥 is invertible or is a unit if there is an element 푦 such that 푥푦 = 푦푥 = 1. If there is such a 푦, it is unit. We write 푥−1 for this unique 푦 if it exists. A zero divisor is an element 푥 such that there exists a non-zero element 푦 with 푥푦 = 0. An element 푥 is nilpotent if there exists a positive integer 푛 such that 푥푛 = 0.

Definition 1.1.7. A field is a commutative ring 푅 such that every non-zero element is invertible. An integral domain is a commutative ring 푅 such that the only zero divisor is 0 and 0 ≠ 1. A commutative ring 푅 is reduced if the only nilpotent element is 0.

Remark 1.1.8. Note that the trivial ring is reduced but not an integral domain.

Examples 1.1.9. • Any field is an integral domain.

• Any integral domain is reduced.

• For a positive integer 푛, ℤ ∕ ⟨푛⟩ is a field if and only if it it is a domain, so if and only if 푛 is prime.

• ℤ ∕ ⟨푛⟩ is reduced if and only if 푛 is square-free.

• The rings ℤ and 푘[푥1, … , 푥푛] are domains but not fields. Definition 1.1.10. Let 퐴 and 퐵 be rings, not necessarily commutative. A ring homomomorphism is a map 푓 ∶ 퐴 → 퐵 such that 푓(푥 + 푦) = 푓(푥) + 푓(푦), 푓(푥 × 푦) = 푓(푥) × 푓(푦), and 푓(1) = 1.

2 1. Rings and homomorphisms

Exercise 1.1.11. Let 푓 ∶ 퐴 → 퐵 be a ring homomorphism. Show that 푓(0) = 0 and 푓(−푥) = −푓(푥). Examples 1.1.12. • If 퐴 is a subring of 퐵, the inclusion 퐴 ↪ 퐵 is a ring homomorphism.

• The identity map id ∶ 퐴 → 퐴 is a ring homomorphism.

• If 푓 ∶ 퐴 → 퐵 and 푔 ∶ 퐵 → 퐶 are ring homomorphisms, then so is the composite map 푔 ∘ 푓 ∶ 퐴 → 퐶. Hence, the class of rings and ring homomorphisms constitute a category.

Definition 1.1.13. A (left) ideal of a ring 푅 is a subgroup 퐼 of the additive group of 푅 such that 푟 × 푥 ∈ 퐼 if 푟 ∈ 푅 and 푥 ∈ 퐼. Examples 1.1.14. • The kernel of any ring homomorphism 푓 ∶ 퐴 → 퐵, regarded as a homomorphism of abelian groups, is an ideal of 퐴.

• The only ideals of a field 푘 are {0} and 푘 itself. • Any ideal containing a unit is the whole ring. In particular, ideals are usually not subrings.

• The ring ℤ is a principal ideal domain, i.e. every ideal is of the form ⟨푛⟩ for some 푛 ∈ ℤ.

• If 퐴 is a ring of functions 푋 → ℝ and 푌 is any subset of 푋, then the set ℐ(푌) = {푓 ∈ 퐴 | ∀푦 ∈ 푌. 푓(푦) = 0} is an ideal of 퐴. Exercise 1.1.15. For a commutative ring 푅, show the following are equivalent: (i) 푅 is a field.

(ii) The only ideals of 푅 are {0} and 푅 itself.

(iii) Any ring homomorphism 푅 → 푆 with 푆 ≠ 0 is injective. Exercise 1.1.16. Show that in any commutative ring 푅, the set of nilpotent elements forms an ideal 픯, called the nilradical of 푅. Show also that 푅 ∕ 픯 is reduced.

3 I. Introduction

2 Modules and linear maps

A module is a linearisation of a ring. To be precise:

Definition 1.2.1. A (left) module over a ring 푅 is an abelian group 푀 together with a map ⋅ ∶ 푅 × 푀 → 푀 such that the following equations hold:

(푟 + 푠) ⋅ 푚 = 푟 ⋅ 푚 + 푠 ⋅ 푚 (푟 × 푠) ⋅ 푚 = 푟 ⋅ (푠 ⋅ 푚) 푟 ⋅ (푚 + 푛) =푟⋅푚+푟⋅푛 1⋅푚=푚

Remark 1.2.2. There is an analogous definition of a right module. For a commutative ring, these two notions are equivalent.

Examples 1.2.3. • If 푘 is a field, a 푘-module is exactly the same thing as a 푘-vector space.

• An abelian group is a ℤ-module in a unique way, and vice versa.

• If 푘 is a field, a 푘[푥]-module is a 푘-vector space 푉 together with aa 푘-linear map 푥 ∶ 푉 → 푉.

• If 푅 is a ring and 퐼 is a (left) ideal, then 퐼 is a (left) 푅-module, and so is 푅 ∕ 퐼.

Definition 1.2.4. Let 푀 and 푁 be (left) 푅-modules. A (left) 푅-module homomorphism, or 푅-linear map, is an abelian group homomorphism 푓 ∶ 푀 → 푁 such that 푓(푟 ⋅ 푚) = 푟 ⋅ 푓(푚).

This yields a category of (left) 푅-modules. If 푅 is commutative and we

are given two 푅-modules 푀 and 푁, the hom set Hom푅(푀, 푁) is naturally an 푅-module, with the evident addition and left action:

(푓 + 푔)(푚) = 푓(푚) + 푔(푚)(푟 ⋅ 푓)(푚) = 푟 ⋅ 푓(푚)

If 푅 is non-commutative, then Hom푅(푀, 푁) is in general only an abelian group.

Definition 1.2.5. A (left) 푅-submodule of an 푅-module 푀 is a subgroup of the additive group of 푀 closed under the (left) action of 푅.

4 3. Prime and maximal ideals

For a (left) 푅-submodule 푁 of a (left) 푅-module 푀, the quotient abelian group 푀 ∕ 푁 is naturally a (left) 푅-module with an evident left action of 푅. For any (left) 푅-module homomorphism 푓 ∶ 푀 → 푁, the kernel of 푓 as an abelian group homomorphism is a (left) 푅-submodule of 푀, and the image of 푓 as an abelian group is a (left) 푅-submodule of 푁. Similarly, the cokernel of 푓, defined by coker 푓 = 푁 ∕ im 푓, is a (left) 푅-module, and the coimage of 푓, defined by coim 푓 = 푀 ∕ ker 푓, is canonically isomorphic to the image of 푓 as a (left) 푅-module.

3 Prime and maximal ideals

Definition 1.3.1. Let 푅 be a commutative ring. A maximal ideal of 푅 is an ideal 퐼 such that 푅 ∕ 퐼 is a field. A prime ideal of 푅 is an ideal 퐼 such that 푅 ∕ 퐼 is an integral domain. A radical ideal of 푅 is an ideal 퐼 such that 푅 ∕ 퐼 is reduced. Hence, by our earlier observations, every maximal ideal is prime and every prime ideal is radical.

Exercise 1.3.2. Let 푅 be a commutative ring.

(i) An ideal of 푅 is maximal if and only if it is proper and not strictly contained in any proper ideal.

(ii) An ideal 퐼 is prime if and only if 퐼 is a proper ideal and 푓푔 ∈ 퐼 implies either 푓 ∈ 퐼 or 푔 ∈ 퐼.

(iii) An ideal 퐼 is radical if and only if 푓푛 ∈ 퐼 implies 푓 ∈ 퐼.

Example 1.3.3. The maximal ideals of ℤ are the ideals ⟨푝⟩, where 푝 varies over the prime number. The prime ideals ℤ are the maximal ideals together with {0}. The reduced ideals of ℤ are the ideals of the form ⟨푛⟩, where 푛 is square-free, 0, or 1. More generally, 푅 is always a radical ideal in any ring 푅.

Example 1.3.4. Let 푘 be a field. The polynomial ring 푘[푥] is a principal ideal domain (indeed, a euclidean domain), so is a unique factorisation domain. Hence, every ideal of 푘[푥] is of the form ⟨푓⟩ for some 푓 in 푘[푥]; moreover, every non- 푒1 푒푛 zero ideal of 푘[푥] is of the form ⟨푓1 ⋯ 푓푛 ⟩ where 푒1, … , 푒푛 are positive integers and 푓1, … , 푓푛 are irreducible elements which are distinct modulo units.

5 I. Introduction

In particular, ⟨푓⟩ is prime if and only if 푓 = 0 or 푓 is irreducible, and ⟨푓⟩ is maximal if and only if 푓 is irreducible. Furthermore, if 푘 is algebraically closed, 푓 is irreducible if and only if it is of the form 푎푥 − 푏, where 푎 ∈ 푘× and 푏 ∈ 푘. In particular, the prime ideals of ℂ[푥] are {0} and ⟨푥 − 푎⟩ for each 푎 in ℂ.

Example 1.3.5. Some examples of prime ideals in ℤ[푥]:

⟨0⟩, ⟨7⟩, ⟨푥⟩, ⟨7, 푥⟩

Of these, only ⟨7, 푥⟩ is a maximal ideal.

Definition 1.3.6. Let 푓 ∶ 퐴 → 퐵 be a homomorphism of rings. If 퐽 is a (left) ideal of 퐵, the contraction of 퐽 in 퐴 is its inverse image 푓−1퐽. If 퐼 is a (left) ideal of 퐴, the extension of 퐼 in 퐵 is the (left) ideal 퐼햾 in 퐵 generated by the image of 퐼.

In particular, if 퐴 is a subring of 퐵, the contraction of 퐽 is just 퐽 ∩ 퐴, and the extension of 퐼 is the ideal 퐵퐼 = {푏푥 | 푏 ∈ 퐵, 푥 ∈ 퐼}.

Lemma 1.3.7. Let 퐴 and 퐵 be commutative rings. For any ring homomorphism 푓 ∶ 퐴 → 퐵, the contraction of a prime ideal of 퐵 is a prime ideal of 퐴.

Proof. Let 퐽 be a prime ideal of 퐵. The residue ring 퐵 ∕ 퐽 is an integral domain, but the kernel of the composite homomorphism 퐴 → 퐵 ↠ 퐵 ∕ 퐽 is 푓−1퐽, so 퐴 ∕ 푓−1퐽 is isomorphic to 퐵 ∕ 퐽 and in particular is an integral domain; hence 푓−1퐽 is a prime ideal of 퐴.

Remark 1.3.8. Maximal ideals do not always pull back to maximal ideals: ⟨0⟩ is maximal in ℚ but not in ℤ. Remark 1.3.9. The trivial ring has no prime, maximal, or proper ideals.

Theorem 1.3.10 (Krull). Every non-trivial commutative ring has at least one maximal ideal.

Proof. Recall Zorn’s lemma: if every chain in a non-empty poset 푆 has an upper bound in 푆, then 푆 has a maximal element. Consider the set 푆 of all proper ideals of a non-trivial commutative ring 푅, partially ordered by inclusion. A maximal ideal of 푅 is precisely a maximal element of 푆. Let (퐼훼) be a chain of ideals of 푅, and let 퐼 = ⋃훼 퐼훼. It is clear that 퐼 is an ideal, and 1 ∉ 퐼 since 1 ∉ 퐼훼 for

6 3. Prime and maximal ideals

any 훼. Thus, 퐼 ∈ 푆 and is an upper bound for the chain (퐼훼). Note that ⟨0⟩ ∈ 푆 since 푅 is non-trivial, so 푆 indeed has a maximal element.

Corollary 1.3.11. Let 푅 be a commutative ring and let 퐼 be an ideal of 푅. If 퐼 is a proper ideal, then there is a maximal ideal 픪 with 퐼 ⊆ 픪.

Proof. The residue ring 푅 ∕ 퐼 has a maximal ideal 픪, so its inverse image 픪 in 푅 is prime and contains 퐼. But we have a commutative diagram of surjections as below: 푅 푅 ∕ 퐼

∼ 푅 ∕ 픪 (푅 ∕ 퐼) ∕ 픪

So 픪 is the required maximal ideal of 푅.

For the rest of this section, we will only consider commutative rings.

Definition 1.3.12. Let 푅 be a ring. The prime spectrum of 푅, denoted by Spec 푅, is the set of all prime ideals of 푅 and is equipped with the Zariski topology, in which a subset 푆 of Spec 푅 is closed if and only if

푆 = 풱(퐼) = {픭 ∈ Spec 푅 | 퐼 ⊆ 픭} for some ideal 퐼 of 푅.

Let 푘 be a field, and suppose 푅 is a ring of functions 푋 → 푘 containing the subfield of constant functions. Each point 푃 of 푋 yields a maximal ideal 픪푃: it is the kernel of the evaluation map 푓 ↦ 푓(푃). For a general ring 푅, we consider homomorphisms of 푅 to arbitrary fields 푘. The kernel of such homomorphisms are prime ideals, and every prime ideal of 푅 arises in this fashion.

Theorem 1.3.13. The Zariski topology on Spec 푅 is indeed a topology.

Proof. It is easy to check that ∅ = 풱(⟨1⟩) and Spec 푅 = 풱(⟨0⟩). Let {퐼훼} be a collection of ideals of 푅. Then,

풱 퐼 = 픭 ∈ Spec 푅 | ∀훼. 퐼 ⊆ 픭 = 풱 퐼 ⋂ ( 훼) { | 훼 } (∑ 훼) 훼 훼

7 I. Introduction

so arbitrary intersections of closed subsets are indeed closed. Finally, if 퐼1 and 퐼2 are two ideals of 푅, then

| 풱(퐼1) ∪ 풱(퐼2) = {픭 ∈ Spec 푅 | 퐼1 ⊆ 픭 or 퐼2 ⊆ 픭} ⊆ 풱(퐼1 ∩ 퐼2) but if 퐼1 ∩ 퐼2 ⊆ 픭, then either 퐼1 ⊆ 픭 or 퐼2 ⊆ 픭: indeed, if 푔 ∈ 퐼2 and 푔 ∉ 픭, then for all 푓 ∈ 퐼1, 푓푔 ∈ 픭, so 푓 ∈ 픭. Hence 풱(퐼1) ∪ 풱(퐼2) = 풱(퐼1 ∩ 퐼2), i.e. finite unions of closed sets are closed.

Example 1.3.14. Spec ℚ is a single point.

Example 1.3.15. Spec ℤ = {⟨0⟩}∪{⟨푝⟩ | 푝 is a prime number}. Note that {⟨푝⟩} is closed for each prime number 푝, but the closure of {⟨0⟩} is Spec ℤ itself. We say ⟨0⟩ is the generic point of Spec ℤ.

Example 1.3.16. Similarly, there is a natural bijection between Spec ℂ[푥] and ℂ ∪ {a generic point}.

Different ideals 퐼 can give the same 풱(퐼).

Theorem 1.3.17. For any ring 푅, the nilradical of 푅 is the intersection of all prime ideals of 푅.

Proof. If 푓푛 = 0 then 푓 ∈ 픭 for any prime ideal 픭: indeed, 푅 ∕ 픭 is an integral domain, so 푓푛 = 푓 = 0 (mod 픭), i.e. 푓 ∈ 픭. Conversely, let 푓 be any non-nilpotent element of 푅. Let 푆 be the poset of all ideals 퐼 of 푅 such that no power of 푓 is contained in 퐼, partially ordered by inclusion. It has a maximal element: if (퐼훼) is a chain in 푆, then ⋃훼 퐼훼 is an ideal not containing any power of 푓, and ⟨0⟩ ∈ 푆, so by Zorn’s lemma we have a maximal element 픭 in 푆. Suppose 푔 ∉ 픭 and ℎ ∉ 픭. Since ℎ ∈ 픭 + ⟨ℎ⟩, the ideal 픭+⟨ℎ⟩ cannot be in 푆, so 푓푛 ∈ 픭+⟨ℎ⟩ for some 푛. Similarly, 푓푚 ∈ 픭+⟨푔⟩ for some 푚. But then 푓푛+푚 ∈ 픭 + ⟨푔ℎ⟩, so 푔ℎ ∉ 픭. Hence 픭 is prime and 푓 ∉ 픭. Thus, if 푓 is contained in every prime ideal of 푅, 푓 must be nilpotent.

Corollary 1.3.18. If 퐼 is an ideal of 푅, then 풱(퐼) = Spec 푅 if and only if 퐼 is contained in the nilradical of 푅.

Definition 1.3.19. Let 퐼 be an ideal of a ring 푅. The radical of 퐼 is the ideal of 푅 defined below: √퐼 = {푓 ∈ 푅 | ∃푛 ∈ ℕ. 푓푛 ∈ 퐼}

8 3. Prime and maximal ideals

Clearly, 퐼 ⊆ √퐼 and √퐼 is radical; in fact, √퐼 is the smallest radical ideal containing 퐼, and √퐼 is the inverse image in 푅 of the nilradical of 푅 ∕ 퐼.

Lemma 1.3.20. If 퐼 is an ideal of a ring 푅, then

√퐼 = 픭 ⋂ 푝∈풱(퐼)

Proof. We know that Spec 푅∕퐼 embeds in Spec 푅 as 풱(퐼) and √퐼 is the inverse image of the nilradical of 푅 ∕ 퐼, so we are done by corollary 1.3.18.

Corollary 1.3.21. If 퐼 and 퐽 are two ideals of 푅, then 풱(퐼) = 풱(퐽) if and only if √퐼 = √퐽. We have a bijection between radical ideals of 푅 and closed subsets of Spec 푅: given a closed subset 푆, the corresponding radical ideal is ⋂픭∈푆 픭; given a radical ideal 퐼, the corresponding closed subset is 풱(퐼).

Example 1.3.22. For 푛 ≠ 0, the closed subset of Spec ℤ corresponding to ⟨푛⟩ is precisely the set {⟨푝⟩ | 푝 is prime and 푝 divides 푛}. For example, 풱(⟨12⟩) is {⟨2⟩, ⟨3⟩}, which is the same as 풱(⟨6⟩).

Definition 1.3.23. Let 퐼 and 퐽 be two ideals of 푅. The product of 퐼 and 퐽 is the ideal of 푅 defined below:

퐼퐽 = {푎푏 | 푎 ∈ 퐼, 푏 ∈ 퐽}

In particular, for an ideal 퐼 and a positive integer 푛,

푛 | 퐼 = 퐼 ⋯ 퐼 = {푓1 ⋯ 푓푛 | 푓1 ∈ 퐼, … , 푓푛 ∈ 퐼}

By convention, we define 퐼0 = 푅.

Clearly, 퐼퐽 ⊆ 퐼 ∩ 퐽. In some examples, 퐼퐽 = 퐼 ∩ 퐽, but this need not be:

⟨4⟩⟨6⟩ = ⟨24⟩ ⊊ ⟨12⟩ = ⟨4⟩ ∩ ⟨6⟩ ⟨2⟩⟨3⟩ = ⟨6⟩ = ⟨2⟩ ∩ ⟨3⟩

Exercise 1.3.24. Show that √퐼퐽 = √퐼 ∩ 퐽.

We previously showed that 풱(퐼) ∪ 풱(퐽) = 풱(퐼 ∩ 퐽), so 풱(퐼) ∪ 풱(퐽) = 풱(퐼퐽). If we know the generators of 퐼 and 퐽, then there is a simple description of the generators of 퐼퐽, but it is not clear how to describe the generators of 퐼 ∩ 퐽.

9 I. Introduction

Theorem 1.3.25. Let 푓 ∶ 퐴 → 퐵 be a ring homomorphism. The induced map Spec 푓 ∶ Spec 퐵 → Spec 퐴 defined by

(Spec 푓)(픭) = 푓−1픭 is continuous. Moreover, if 푓 is surjective, then Spec 푓 is injective and is a homeomorphism onto its image.

Proof. For convenience, let 푔 = Spec 푓. Let 퐼 be an ideal of 퐴. We need to show that 푔−1풱(퐼) = 풱(퐽) for some ideal 퐽 of 퐵. Let 퐽 be the extension of 퐼 in 퐵. We wish to show that a prime 픮 of 퐵 contains 퐽 if and only if 푔(픮) contains 퐼, i.e.

퐽 ⊆ 픮 ⟺ 퐼 ⊆ 푓−1픮 but 퐽 ⊆ 픮 if and only if 픮 contains the image of 퐼. Hence, 푔 is indeed continuous. If 푓 is surjective, then 푔 is injective, since different ideals of 퐵 will pull back to different ideals of 퐴. Under this hypothesis, it is easy to check that the image of a closed subset of Spec 퐵 is closed in Spec 퐴, so we are done.

Definition 1.3.26. An affine scheme is a topological space 푋 and a ring 퐴 such that 푋 is homeomorphic to Spec 퐴. The ring of regular functions on 푋 is denoted by 풪(푋) and is defined to be (any ring isomorphic to) 퐴.

Example 1.3.27. For any field 푘, Spec 푘 is just a point; but Spec 푘 as an affine scheme remembers 푘.

Definition 1.3.28. Let 푅 be a ring and let 푛 be a positive integer. Affine 푛-space 푛 over 푅 is denoted by 픸푅 and is the affine scheme Spec 푅[푥1, … , 푥푛]. Recall that a continuous map 푓 ∶ 푋 → 푌 induces a ring homomorphism (−) ∘ 푓 ∶ 퐶(푌) → 퐶(푋), where 퐶(푌) denotes the ring of continuous functions 푌 → ℝ. By analogy:

Definition 1.3.29. A morphism of affine schemes 푓 ∶ 푋 → 푌 is a ring homomorphism 푓♯ ∶ 풪(푌) → 풪(푋).

Lemma 1.3.30. Let 푅 be an integral domain. The closure of {⟨0⟩} in Spec 푅 is Spec 푅 itself, i.e. ⟨0⟩ is the generic point of Spec 푅.

Proof. Let 퐼 be an ideal of 푅 and suppose ⟨0⟩ ∈ 풱(퐼). Then 퐼 ⊆ ⟨0⟩, so 퐼 = ⟨0⟩, so 풱(퐼) = Spec 푅.

10 3. Prime and maximal ideals

Corollary 1.3.31. For any ring 푅 and any prime 픭 of 푅, the closure of {픭} is 풱(픭).

Proof. We know that Spec 푅 ∕ 픭 is homeomorphic to 풱(픭) by theorem 1.3.25, so the claim follows from the lemma above.

Definition 1.3.32. A connected topological space is a topological space 푋 such that 푋 is never the union of two non-empty disjoint closed subsets. An irreducible topological space is a topological space 푋 such that 푋 is never the union of two non-empty closed proper subsets.

Lemma 1.3.33. For any ring 푅, there is a commutative triangle of bijections between the following sets:

(i) The set of prime ideals of 푅.

(ii) The set of points of Spec 푅.

(iii) The set of irreducible closed subsets of Spec 푅.

Proof. (i) ⇒ (ii). This is by definition. (ii) ⇒ (iii). Given a prime 픭 of 푅, {픭} is irreducible: if 풱(픭) = 풱(픞) ∪ 풱(픟), then either 픭 ∈ 풱(픞) or 픭 ∈ 풱(픟); but 풱(픭) = {픭}, so either 풱(픭) = 풱(픞) or 풱(픭) = 풱(픟). (iii) ⇒ (i). Suppose 풱(픞) is irreducible and 픞 = √픞. The ideal 픞 must be prime: if 푎 ∉ 픞 and 푏 ∉ 픞 then 풱(픞 + ⟨푎⟩) ⊊ 풱(픞) and 풱(픞 + ⟨푏⟩) ⊊ 풱(픞) since 픞 is radical; but 풱(픞 + ⟨푎푏⟩) = 풱(픞 + ⟨푎⟩) ∪ 풱(픞 + ⟨푏⟩), so 푎푏 ∉ 픞. Bijectivity follows from corollary 1.3.21 Exercise 1.3.34. Show that the above bijection restricts to a bijection between the closed points of Spec 푅 and the maximal ideals of 푅.

II Modules

1 Exactness

Definition 2.1.1. Let {푀훼} be a set of (left) 푅-modules. The direct product of {푀훼} is the set ∏훼 푀훼 equipped with the evident (left) 푅-module structure. The direct sum of {푀훼} is the (left) 푅-submodule ⨁훼 푀훼 consisting of the elements of ∏훼 푀훼 with only finitely many non-zero components. Definition 2.1.2. A free (left) 푅-module is a (left) 푅-module isomorphic to the direct sum of a set of copies of 푅.

If 푆 is a set, we write 푅⊕푆 for the direct sum of 푆-many copies of 푅. It contains one copy of 푅 for each element of 푆, so every element of 푅⊕푆 is a finite 푅-linear combination of elements of 푆, regarded as a basis of 푅⊕푆. The free module 푅⊕푆 has the following universal property: if 푀 is any (left) 푅-module, there is a natural bijection between the set of all 푅-linear maps 푅⊕푆 → 푀 and the set of all maps 푆 → 푀.

Definition 2.1.3. An exact sequence of 푅-linear maps is a sequence of the form

휕푖+1 휕푖 ⋯ 푀푖+1 푀푖 푀푖−1 ⋯

where each 휕푖 ∶ 푀푖 → 푀푖−1 is a homomorphism of (left) 푅-modules such that

im 휕푖+1 = ker 휕푖

for all integers 푖 where this makes sense.

Examples 2.1.4. Let 푓 ∶ 푀 → 푁 be a homomorphism of (left) 푅-modules.

13 II. Modules

푓 • The sequence 0 → 푀 → 푁 is exact if and only if 푓 is injective.

푓 • The sequence 푀 → 푁 → 0 is exact if and only if 푓 is surjective.

푓 • The sequence 0 → 푀 → 푁 → 0 is exact if and only if 푓 is an isomorphism.

Definition 2.1.5. A short exact sequence if an exact sequence of the form below: 0 푀 푁 푃 0

Note that this means 푀 is isomorphic to a submodule of 푁 and 푃 is isomorphic to the quotient module 푁 ∕ 푀.

Definition 2.1.6. A (left) 푅-module 푀 is generated by a subset 푆 if 푀 is the smallest submodule of 푀 containing 푆.A finitely-generated (left) 푅-module is a left 푅-module which can be generated by a finite subset.

Equivalently, a (left) 푅-module 푀 is generated by a set 푆 if and only there ⊕푆 ⊕푆 is a surjection 푅 ↠ 푀. Let 퐾 = ker (푅 ↠ 푀). If 퐾 is generated by a set 푇, then we have an exact sequence of the form below:

푅⊕푇 푅⊕푆 푀 0

Such an exact sequence is called a presentation of 푀. Note that every (left) 푅-module has a presentation: take 푆 = 푀 and 푇 = 퐾.

Example 2.1.7. The ℤ-module generated by the set {푒1, 푒2} subject to the relation 2푒1 − 2푒2 = 0 is isomorphic to ℤ ⊕ (ℤ ∕ 2ℤ).

Definition 2.1.8. A projective (left) 푅-module is a (left) 푅-module 푃 such that there is a (left) 푅-module 푁 with 푁 ⊕ 푃 free.

Example 2.1.9. Any free (left) 푅-module is certainly projective.

Lemma 2.1.10. Let 푃 be a (left) 푅-module. The following are equivalent:

(i) 푃 is a projective (left) 푅-module.

14 2. Direct limits

(ii) Any short exact sequence of the form

0 푀 푁 푃 0

splits, i.e. there is an 푅-linear map 푃 → 푁 such that the composite homomorphism 푃 → 푁 → 푃 is the identity map. (In this case, 푁 ≅ 푀 ⊕ 푃.)

(iii) For any surjective homomorphism of (left) 푅-modules 푓 ∶ 퐴 ↠ 퐵 and any 푅-linear map ℎ ∶ 푃 → 퐵, there is at least one 푅-linear map 푔 ∶ 푃 → 퐴 such that ℎ = 푓 ∘ 푔.

Proof. (iii) ⇒ (ii). Take 퐴 = 푁, 퐵 = 푃, and 푔 = id. (ii) ⇒ (i). Let 푆 be a set of generators for 푃, so that we have a short exact sequence: 0 퐾 푅⊕푆 푃 0

Since this sequence splits, we have 푅⊕푆 ≅ 퐾 ⊕ 푃 as required. (i) ⇒ (iii). Suppose 푁 ⊕ 푃 ≅ 푅⊕푆. We have a projection 푝 ∶ 푅⊕푆 ↠ 푃 which splits the inclusion 푖 ∶ 푃 ↪ 푅⊕푆. For each 푠 in 푆, pick an 푎 in 퐴 such that 푓(푎) = ℎ(푝(푠)); this defines a 푅-linear map 푔′ ∶ 푅⊕푆 → 퐴 such that ℎ∘푝 = 푓∘푔′; by composing with 푖, we get the required 푔.

Example 2.1.11. The ℤ-module ℤ ∕ 2ℤ is not projective, since the canonical sequence

0 2ℤ ℤ ℤ ∕ 2ℤ 0

does not split.

We may think of a finitely-generated projective 푅-module over a commutative noetherian ring 푅 as a vector bundle over Spec 푅.

Exercise 2.1.12. Show that a finitely-generated projective 푅-module is a direct summand of 푅⊕푛 for a natural number 푛.

2 Direct limits

Definition 2.2.1. A directed set is a poset 푆 with the following property: for any 푎 and 푏 in 푆, there is a 푐 in 푆 such that 푎 ≤ 푐 and 푏 ≤ 푐.

15 II. Modules

Definition 2.2.2. A directed system of sets comprises the following data:

• a directed system 푆, | • a family of sets (퐴푠 | 푠 ∈ 푆), and

• a map 푓푠,푡 ∶ 퐴푠 → 퐴푡 whenever 푠 ≤ 푡 in 푆, satisfying the following axioms:

• the map 푓푠,푠 is the identity id ∶ 퐴푠 → 퐴푠, and

• when 푠 ≤ 푡 ≤ 푢, the composite 푓푠,푡 ∘ 푓푡,푢 is equal to 푓푠,푢. In other words, it is a functor 푆 → Set, when 푆 is regarded as category.

Definition 2.2.3. The direct limit of a directed system of sets (푆, 퐴•, 푓•,•) is the set ∐푠∈푆 퐴푠/∼, where ∼ is the equivalence relation such that 푎푠 ∼ 푎푡 if and only if 푎푠 ∈ 퐴푠, 푎푡 ∈ 퐴푡, and 푓푠,푢(푎푠) = 푓푡,푢(푎푡) for some 푢 in 푆 with 푠 ≤ 푢 and 푡 ≤ 푢. We write lim 퐴 for the direct limit of this system. −−→푠∈푆 푠 We can use the same definition for directed systems of groups, rings, modules, etc. For example, if (푆, 퐴•, 푓•,•) is a directed system of (left) 푅-modules, then 푎 + 푎 = 푓 푎 + 푓 푎 in lim 퐴 , where 푎 ∈ 퐴 and 푎 ∈ 퐴 , [ 푠] [ 푡] [ 푠,푢( 푠) 푡,푢( 푡)] −−→푠∈푆 푠 푠 푠 푡 푡 where 푠 ≤ 푢 and 푡 ≤ 푢. This is well-defined by the transitivity condition on 푓•,•. Exercise 2.2.4. Show that the direct limit has the following universal property: for a directed system (푆, 퐴•, 푓•) of (left) 푅-modules, there is an isomorphism

Hom lim 퐴 , 푀 푅(−−→푠∈푆 푠 ) Hom | for all ≅ {(푔푠) ∈ ∏푠∈푆 푅(퐴푠, 푀) | 푔푠 = 푔푡 ∘ 푓푠,푡 푠 ≤ 푡} which is natural in 푀. Example 2.2.5. The direct limit of the system

2 ⋅ − 2 ⋅ − ℤ ℤ ℤ ⋯ is ℤ⟨2⟩ = ℤ[1 ∕ 2], because it is isomorphic to the directed system of inclusions

1 1 ℤ 2 ℤ 4 ℤ ⋯ Example 2.2.6. The direct limit of the system below is the zero object:

0 0 ℤ ℤ ℤ ⋯

16 3. Tensor products

3 Tensor products

Definition 2.3.1. Let 푅 be a ring, not necessarily commutative. Let 푀 be a right 푅-module, let 푁 be a left 푅-module, and let 푃 be an abelian group. An 푅-balanced map is a map 푓 ∶ 푀 × 푁 → 푃 such that the following hold:

′ ′ 푓(푚 + 푚 , 푛) = 푓(푚, 푛) + 푓(푚 , 푛) ′ ′ 푓(푚, 푛 + 푛 ) = 푓(푚, 푛) + 푓(푚, 푛 ) 푓(푚 ⋅ 푟, 푛) = 푓(푚, 푟 ⋅ 푛)

If 푅 is commutative and 푃 is also an 푅-module, then a 푅-bilinear map is a 푅-balanced map 푓 ∶ 푀 × 푁 → 푃 which further satisfies the equation below:

푓(푚 ⋅ 푟, 푛) = 푟 ⋅ 푓(푚, 푛) = 푓(푚, 푟 ⋅ 푛)

Theorem 2.3.2. For any right 푅-module 푀 and left 푅-module 푁, there is an

abelian group 푀 ⊗푅 푁 and an 푅-balanced map ℎ ∶ 푀 × 푁 → 푀 ⊗푅 푁 with the following universal property: for any 푅-balanced map 푓 ∶ 푀 × 푁 → 푃,

there is a unique homomorphism of abelian groups 푔 ∶ 푀 ⊗푅 푁 → 푃 such that 푓 = 푔∘ℎ. The tensor product of 푀 and 푁 over 푅 is any abelian group 푀⊗푅 푁 with this universal property.

Moreover, if 푅 is commutative, then 푀 ⊗푅 푁 is naturally an 푅-module, the map ℎ ∶ 푀 × 푁 → 푀 ⊗푅 푁 above is 푅-bilinear, and for every 푅-bilinear map 푓 ∶ 푀 × 푁 → 푃, there is a unique 푅-linear map 푔 ∶ 푀 ⊗푅 푁 → 푃 such that 푓 = 푔 ∘ ℎ.

Proof. Consider the free abelian group 퐹 generated by symbols of the form 푚⊗푛

for 푚 ∈ 푀 and 푛 ∈ 푁. The tensor product 푀 ⊗푅 푁 is the quotient of 퐹 by the submodule generated by elements of the following forms:

푚 ⊗ (푟 ⋅ 푛) − (푚 ⋅ 푟) ⊗ 푚 for 푟 ∈ 푅, 푚 ∈ 푀, 푛 ∈ 푁 ′ ′ ′ (푚 + 푚 ) ⊗ 푛 − 푚 ⊗ 푛 − 푚 ⊗ 푛 for 푚 ∈ 푀, 푚 ∈ 푀, 푛 ∈ 푁 ′ ′ ′ 푚 ⊗ (푛 + 푛 ) − 푚 ⊗ 푛 − 푚 ⊗ 푛 for 푚 ∈ 푀, 푛 ∈ 푁, 푛 ∈ 푁

Clearly, the map (푚, 푛) ↦ 푚 ⊗ 푛 is 푅-balanced, and given a 푅-balanced map

푓 ∶ 푀 × 푁 → 푃, we define the homomorphism 푔 ∶ 푀 ⊗푅 푁 → 푃 by setting 푔(푚 ⊗ 푛) = 푓(푚, 푛). This is well-defined since 푓 is 푅-balanced and 푀 ⊗푅 푁 is generated by {푚 ⊗ 푛 | 푚 ∈ 푀, 푛 ∈ 푁}, and clearly 푓 = 푔 ∘ ℎ.

17 II. Modules

If 푅 is commutative, then there is an action of 푅 on 푀 ⊗푅 푁 as follows:

푟 ⋅ (푚 ⊗ 푛) = 푚 ⊗ (푟 ⋅ 푛)

This makes the map ℎ above 푅-bilinear, and the rest is easy to check.

By construction, any element of 푀 ⊗푅 푁 is a finite ℤ-linear combination of elements of the form 푚 ⊗ 푛 for some 푚 in 푀 and 푛 in 푁. Elements of the form

푚 ⊗ 푛 are called decomposable, but in general not every element 푀 ⊗푅 푁 is decomposable. ′ ′ It can be hard to tell whether 푚 ⊗ 푛 = 푚 ⊗ 푛 in 푀 ⊗푅 푁. For example, ℚ ⊗ℤ (ℤ ∕ 2ℤ) = 0, since e.g. 1 1 1 1 ⊗ 1 = ⋅ 2 ⊗ 1 = ⊗ (2 ⋅ 1) = ⊗ 0 = 0 (2 ) 2 2

If 푅 is non-commutative, 푀 ⊗푅 푁 is in general not an 푅-module. However, if 퐴 is the centre of 푅 (i.e. the subring {푟 ∈ 푅 | ∀푠 ∈ 푅. 푟푠 = 푠푟}), then 푀 ⊗푅 푁 is naturally an 퐴-module.

Exercise 2.3.3. Show that 푀 ⊗푅 푁 is functorial in both 푀 and 푁.

Theorem 2.3.4. Let 푅 be a commutative ring, and let 퐴, 퐵, and 퐶 be 푅-modules. We have the following natural isomorphisms:

퐴 ⊗푅 퐵 ≅ 퐵 ⊗푅 퐴

(퐴 ⊗푅 퐵) ⊗푅 퐶 ≅ 퐴 ⊗푅 (퐵 ⊗푅 퐶)

(퐴 ⊕ 퐵) ⊗푅 퐶 ≅ (퐴 ⊗푅 퐶) ⊕ (퐵 ⊗푅 퐶)

푅 ⊗푅 퐴 ≅ 퐴

0 ⊗푅 퐴 ≅ 0

Proof. Use the universal property of the tensor product and these maps:

푎 ⊗ 푏 ↦ 푏 ⊗ 푎 (푎 ⊗ 푏) ⊗ 푐 ↦ 푎 ⊗ (푏 ⊗ 푐) (푎, 푏) ⊗ 푐 ↦ (푎 ⊗ 푐, 푏 ⊗ 푐) 푟 ⊗ 푎 ↦ 푟푎 푟 ⊗ 0 ↦ 0

18 3. Tensor products

⊕푆 ⊕푇 ⊕(푆×푇) Corollary 2.3.5. If 푆 and 푇 are sets, then 푅 ⊗푅 푅 ≅ 푅 .

| ⊕푆 | ⊕푇 Proof. Let {푒푠 | 푠 ∈ 푆} be a basis of 푅 and {푓푡 | 푡 ∈ 푇} be a basis of 푅 . It ⊕푆 ⊕푇 | is easy to check that 푅 ⊗푅 푅 is freely generated by {푒푠 ⊗ 푓푡 | 푠 ∈ 푆, 푡 ∈ 푇}.

Lemma 2.3.6. Let 푀 be a right 푅-module. If the sequence

퐴 퐵 퐶 0 is a right exact sequence of left 푅-modules, then the tensored sequence

푀 ⊗푅 퐴 푀 ⊗푅 퐵 푀 ⊗푅 퐶 0 is a right exact sequence of abelian groups.

Proof. By considering decomposable elements, we see that 푀⊗푅 퐵 → 푀⊗푅 퐶 is surjective. It is clear that the composite 푀 ⊗푅 퐴 → 푀 ⊗푅 퐵 → 푀 ⊗푅 퐶 is the zero homomorphism. It remains to be shown that

im (푀 ⊗푅 퐴 → 푀 ⊗푅 퐵) = ker (푀 ⊗푅 퐵 → 푀 ⊗푅 퐶) but this follows readily from the universal property of 퐶 as a cokernel: indeed, ′ given 푀⊗푅 퐵 → 퐶 such that the composite with 푀⊗푅 퐴 → 푀⊗푅 퐵 vanishes, we obtain a unique factorisation through 푀 ⊗푅 퐵 → 푀 ⊗푅 퐶, i.e. 푀 ⊗푅 퐶 is the cokernel of 푀 ⊗푅 퐴 → 푀 ⊗푅 퐵 as required.

Example 2.3.7. Let 푓 ∈ 푅. We have an exact sequence

− ⋅ 푓 푅 푅 푅 ∕ ⟨푓⟩ 0 of left 푅-modules, so by tensoring with 푀, we have a natural isomorphism:

푀/(푀푓) ≅ 푀 ⊗푅 (푅/⟨푓⟩)

This can be used to understand tensor products of finitely-generated modules over principal ideal domains, via the classification of such modules.

19 II. Modules

More generally, the lemma implies that, for any right 푅-module 푀 generated by {푒1, … , 푒푚} and any left 푅-module 푁 generated by {푓1, … , 푓푛}, the abelian group 푀 ⊗푅 푁 is generated by {푒푖 ⊗ 푓푗} subject to the following relations:

푒푖 ⊗ 푠 = 0 for each relation 푠 of 푁

푟 ⊗ 푓푗 = 0 for each relation 푟 of 푀

We have shown that, for any right 푅-module 푀, the functor 푀 ⊗푅 (−) preserves right exact sequences of left 푅-modules. However, it does not in general preserve exact sequences.

2 0 Example 2.3.8. The exact sequence 0 → ℤ → ℤ becomes 0 → ℤ ∕ 2ℤ → ℤ ∕ 2ℤ, which is not exact.

Definition 2.3.9. A right 푅-module 푀 is flat just if 푀⊗푅 (−) preserves all exact sequences. Similarly, a left 푅-module 푁 is flat just if (−) ⊗푅 푁 preserves all exact sequences.

Clearly, ℤ ∕ 2ℤ is not a flat ℤ-module.

Example 2.3.10. For any ring 푅, 푅 ⊗푅 (−) is isomorphic to the identity functor, so 푅 is a flat right 푅-module. It is also a flat left 푅-module.

The direct sum of arbitrarily many flat modules is flat, since (−) ⊗푅 푁 preserves direct sums, and the direct sum of arbitrarily many exact sequences is exact.[1] Hence, every free module is flat.

Theorem 2.3.11. Let 푅 be a ring, and let 푀 be a right 푅-module. The following are equivalent:

(i) 푀 is flat.

(ii) 푀 ⊗푅 (−) preserves injections.

(iii) For any left ideal 퐼 of 푅, the induced homomorphism 푀 ⊗푅 퐼 → 푀 ⊗푅 푅 is injective.

[1] Caution: This does not hold in general abelian categories! However, an abelian category satisfying axiom (AB4) will have this property.

20 3. Tensor products

Proof. (i) ⇒ (ii). Injectivity can be detected by exactness: see examples 2.1.4. (ii) ⇒ (iii). Clear. (ii) ⇒ (i). Given an exact sequence of left 푅-modules

푓1 푓2 푁1 푁2 푁3 we obtain a right exact sequence

푓1 푁1 푁2 im (푓2) 0 and so the sequence of abelian groups

푀 ⊗푅 푁1 푀 ⊗푅 푁2 푀 ⊗푅 im (푓2) 0 is also right exact; but 푀 ⊗푅 im (푓2) → 푀 ⊗푅 푁3 remains injective, so the sequence

푀 ⊗푅 푁1 푀 ⊗푅 푁2 푀 ⊗푅 푁3 is an exact sequence of abelian groups, as required. 푅 (iii) ⇒ (ii). We will use Tor1 : see definition 4.2.6. Suppose

0 퐼 푅 푅 ∕ 퐼 0 is a short exact sequence of left 푅-modules. Then, there is a long exact sequence

푅 푅 ⋯ Tor1 (푀, 푅) Tor1 (푀, 푅 ∕ 퐼) 푀 ⊗푅 퐼 푀 ⊗푅 푅 ⋯

푅 of abelian groups, but 푅 is flat, so 1Tor (푀, 푅) = 0, and 푀 ⊗푅 퐼 → 푀 ⊗푅 푅 is 푅 injective, so Tor1 (푀, 푅 ∕ 퐼) = 0 as well. Now, let 푁 be a finitely-generated left 푅-module, say 푁 = 푅⟨푥1, … , 푥푟⟩. Define submodules 푁푗 = 푅⟨푥1, … , 푥푗⟩; then we have a filtration

0 = 푁0 ⊆ 푁1 ⊆ 푁2 ⊆ ⋯ ⊆ 푁푟 = 푁 with 푁푗/푁푗−1 ≅ 푅/퐼푗 for some left ideals 퐼1, … , 퐼푟 of 푅. Thus, we have short exact sequences of left 푅-modules

0 푁푗−1 푁푗 푅/퐼푗 0

21 II. Modules

푅 and since Tor1 (푀, 푅/퐼푗 ) = 0, we get a right exact sequence of abelian groups

푅 푅 푅 Tor2 (푀, 푅/퐼푗 ) Tor1 (푀, 푁푗−1) Tor1 (푀, 푁푗) 0

푅 푅 If Tor1 (푀, 푁푗−1) = 0, then Tor1 (푀, 푁푗) = 0 as well. Thus, by induction, 푅 Tor1 (푀, 푁) = 0 for all finitely-generated left 푅-modules 푁. Finally, let 푁 be any left 푅-module. 푁 is the direct limit of all its finitely-generated submodules, 푅 [2] 푅 and Tor1 (푀, −) commutes with direct limits, so Tor1 (푀, 푁) = 0. Given an injective 푅-linear map 퐴 ↣ 퐵, we may form a short exact sequence

0 퐴 퐵 퐶 0

and so there is a long exact sequence of abelian groups

푅 Tor1 (푀, 퐶) 푀 ⊗푅 퐴 푀 ⊗푅 퐵 푀 ⊗푅 퐶 0

푅 but Tor1 (푀, 퐶) = 0, so 푀 ⊗푅 퐴 → 푀 ⊗푅 퐵 is injective as required. Exercise 2.3.12. Let 푅 be an integral domain.

(i) Show that any flat 푅-module is torsion free, i.e. for 푟 ∈ 푅 and 푚 ∈ 푀, 푟 ⋅ 푚 = 0 if and only if 푟 = 0 or 푚 = 0.

Now let 푅 be a principal ideal domain.

(ii) Show that any torsion-free 푅-module is flat.

4 Algebras

Definition 2.4.1. Let 퐵 be a commutative ring. A 퐵-algebra is a ring 퐴 together with a ring homomorphism 퐵 → Z(퐴), where Z(퐴) is the centre of 퐴.A commutative 퐵-algebra is a 퐵-algebra whose underlying ring is commutative. Let 퐴 and 퐴′ be 퐵-algebras. A homomorphism of 퐵-algebras is a ring homomorphism 퐴 → 퐴′ which commutes with the structural homomorphisms 퐵 → Z(퐴) and 퐵 → Z(퐴′).

Examples 2.4.2. Let 푘 be a field. [2] See question 9 on examples sheet 2

22 4. Algebras

• The matrix ring M푛(푘) is a non-commutative 푘-algebra.

• The group algebra 푘퐺 is a non-commutative 푘-algebra.

Proposition 2.4.3. The polynomial ring 퐵[푥1, … , 푥푛] has the following universal property: for any commutative 퐵-algebra 퐴, there is a natural bijection between the set of 퐵-algebra homomorphisms 퐵[푥1, … , 푥푛] → 퐴 and the set of maps {푥1, … , 푥푛} → 퐴.

Proof. Easy.

Henceforth, 퐵-algebras will be commutative unless otherwise stated.

Definition 2.4.4. Let 퐵 be a commutative ring. A 퐵-algebra of finite type is a 퐵-algebra 퐴 such that there is a natural number 푛 and a surjective 퐵-algebra homomorphism 퐵[푥1, … , 푥푛] → 퐴. Let 푋 and 푌 be affine schemes. A morphism of schemes 푓 ∶ 푋 → 푌 is of finite type just if the corresponding homomorphism of rings 푓♯ ∶ 풪(푌) → 풪(푋) makes 풪(푋) into a 풪(푌)-algebra of finite type.

Notice that if 푓 ∶ 푋 → 푌 is of finite type, then 푋 is isomorphic to a closed 푛 subscheme of 픸푌 for some natural number 푛.

Definition 2.4.5. Let 푘 be a field. An affine variety over 푘 is an affine scheme 푋 and a morphism of schemes 푋 → Spec 푘 which is of finite type, such that 풪(푋) is an integral domain.

Examples 2.4.6. 푛 • Affine 푘-space, 픸푘, is an affine variety over 푘.

• The vanishing of a polynomial irreducible over 푘 is also an affine variety over 푘.

If 퐴 is a 퐵-algebra, then there is a functor taking 퐴-modules to 퐵-modules: given the structural homomorphism 푓 ∶ 퐵 → 퐴 and an 퐴-module 푀, we make 푀 into a 퐵-module by setting 푏 ⋅ 푚 = 푓(푏) ⋅ 푚. However, there is also a functor taking 퐵-modules to 퐴-modules: given a 퐵-module 푁, we make 퐴 ⊗퐵 푁 into an 퐴-module by setting 푟 ⋅ (푎 ⊗ 푛) = (푟푎) ⊗ 푛.

23 II. Modules

⊕푛 ⊕푛 ⊕푛 Example 2.4.7. If 푀 = 퐵 , then 퐴 ⊗퐵 푀 ≅ 퐴 . More generally, given a presentation of 푀, then we can obtain a presentation of 퐴 ⊗퐵 푀: indeed, if

퐵⊕푇 퐵⊕푆 푀 0 is an exact sequence of 퐵-modules, then the tensored sequence

⊕푇 ⊕푆 퐴 퐴 퐴 ⊗퐵 푀 0 is an exact sequence of 퐴-modules.

Example 2.4.8. The abelian group 퐺 = ℤ⊕(ℤ ∕ 2ℤ) has two generators 푒1 and 푒2 and a single relation 2푒1 = 2푒2, so we obtain

ℚ ⊗ℤ 퐺 ≅ ℚ⟨푒1, 푒2⟩ ∕ ℚ⟨2푒1 − 2푒2⟩ ≅ ℚ while on the other hand we have

⊕2 픽2 ⊗ℤ 퐺 ≅ 픽2⟨푒1, 푒2⟩ ∕ 픽2⟨2푒1 − 2푒2⟩ ≅ 픽2

Lemma 2.4.9. If 퐴 and 퐶 are 퐵-algebras, then 퐴⊗퐵 퐶 is naturally a 퐵-algebra.

Proof. Easy: set (푎1 ⊗ 푐1) × (푎2 ⊗ 푐2) = (푎1 × 푎2) ⊗ (푐1 × 푐2) and extend bilinearly.

24 III Local algebra

In this chapter, all rings are commutative.

1 Localisation of rings

We may think of ℂ[푥] as a ring of functions ℂ → ℂ. The field of rational functions, ℂ(푥), is then a ring of partial functions ℂ ⇁ ℂ, defined on cofinite

subsets of ℂ. A typical localisation of ℂ[푥] is ℂ[푥]⟨푥⟩, which is the subring of ℂ(푥) comprising all the partial functions which are defined at 0.

Definition 3.1.1. A multiplicatively closed subset of a ring 푅 is a subset 푆 which is a submonoid of 푅 under multiplication.

Theorem 3.1.2. Let 푅 be a ring and let 푆 be a multiplicatively closed subset of −1 −1 푅. There exists a ring 푅[푆 ] and a ring homomorphism ℎ ∶ 푅 → 푅[푆 ] with the following properties:

• If 푠 is in 푆, then ℎ(푠) is invertible.

• For all rings 퐵 and all ring homomorphisms 푓 ∶ 푅 → 퐵 such that 푓(푠) is −1 invertible for all 푠 in 푆, there is a unique homomorphism 푔 ∶ 푅[푆 ] → 퐵 such that 푓 = 푔 ∘ ℎ.

−1 −1 The ring 푅[푆 ] and homomorphism ℎ ∶ 푅 → 푅[푆 ] are moreover unique up to unique isomorphism.

−1 Proof. Uniqueness of 푅[푆 ] follows from a standard argument.

25 III. Local algebra

−1 The ring 푅[푆 ] is defined to be the set of symbols of the form 푟 ∕ 푠, where 푟 ∈ 푅 and 푠 ∈ 푆, modulo the following equivalence relation:

푟1/푠1 = 푟2/푠2 just if 푢 (푠2푟1 − 푠1푟2) = 0 for some 푢 in 푆 This is indeed an equivalence relation: reflexivity and symmetry are clear, and if 푟1/푠1 = 푟2/푠2 and 푟2/푠2 = 푟3/푠3 , then there are 푢 and 푣 in 푆 such that

푢 (푠2푟1 − 푠1푟2) = 0 푣 (푠3푟2 − 푠2푟3) = 0 from which it follows that

푢푣푠2 (푠3푟1 − 푠1푟3) = 0 −1 but 푢푣푠2 ∈ 푆, so 푟1/푠1 = 푟3/푠3 as required. The algebraic operations on 푅[푆 ] are defined in the obvious way, and the verification of the details isleftasan exercise.

−1 Lemma 3.1.3. For 푟 ∈ 푅, 푟 ∕ 1 = 0 in 푅[푆 ] if and only if 푟푠 = 0 for some some 푠 in 푆. Example 3.1.4. If 푅 is an integral domain, then 푆 = 푅 ⧵ {0} is multiplicatively −1 closed, and 푅[푆 ] = Frac 푅 is a field, called the field of fractions of 푅. Note that the homomorphism 푅 → Frac 푅 is injective in this case. For example, Frac ℤ = ℚ, and Frac 푘[푥1, … , 푥푛] = 푘(푥1, … , 푥푛). More generally, if 푅 is an integral domain and 푆 is any multiplicatively −1 closed subset of 푅 not containing 0, 푅[푆 ] is a subring of Frac 푅. On the other −1 hand, if 푆 contains 0, then 푅[푆 ] is the trivial ring. −1 Definition 3.1.5. If 푅 is a ring and 푓 ∈ 푅, we write 푅[1 ∕ 푓] for 푅[푆 ] where 2 푆 = {1, 푓, 푓 ,…}.A principal localisation of 푅 is a ring of this form. Definition 3.1.6. If 픭 is a prime ideal of 푅, 푆 = 푅 ⧵ 픭 is multiplicatively closed, −1 and we write 푅픭 for 푅[푆 ]. This is the localisation of 푅 at the prime 픭. Example 3.1.7. Let 푝 be a prime number. The principal localisation ℤ[1 ∕ 푝] comprises all rational numbers which can be written with a 푝-power denominator. On the other hand, the localisation of ℤ at ⟨푝⟩ comprises all rational numbers which can be written with a denominator coprime to 푝. The ring of Laurent polynomials over a field 푘 is the ring 푘[푥, 1 ∕ 푥]. This 1 may be viewed as the ring of regular functions on 픸푘 ⧵ {0}. On the other hand, the ring 푘[푥]⟨푥⟩ can be thought of as the ring of germs of functions defined at 0.

26 1. Localisation of rings

Definition 3.1.8. A local ring is a ring 푅 with a unique maximal ideal 픪. The residue field of a local ring 푅 is 푅 ∕ 픪.

Lemma 3.1.9. A ring is a local ring if and only if the set of non-invertible elements is an ideal.

Proof. Let 푅 be a local ring with maximal ideal 픪. If 푎 ∈ 픪, then 푎 is not a unit, since 1 ∉ 픪. Conversely, if 푎 ∉ 픪, then ⟨푎⟩ = 푅, so 푎 must be a unit. Now, let 푅 be a ring, and let 픪 be the set of non-invertible elements. Suppose 픪 is an ideal. Any proper ideal of 푅 contains only non-invertible elements, so must be contained in 픪, hence 픪 is the unique maximal ideal of 푅.

Example 3.1.10. If 푘 is a field, then the power series ring 푘 푥 , … , 푥 is a local J 1 푛K ring with maximal ideal ⟨푥1, … , 푥푛⟩.

Theorem 3.1.11. Let 푅 be a ring, and let 푆 be a multiplicatively closed subset −1 of 푅. The prime ideals of 푅[푆 ] are in natural bijection with the prime ideals −1 of 푅 not meeting 푆, i.e. Spec 푅[푆 ] ≅ {픭 ∈ Spec 푅 | 픭 ∩ 푆 = ∅}

−1 Proof. Certainly, if ℎ ∶ 푅 → 푅[푆 ] is the canonical homomorphism, then ∗ −1 ∗ ℎ ∶ Spec 푅[푆 ] → Spec 푅 is continuous, and ℎ (픓) ∩ 푆 = ∅ because 픓 does −1 not contain any units in 푅[푆 ]. −1 ∗ If 픓 ∈ Spec 푅[푆 ] and 푎∕푠 ∈ 픭, then 푎∕1 ∈ 픓, so 푎 ∈ ℎ (픓); conversely, if 푎 ∈ ℎ∗(픓) then 푎 ∕ 1 ∈ 픓, so 푎 ∕ 푠 ∈ 픓. Thus, ℎ∗ is injective. It remains to be shown that any 픭 in Spec 푅 not meeting 푆 lies under a prime −1 of 푅[푆 ]. Consider the canonical homomorphism 푅 ↠ 푅 ∕ 픭 ↪ Frac (푅 ∕ 픭). −1 By the universal property of 푅[푆 ], there is a unique homomorphism 푔 making the diagram below commute:

ℎ −1 푅 푅[푆 ] 푔

푅 ∕ 픭 Frac (푅 ∕ 픭)

The ideal 픓 = ker ℎ is prime, but ker (ℎ ∘ 푔) = ℎ∗(픓) = 픭 by construction.

Corollary 3.1.12. If 픭 is a prime ideal of a ring 푅, then the localisation 푅픭 is a local ring.

27 III. Local algebra

Proof. By the above theorem, the primes of 푅픭 correspond to the primes contained in 픭; in particular, the extension of 픭 is the unique maximal ideal of

푅픭. Examples 3.1.13. • Any field is a local ring, trivially.

• For a prime number 푝, ℤ⟨푝⟩ is a local ring with residue field 픽푝.

• 푘[푥]⟨푥⟩ is a local ring, with residue field 푘.

Exercise 3.1.14. Describe ℂ[푥, 푦]⟨푥⟩ and its residue field.

−1 In general, 푅[푆 ] need not be a local ring. For example, ℤ[1 ∕ 푝] is not local.

Definition 3.1.15. Let 푅 be a ring. A basic open subset of Spec 푅 is any subset of the form

풟(푓) = Spec 푅 ⧵ 풱(푓) = {픭 ∈ Spec 푅 | 푓 ∉ 픭} for some 푓 in 푅.

Exercise 3.1.16. Show that 풟(푓) is homeomorphic to Spec 푅[1 ∕ 푓]. We can think of Spec ℤ as

Spec ℚ ∪ Spec 픽2 ∪ Spec 픽3 ∪ Spec 픽5 ∪ ⋯

whereas Spec ℤ⟨3⟩ is like Spec ℚ ∪ Spec 픽3. We may think of a ℤ-module as a bundle over Spec ℤ; for example, the ℤ-module ℤ ⊕ (ℤ ∕ 2ℤ) has a two-

dimensional fibre over Spec 픽2 but a one-dimensional fibre elsewhere.

2 Localisation of modules

Definition 3.2.1. Let 푅 be a ring and let 푆 be a multiplicatively closed subset −1 −1 of 푅. The localisation 푀[푆 ] of an 푅-module 푀 is the 푅[푆 ]-module comprising fractions with numerators in 푀 and denominators in 푆, where 푚1/푠1 = 푚2/푠2 just if 푢 (푠2푚1 − 푠1푚2) = 0 for some 푢 in 푆; addition and scalar multiplication are defined in the obvious way. −1 2 As before, we write 푀[1 ∕ 푓] for 푀[푆 ] when 푆 = {1, 푓, 푓 ,…} and 푀픭 −1 for 푀[푆 ] when 푆 = 푅 ⧵ 픭.

28 2. Localisation of modules

−1 Any 푅-linear map 푓 ∶ 푀 → 푁 determines a natural 푅[푆 ]-linear map −1 −1 푓̃ ∶ 푀[푆 ] → 푁[푆 ], namely 푚 ∕ 푠 ↦ 푓(푚) ∕ 푠. It is easy to check that we −1 get a functor Mod(푅) → Mod(푅[푆 ]). Theorem 3.2.2. The localisation functor defined above is exact: given an exact sequence of 푅-modules

푓1 푓2 푀1 푀2 푀3

−1 the induced sequence of 푅[푆 ]-modules is exact:

̃ ̃ −1 푓1 −1 푓2 −1 푀1[푆 ] 푀2[푆 ] 푀3[푆 ]

Proof. The localisation functor is clearly additive, so 푓2̃ ∘ 푓1̃ = 0. It suffices now to show im 푓1̃ ⊇ ker 푓2̃ . Suppose 푓2̃ (푚 ∕ 푠) = 0. Then, 푓2(푚) ∕ 푠 = 0, so 푡푓2(푚) = 0 for some 푡 in 푆. But 푡푓2(푚) = 푓2(푡푚), so 푡푚 = 푓1(푛) for some 푛, and 푓1̃ (푛) ∕ 푡 = 푚 ∕ 1, so 푓1̃ (푛 ∕ (푠푡)) = 푚 ∕ 푠 as required. Definition 3.2.3. A flat 퐵-algebra is a 퐵-algebra which is flat as a 퐵-module.

Theorem 3.2.4. Let 푅 be a ring, let 푆 be a multiplicatively closed subet, and let −1 푀 be an 푅-module. There is a natural isomorphism of 푅[푆 ]-modules:

−1 −1 푅[푆 ] ⊗푅 푀 ≅ 푀[푆 ] Proof. Omitted.

−1 Corollary 3.2.5. The localisation 푅[푆 ] is a flat 푅-algebra. Example 3.2.6. Consider the ℤ-module below:

푀 = ℤ ⊕ ℤ ⊕ (ℤ ∕ 2ℤ) ⊕ (ℤ ∕ 8ℤ) ⊕ (ℤ ∕ 5ℤ)

We have 푀⟨0⟩ ≅ ℚ ⊕ ℚ, while 푀⟨2⟩ ≅ ℤ⟨2⟩ ⊕ ℤ⟨2⟩ ⊕ (ℤ ∕ 2ℤ) ⊕ (ℤ ∕ 8ℤ) and 푀⟨5⟩ ≅ ℤ⟨5⟩ ⊕ ℤ⟨5⟩ ⊕ (ℤ ∕ 5ℤ). For a prime 푝 other than 2 or 5, 푀⟨푝⟩ ≅ ℤ⟨푝⟩ ⊕ ℤ⟨푝⟩. Definition 3.2.7. Let 푃 be a property of rings. We say that 푃 is local if the following rule of inference is valid: a ring 푅 has property 푃 if and only if 푅픭 has property 푃 for all primes 픭 of 푅. We define local properties of 푅-modules in a similar way.

29 III. Local algebra

Definition 3.2.8. Let 푀 be a (left) 푅-module. The annihilator of 푀 is the (left) ideal of 푅 defined below:

Ann푅(푀) = {푟 ∈ 푅 | ∀푚 ∈ 푀. 푟푚 = 0}

If 푚 ∈ 푀, the annihilator of 푚 is the (left) ideal of 푅 defined below:

Ann푅(푚) = {푟 ∈ 푅 | 푟푚 = 0}

This is only a small abuse of notation since Ann푅(푚) = Ann푅(푅⟨푚⟩), where 푅⟨푚⟩ is the 푅-cyclic (left) submodule of 푀 generated by 푚.

Lemma 3.2.9. Let 푅 be a ring and let 푀 be an 푅-module. The following are equivalent:

(i) 푀 = 0.

(ii) 푀픭 = 0 for all prime ideals 픭 of 푅.

(iii) 푀픪 = 0 for all maximal ideals 픪 of 푅.

Proof. (i) ⇒ (ii) ⇒ (iii). Clear. (iii) ⇒ (i). Let 푚 ∈ 푀, and suppose 푚 ≠ 0. Then, there is a maximal ideal

픪 containing Ann푅(푚), so 0 ≠ 푚 ∕ 1 ∈ 푀픪, since 푚 ∕ 1 = 0 if and only if 푠푚 = 0 for some 푠 ∉ 픪, but Ann푅(푚) ⊆ 픪.

Lemma 3.2.10. Let 푓 ∶ 푀 → 푁 be a homomorphism of 푅-modules. The following are equivalent:

(i) The map 푓 ∶ 푀 → 푁 is injective (resp. surjective or bijective).

(ii) The map 푓픭 ∶ 푀픭 → 푁픭 is injective (resp. surjective or bijective) for all prime ideals 픭 of 푅.

(iii) The map 푓 ∶ 푀픪 → 푁픪 is injective (resp. surjective or bijective) for all maximal ideals 픪 of 푅.

Proof. (i) ⇒ (ii). We have an exact sequence of 푅-modules

푓 0 퐾 푀 푁 퐶 0

30 2. Localisation of modules

and 푅픭 is flat over 푅, so we get an exact sequence of 푅픭-modules

푓픭 0 퐾픭 푀픭 푁픭 퐶픭 0 and the claim follows. (ii) ⇒ (iii). Clear. (iii) ⇒ (i). Use the previous lemma and the above exact sequences. Lemma 3.2.11. Flatness is a local property. To be precise, the following are equivalent:

(i) 푀 is a flat 푅-module.

(ii) 푀픭 is a flat 푅픭-module for every prime ideal 픭 of 푅.

(iii) 푀픪 is a flat 푅픪-module for every maximal ideal 픪 of 푅.

Proof. (i) ⇒ (ii) ⇒ (iii). Flatness is preserved by scalar extension: if 푆 is any

푅-algebra and 푀 is a flat 푅-module, then 푆 ⊗푅 푀 is a flat 푆-module; indeed, if 푁 is any 푆-module, we have natural isomorphisms as below:

푁 ⊗푆 (푆 ⊗푅 푀) ≅ (푁 ⊗푆 푆) ⊗푅 푀 ≅ 푁 ⊗푅 푀

(iii) ⇒ (i). Let 퐴 ↣ 퐵 be an injective homomorphism of 푅-modules, and suppose is flat for each . Then, is injective 푀픪 픪 퐴픪 ⊗푅픪 푀픪 → 퐵픪 ⊗푅픪 푀픪 for each 픪, but this is the localisation of 퐴 ⊗푅 푀 → 퐵 ⊗푅 푀 at 픪, so we are done by lemma 3.2.10 and theorem 2.3.11.

Lemma 3.2.12 (Nakayama). Let 푅 be a local ring with maximal ideal 픪 and residue field 푘. If 푀 is a finitely-generated 푅-module and 푘 ⊗푅 푀 = 0, then 푀 = 0.

Proof. Let {푥1, … , 푥푛} generate 푀 as an 푅-module, and suppose 푛 is minimal and positive. If 픪푀 = 푀, then 푥푛 = 푎1푥1 + ⋯ + 푎푛푥푛 for some 푎푖 in 픪. But then (1 − 푎푛) 푥푛 = 푎1푥1 + ⋯ + 푎푛−1푥푛−1 and (1 − 푎푛) is invertible in 푅, so {푥1, … , 푥푛−1} generates 푀 as an 푅-module — a contradiction. So 푛 = 0, and 푀 = 0.

Beware: Nakayama’s lemma can fail if 푀 is not finitely generated. For example, take , : we have but . 푅 = ℤ⟨2⟩ 푀 = ℚ 픽2 ⊗ℤ⟨2⟩ 푀 = 0 푀 ≠ 0

31 III. Local algebra

Corollary 3.2.13. Let 푅 be a local ring with maximal ideal 픪 and residue field 푘. Let 푀 be a finitely-generated 푅-module. A finite subset of 푀 generates 푀 as an 푅-module if and only if its image in 푘⊗푅 푀 spans the 푘-vector space 푘⊗푅 푀.

Proof. The forward direction is trivial. Let {푥1, … , 푥푛} ⊆ 푀, and write 푥푖 for the image of 푥푖 in 푘 ⊗푅 푀. Let 푄 = 푀/푅⟨푥1, … , 푥푛⟩. We have a right exact sequence of finitely-generated 푅-modules as below:

푅⊕푛 푀 푄 0

It is enough to show that 푄 = 0 to conclude that {푥1, … , 푥푛} generates 푀. By tensoring with 푘, we obtain a right exact sequence of 푘-vector spaces:

⊕푛 푘 푘 ⊗푅 푀 푘 ⊗푅 푄 0

It follows from the hypothesis that 푘⊗푅 푄 = 0. Nakayama’s lemma then implies 푄 = 0, as required.

32 IV Homological algebra

1 Chain complexes

Let 푅 be a ring, not necessarily commutative.

Definition 4.1.1. A chain complex 푀• is a sequence of (left) 푅-modules and 푅-linear maps

휕푖+1 휕푖 ⋯ 푀푖+1 푀푖 푀푖−1 ⋯

such that 휕푖 ∘휕푖+1 = 0 for all integers 푖. The homology groups of a chain complex 푀• are the (left) 푅-modules defined below:

퐻푖(푀•) = ker 휕푖/im 휕푖+1

This makes sense because im 휕푖+1 ⊆ ker 휕푖 in a chain complex. Notice that 퐻푖(푀•) = 0 if and only if the sequence is exact at 푀푖.

Definition 4.1.2. Let 푀• and 푁• be two chain complexes. A homomorphism of chain complexes, or chain map, 푓• ∶ 푀• → 푁• is a sequence of 푅-linear maps 푓푖 ∶ 푀푖 → 푁푖 such that 푓푖 ∘ 휕푖 = 휕푖 ∘ 푓푖 for all 푖, i.e. the diagram below commutes for all integers 푖:

휕푖 푀푖 푀푖−1

푓푖 푓푖−1

푁푖 푁푖−1 휕푖

33 IV. Homological algebra

Exercise 4.1.3. Show that chain maps induce (non-trivial!) homomorphisms between the homology groups making 퐻푖(−) into a functor.

Definition 4.1.4. Let 푓•, 푔• ∶ 푀• → 푁• be chain maps. A chain homotopy 훼• ∶ 푓• ⇒ 푔• is a sequence of 푅-linear maps 훼푖 ∶ 푀푖 → 푁푖+1 such that 휕푖+1 ∘ 훼푖 + 훼푖−1 ∘ 휕푖 = 푔푖 − 푓푖 for all integers 푖. We say 푓 and 푔 are homotopic if there is such a chain homotopy. Exercise 4.1.5. Show that two homotopic chain maps induce the same homomorphisms between homology groups.

Definition 4.1.6. Let 푀• and 푁• be a pair of chain maps. A chain homotopy equivalence is a pair of chain maps 푓• ∶ 푀• → 푁• and 푔• ∶ 푁• → 푀• such that there are chain homotopies id and id . 휀• ∶ 푓• ∘ 푔• ⇒ 푁• 휂• ∶ 푀• ⇒ 푔• ∘ 푓• Exercise 4.1.7. Show that a chain homotopy equivalence induces isomorphisms of homology groups.

Definition 4.1.8. A left resolution of a (left or right) 푅-module is a chain complex 푃• and a homomorphism ℎ ∶ 푃0 → 푀 with the following properties:

• For each negative integer 푖, 푃푖 = 0.

• For each positive integer 푖, 퐻푖(푃•) = 0.

• The homomorphisms 휕1 ∶ 푃1 → 푃0 and ℎ ∶ 푃0 → 푀 fit into a right exact sequence:

휕1 ℎ 푃1 푃0 푀 0

A projective resolution (resp. free resolution) is a left resolution 푃• such that 푃푖 is projective (resp. free) for each natural number 푖. Remark 4.1.9. Every module has a free resolution, hence a projective resolution: if 푇 generates 푀, then we have a surjective homomorphism 푅⊕푇 ↠ 푀, so set ⊕푇 ⊕푇 푃0 = 푅 , consider ker (푅 ↠ 푀) and continue inductively. Example 4.1.10. The ℤ-module ℤ has a free resolution:

⋯ 0 0 0 ℤ

ℤ

34 2. Derived functors

The ℤ-module ℤ/푛ℤ has a free resolution:

⋯ 0 0 푛ℤ ℤ

ℤ/푛ℤ

This generalises to 푅/푅⟨푓⟩ for any ring 푅 and any non-zerodivisor 푓. Theorem 4.1.11. Any two projective resolutions of the same (left) 푅-module 푀 are chain homotopy equivalent.

Proof. See [Lang, 2002, Ch. XX, Lem. 5.2].

2 Derived functors

Definition 4.2.1. Let 푅 and 푆 be rings. An additive functor is a functor 푇 ∶ Mod(푅) → Mod(푆) such that 푇(푓 + 푔) = 푇(푓) + 푇(푔) for all 푅-linear maps 푓, 푔 ∶ 푀 → 푁.A right exact functor is a functor 푇 ∶ Mod(푅) → Mod(푆) such that every right exact sequence of (left) 푅-modules

푓 푔 퐴 퐵 퐶 0

is carried to a right exact sequence of (left) 푆-modules:

푇(푓) 푇(푔) 푇(퐴) 푇(퐵) 푇(퐶) 0

Similarly, a left exact functor preserves left exact sequences, and an exact functor preserves short exact sequences.

Remark 4.2.2. Any left or right exact functor is automatically additive: see [Freyd, 1964, Thm 3.11].

Example 4.2.3. The functor 푀 ⊗푅 (−) ∶ Mod(푅) → Ab is additive and right exact. It is exact if and only if 푀 is a flat right 푅-module.

Definition 4.2.4. Let 푇 ∶ Mod(푅) → Mod(푆) be a right exact functor. The 푖-th

left derived functor of 푇 is the functor 퐿푖푇 ∶ Mod(푅) → Mod(푆) defined as follows: for each (left) 푅-module 푀, choose a projective resolution 푃• and set 퐿푖푇(푀) = 퐻푖(푇(푃•)).

35 IV. Homological algebra

Remark 4.2.5. Since 푇 is right exact, 퐿0푇 is naturally isomorphic to 푇. The module 퐿푖푇(푀) is independent of the choice of 푃•, since any two projective resolutions of 푀 are chain homotopy equivalent.

푅 Definition 4.2.6. For a left 푅-module 푁, Tor푖 (−, 푁) is the 푖-th left derived functor of (−) ⊗푅 푁.

Remark 4.2.7. 푅 • Tor푖 (푀, 푁) ≅ 푀⊗푅 푁 for all right 푅-modules 푀 and left 푅-modules 푁.

푅 • If 푀 is projective, then Tor푖 (푀, 푁) = 0 for all left 푅-modules 푁 and all 푖 > 0.

푅 • If 푁 is flat, then 푖Tor (푀, 푁) = 0 for all right 푅-modules 푀 and all 푖 > 0.

• If 푓 is a non-zerodivisor in 푅, then we have a free resolution of the right 푅-module 푅/⟨푓⟩푅:

푓 ⋅ − ⋯ 0 0 푅 푅

푅/⟨푓⟩푅

푅 But Tor푖 (푅/⟨푓⟩푅 , 푁) is the homology of the complex of abelian groups

푓 ⋅ − ⋯ 0 0 푁 푁

so we obtain

푅 푅 Tor0 (푅/⟨푓⟩푅 , 푁) ≅ 푁/⟨푓⟩푁 Tor1 (푅/⟨푓⟩푅 , 푁) ≅ 푁[푓]

where 푁[푓] = {푥 ∈ 푁 | 푓 ⋅ 푥 = 0}.

푖 Definition 4.2.8. For a (left) 푅-module 푁, Ext푅(−, 푁) is the 푖-th right derived functor of Hom푅(−, 푁): if 푃• is a projective resolution of a (left) 푅-module 푀, 푖 then Ext푅(푀, 푁) is the 푖-th cohomology group of the cochain complex below:

Hom푅(푃0, 푁) Hom푅(푃1, 푁) Hom푅(푃2, 푁) ⋯

36 2. Derived functors

Example 4.2.9. Let 푓 be a non-zerodivisor in 푅. Using the evident projective resolution of 푅/푅⟨푓⟩, we find that

0 1 Ext푅(푅/푅⟨푓⟩ , 푁) ≅ 푁[푓] Ext푅(푅/푅⟨푓⟩ , 푁) ≅ 푁/⟨푓⟩푁

This is like Poincaré duality on the circle.

1 Remark 4.2.10. Ext푅(푀, 푁) can be regarded as classifying extensions, i.e. short exact sequences of (left) 푅-modules of the form

0 푁 퐸 푀 0

1 The trivial extension with 퐸 = 푀 ⊕ 푁 corresponds to 0 in Ext푅(푀, 푁).

Lemma 4.2.11 (Snake lemma). If we have a commutative diagram of (left) 푅-modules and 푅-linear maps of the form

0 퐴1 퐴2 퐴3 0

푓1 푓2 푓3

0 퐵1 퐵2 퐵3 0 with exact rows, then there is an exact sequence of the form

0 ⟶ ker 푓1 ⟶ ker 푓2 ⟶ ker 푓3

⟶ coker 푓1 ⟶ coker 푓2 ⟶ coker 푓3 ⟶ 0 and this is natural in the data.

Proof. Exercise.

Lemma 4.2.12 (Zig-zag lemma). If we have a short exact sequence of chain complexes

푓• 푔• 0 퐴• 퐵• 퐶• 0

37 IV. Homological algebra i.e. a commutative diagram of the form

⋮ ⋮ ⋮

푓푖+1 푔푖+1 0 퐴푖+1 퐵푖+1 퐶푖+1 0

푓푖 푔푖 0 퐴푖 퐵푖 퐶푖 0

푓푖−1 푔푖−1 0 퐴푖−1 퐵푖−1 퐶푖−1 0

⋮ ⋮ ⋮ then we get a natural long exact sequence of homology groups:

퐻푖+1(푔•) 퐻푖(푓•) ⋯ 퐻푖+1(퐵•) 퐻푖+1(퐶•) 퐻푖(퐴•) 퐻푖(퐵•) ⋯

Proof. See [Lang, 2002, Ch. XX, Thm 2.1].

Corollary 4.2.13. Given a short exact sequence of left 푅-modules

0 퐴 퐵 퐶 0 and a right 푅-module 푀, we obtain a natural long exact sequence of abelian groups as below:

푅 푅 ⋯ ⟶ Tor푖+1(푀, 퐵) ⟶ Tor푖+1(푀, 퐶) 푅 푅 ⟶ Tor푖 (푀, 퐴) ⟶ Tor푖 (푀, 퐵) ⟶ ⋯

Proof. Let 푃• be a projective resolution of 푀. Projective modules are flat, so we get a short exact sequence of chain complexes

0 푃• ⊗푅 퐴 푃• ⊗푅 퐵 푃• ⊗푅 퐶 0 and the claim follows from the zig-zag lemma.

38 2. Derived functors

Theorem 4.2.14. Let 푀 be a right 푅-module and let 푁 be a left 푅-module. The 푅 groups Tor∗ (푀, 푁) can be computed by taking a flat resolution of 푁: i.e. given 푅 a flat resolution 퐹• of 푁, Tor푖 (푀, 푁) ≅ 퐻푖(푀 ⊗푅 퐹•) for all 푖.

Proof. Let 퐵−1 = 푁, and for each natural number 푗, let 퐵푗 = im (퐹푗+1 → 퐹푗). Since 퐹• is a resolution of 푁, we have a short exact sequence of left 푅-modules for each natural number 푗:

0 퐵푗 퐹푗 퐵푗−1 0 We thus obtain a long exact sequence of abelian groups as below:

푅 푅 ⋯ ⟶ Tor푖+1(푀, 퐹푗) ⟶ Tor푖+1(푀, 퐵푗−1) 푅 푅 ⟶ Tor푖 (푀, 퐵푗) ⟶ Tor푖 (푀, 퐹푗) ⟶ ⋯ 푅 But Tor푖 (푀, 퐹푗) = 0 for all 푖 > 0, so we have isomorphisms 푅 푅 푅 Tor푖+1(푀, 퐵푗−1) ≅ Tor푖 (푀, 퐵푗) ≅ ⋯ ≅ Tor1 (푀, 퐵푖+푗−1) for all natural numbers 푗 and all 푖 > 0; in particular, 푅 푅 Tor푖+1(푀, 푁) ≅ Tor1 (푀, 퐵푖−1) and we also have an exact sequence for each natural number 푗:

푅 0 Tor1 (푀, 퐵푗−1) 푀 ⊗푅 퐵푗 푀 ⊗푅 퐹푗 푀 ⊗푅 퐵푗−1 0

Since 푀 ⊗푅 (−) is right exact, it can be shown that there is an exact sequence

0 퐻푗(푀 ⊗푅 퐹•) 푀 ⊗푅 퐵푗 푀 ⊗푅 퐹푗 푀 ⊗푅 퐵푗−1 0

푅 푅 and so 퐻푗(푀 ⊗푅 퐹•) ≅ Tor1 (푀, 퐵푗−1). Hence, Tor푖 (푀, 푁) ≅ 퐻푖(푀 ⊗푅 퐹•) as required.

푅 Thus, we may compute Tor∗ (푀, 푁) by a projective resolution of 푀 or of 푁, and a short exact sequence in either the first or second variable induces along exact sequence. When 푅 is commutative, 푀⊗푅푁 ≅ 푁⊗푅푀, and it follows that 푅 푅 Tor푖 (푀, 푁) ≅ Tor푖 (푁, 푀); thus we can use a flat resolution of 푀 to compute 푅 푅 Tor푖 (푀, 푁). In fact, we can use a flat resolution of 푀 to compute Tor∗ (푀, 푁) even when 푅 is non-commutative, but this requires a more subtle argument. 푛 푛 Ext푅(푀, 푁) is in general not related to Ext푅(푁, 푀). Nonetheless, we may ∗ compute Ext푅(푀, 푁) by taking a projective resolution of 푀 or an injective resolution of 푁.

39 IV. Homological algebra

Definition 4.2.15. Let 푅 be a ring. An injective (left) 푅-module is a (left) 푅-module 퐶 with the following property: for any injective homomorphism of (left) 푅-modules 푓 ∶ 퐴 ↣ 퐵 and any homomorphism ℎ ∶ 퐴 → 퐶, there is at least one homomorphism 푔 ∶ 퐵 → 퐶 such that ℎ = 푔 ∘ 푓.

Definition 4.2.16. A right resolution of a (left) 푅-module 푀 is a cochain com- • plex 푄 and a homomorphism ℎ ∶ 푀 → 푄0 with the following properties:

(i) For each negative integer 푖, 푄푖 = 0.

(ii) For each positive integer 푖, 퐻푖(푄•) = 0.

(iii) The homomorphisms d0 ∶ 푄0 → 푄1 and ℎ ∶ 푀 → 푄0 fit into a left exact sequence: ℎ d0 0 푀 푄0 푄1

A injective resolution is a right resolution 푄• such that 푄푖 is injective for each natural number 푖. We have long exact sequences in both variables: that is, given a long exact sequence of (left) 푅-modules

0 퐴 퐵 퐶 0 for any (left) 푅-module 푀, there is a covariant long exact sequence of abelian groups:

푖 푖 ⋯ ⟶ Ext푅(푀, 퐵) ⟶ Ext푅(푀, 퐶) 푖+1 푖+1 ⟶ Ext푅 (푀, 퐴) ⟶ Ext푅 (푀, 퐵) ⟶ ⋯ and for any (left) 푅-module 푁, there is a contravariant long exact sequence of abelian groups:

푖 푖 ⋯ ⟶ Ext푅(퐵, 푁) ⟶ Ext푅(퐴, 푁) 푖+1 푖+1 ⟶ Ext푅 (퐶, 푁) ⟶ Ext푅 (퐵, 푁) ⟶ ⋯

40 V Finiteness conditions

1 Noetherian rings

Definition 5.1.1. Let 푅 be a ring. A noetherian (left) 푅-module is a (left) 푅-module 푀 satisfying the ascending chain condition: every strictly increasing chain of submodules of 푀 is of finite length. An artinian (left) 푅-module is a (left) 푅-module 푀 satisfying the descending chain condition: every strictly decreasing chain of submodules of 푀 is of finite length.

Definition 5.1.2. A left noetherian ring is a ring 푅 which satisfies the ascending chain condition as a left 푅-module. Similarly, a right noetherian ring is a ring 푅 which satisfies the ascending chain condition as aright 푅-module. A noetherian ring is a ring 푅 which satisfies the ascending chain condition as aleft 푅-module and as a right 푅-module.

Definition 5.1.3. A left artinian ring is a ring 푅 which satisfies the descending chain condition as a left 푅-module. We define a right artinian ring and an artinian ring analogously.

Theorem 5.1.4. Let 푅 be a ring. The following are equivalent:

(i) 푅 is left noetherian (resp. right noetherian).

(ii) Every left ideal (resp. right ideal) of 푅 is finitely-generated.

Proof. (i) ⇒ (ii). Suppose 퐼 is a left ideal of 푅 which is not finitely generated.

Let 퐼0 = {0}. For each natural number 푛, since 퐼 is not finitely generated, 퐼⧵퐼푛 is non-empty and contains an element 푥푛+1, and 퐼푛+1 = 푅⟨푥1, … , 푥푛+1⟩ is finitely

41 V. Finiteness conditions generated. By induction, we obtain an infinite strictly ascending chain of left ideals:

0 ⊊ 푅⟨푥1⟩ ⊊ 푅⟨푥1, 푥2⟩ ⊊ 푅⟨푥1, 푥2, 푥3⟩ ⊊ ⋯ Hence, 푅 is not left noetherian in this case.

(ii) ⇒ (i). Let 퐼0 ⊆ 퐼1 ⊆ 퐼2 ⊆ ⋯ be an ascending chain of left ideals. Let

퐽 = ⋃푛 퐼푛. This is a left ideal, and if 퐽 is generated by {푥1, … , 푥푚}, then there is an 푁 such that {푥1, … , 푥푚} ⊆ 퐼푁, but then 퐼푁 = 퐼푁+1 = ⋯ = 퐽. Examples 5.1.5. • Every field is both noetherian and artinian.

• The ring ℤ is noetherian but not artinian:

⟨2⟩ ⊋ ⟨4⟩ ⊋ ⟨8⟩ ⊋ ⋯

Similarly, the ring 푘[푥] is noetherian but not artinian.

• The polynomial ring 푘[푥1, 푥2, 푥3,…] is neither noetherian nor artinian. Nonetheless, its fraction field is both noetherian and artinian: so a subring of a noetherian (resp. artinian) ring need not be noetherian (resp. artinian).

• Every artinian ring is noetherian, but this is a non-trivial fact.

Lemma 5.1.6. Let 푅 be a ring. If 푅 is left noetherian (resp. left artinian), then every quotient of 푅 is a noetherian (resp. artinian) left 푅-module. In particular, if 푅 is noetherian (resp. artinian) commutative ring, then every quotient of 푅 is a noetherian (resp. artinian) commutative ring.

Proof. There is an inclusion-preserving bijection between left 푅-submodules of of 푅 ∕ 퐼 and left ideals of 푅 containing 퐼. In the commutative case, every ideal of 푅 ∕ 퐼 is automatically also a 푅-submodule of 푅 ∕ 퐼.

Theorem 5.1.7 (Hilbert’s basis theorem). If 푅 is a noetherian commutative ring, then 푅[푥] is also a noetherian commutative ring.

Proof. Let 퐼 be an ideal of 푅[푥]. We will show that 퐼 is finitely generated. Consider the following ideal of 푅:

| 푗 퐼푗 = {푎푗 | 푎푗푥 + ⋯ + 푎0 ∈ 퐼, 푎푖 ∈ 푅}

42 1. Noetherian rings

One easily verifies that we have an ascending chain of ideals:

퐼0 ⊆ 퐼1 ⊆ 퐼2 ⊆ ⋯

By hypothesis, there must be a natural number 푁 such that 퐼푁 = 퐼푁+1 = ⋯, and by theorem 5.1.4, for , there are elements of such 0 ≤ 푗 ≤ 푁 푏1,푗, … , 푏푘푗,푗 푅 that 퐼 = 푏 , … , 푏 . For each 푏 , let 푝 be an element of 푅[푥] such that 푗 ⟨ 1,푗 푘푗,푗⟩ 푖,푗 푖,푗 deg 푝푖,푗 ≤ 푗, the 푗-th coefficient of 푝푖,푗 is 푏푖,푗 and 푝푖,푗 ∈ 퐼. Let 푝 ∈ 퐼, deg 푝 = 푛, and let 푎 be the 푛-th coefficient of 푝. If 푛 > 푁, then there are elements in such that , and 푐1, … , 푐푘푁 푅 푎푛 = 푐1푏1,푁 + ⋯ + 푐푘푁 푏푘푁,푁

푛−푁 푝 − (푐1푝1,푁 + ⋯ + 푐푘푁 푝푘푁,푁) 푥 is of strictly lower degree than . If , then there are elements in 푝 푛 ≤ 푁 푐1, … , 푐푘푛 such that , and 푅 푎푛 = 푐1푏1,푛 + ⋯ + 푐푛푏푘푛,푛

푝 − 푐1푝1,푛 + ⋯ + 푐푛푝푘푛,푛 is also of strictly lower degree than 푝. By induction, it follows that 푝 is a 푅-linear combination of | . So is finitely generated. {푝푖,푗 | 1 ≤ 푖 ≤ 푘푗, 0 ≤ 푗 ≤ 푁} 퐼 Corollary 5.1.8. If 푅 is a noetherian commutative ring, then any 푅-algebra of finite type is also a noetherian commutative ring.

Proof. Use theorem 5.1.7 and lemma 5.1.6.

Lemma 5.1.9. Let 푅 be a noetherian commutative ring, and let 푆 be a multi- −1 plicatively closed subset. Then 푅[푆 ] is also a noetherian commutative ring.

−1 Proof. Let ℎ ∶ 푅 → 푅[푆 ] be the canonical homomorphism. Then, every −1 −1 −1 −1 ideal 퐼 of 푅[푆 ] satisfies 퐼 = 푅[푆 ]ℎ 퐼, and since ℎ 퐼 is finitely generated, so too must 퐼.

Lemma 5.1.10. Let 푅 be a ring. If we are given a short exact sequence of (left) 푅-modules 0 퐴 퐵 퐶 0 then 퐵 is noetherian if and only if 퐴 and 퐶 are.

Proof. Exercise.

43 V. Finiteness conditions

Corollary 5.1.11. Let 푅 be a left noetherian ring. Every submodule of a finitely- generated left 푅-module is finitely generated and noetherian. Proof. 푅 is a noetherian left 푅-module, and by induction 푅⊕푛 is a noetherian left 푅-module for each 푛 > 0, hence, any finitely-generated left 푅-module is noetherian. But every left 푅-submodule of a noetherian left 푅-module is finitely generated and noetherian. Henceforth, we will only consider commutative noetherian rings.

Theorem 5.1.12. Let 푅 be a noetherian ring. Then, Spec 푅 is a union of finitely many irreducible closed subsets, say Spec 푅 = ⋃푖 푋푖, where 푋푖 ⊈ 푋푗 for 푖 ≠ 푗. This decomposition is unique up to reordering. The irreducible components of Spec 푅 are these subsets. The same is true for any subset of Spec 푅. Proof. Under the hypotheses, Spec 푅 is a noetherian topological space, i.e. satisfies the descending chain condition for closed subsets. If 푋 is a subspace of Spec 푅, since any strictly descending chain of closed subsets of 푋 lifts to a strictly descending chain of closed subsets of Spec 푅, 푋 must be also be a noetherian topological space. Suppose 푋 is a topological space and is not the union of finitely many irreducible closed subsets. Then, 푋 is itself not irreducible and there are closed proper subsets 푌 and 푍 such that 푋 = 푌 ∪ 푍. Continuing inductively, we obtain an infinite strictly descending chain of closed subsets, so 푋 is not noetherian in this case. Uniqueness of the decomposition is left as an exercise. Since closed subsets of Spec 푅 are in one-to-one correspondence with radical ideals of 푅, we may state the theorem algebraically: Corollary 5.1.13. Let 푅 be a noetherian ring, and let 퐼 be an ideal of 푅. Then, 푛 √퐼 = ⋂푖=1 픭푖 for some prime ideals 픭1, … , 픭푛 such that 픭푖 ⊈ 픭푗 for 푖 ≠ 푗; these are precisely the minimal prime ideals containing 퐼. Remark 5.1.14. In the case 푅 = ℤ, the irreducible components of Spec ℤ ∕ ⟨푛⟩ are Spec 픽푝 for each prime 푝 dividing 푛. There may be many ideals with the same radical: for example, if 푅 = ℂ[푥, 푦] and 퐼 is an ideal of 푅 with √퐼 = ⟨푥, 푦⟩, then ⟨푥, 푦⟩푁 ⊆ 퐼 ⊆ ⟨푥, 푦⟩ for some positive integer 푁, but there are many such 퐼, e.g. any ideal generated by monomials. There can also be continuous families of ideals with the same radical.

44 2. Integral extensions

Lemma 5.1.15. Let 푅 be a noetherian ring, and let 퐼 be an ideal of 푅. Then, 푁 √퐼 ⊆ 퐼 ⊆ √퐼 for sufficiently large positive integers 푁.

Proof. By theorem 5.1.4, √퐼 = ⟨푥1, … , 푥푛⟩ for some 푥1, … , 푥푛 in 푅. Hence, 푒1 푒푛 {푥1 , … , 푥푛 } ⊆ 퐼 for some positive integers 푒1, … , 푒푛. Hence, it is enough to take 푁 ≥ 푛 max {푒1, … , 푒푛}. Theorem 5.1.16. Let 푅 be a noetherian ring and let 푀 be a finitely-generated 푅-module. Then, there is a finite filtration of 푀 of the form

0 = 푀0 ⊊ 푀1 ⊊ ⋯ ⊊ 푀푟 = 푀

such that each 푀푖/푀푖−1 is of the form 푅 ∕ 픭푖 for some prime ideal 픭푖 of 푅. Proof. For any non-zero module 푀, there is a submodule isomorphic to 푅 ∕ 픭:

indeed, since 푅 is noetherian, there is an element 푥 of 푀 such that Ann푅(푥) is maximal among all such annihilators, and Ann푅(푥) is a prime ideal of 푅 in that case.[1] The conclusion of the theorem follows by induction on the number of generators of 푀.

Remark 5.1.17. Such a filtration of 푀 is far from unique; even the primes 픭푖 are not uniquely determined.

Exercise 5.1.18. Let 푅 be a noetherian ring and let 푀 be a finitely-generated

푅-module. Show that any such filtration has 픭1 ∩ ⋯ ∩ 픭푟 = √Ann푅(푀). The closed subset 풱(Ann푅(푀)) is called the support of 푀: it is the set of all primes 픭 such that 푀픭 ≠ 0.

2 Integral extensions

In this section we only consider commutative rings.

Definition 5.2.1. Let 퐴 be a ring with a subring 퐵. An element 푎 of 퐴 is integral

over 퐵 just if there are 푏0, … , 푏푛−1 in 퐵 such that 푛−1 푛 푏0 + 푏1푎 + ⋯ + 푏푛−1푎 + 푎 = 0 In other words, 푎 is a zero of a monic polynomial over 퐵. [1] See question 2 on examples sheet 2.

45 V. Finiteness conditions

Example 5.2.2. If 푎 ∈ 퐵, then 푎 is integral over 퐵.

Example 5.2.3. If 퐾 | ℚ is a finite field extension, and

풪퐾 = {푥 ∈ 퐾 | 푥 is integral over ℤ} then 풪퐾 is the ring of integers in 퐾.

Exercise 5.2.4. Using the fact that ℤ is a unique factorisation domain, show that

풪ℚ = ℤ.

Definition 5.2.5. A finite 퐵-algebra is an 퐵-algebra which is finitely generated as a 퐵-module. If 푋 and 푌 are affine schemes, a morphism 푓 ∶ 푋 → 푌 is a finite morphism of schemes just if 푓♯ ∶ 풪(푌) → 풪(푋) makes 풪(푋) into a finite 풪(푌)-algebra.

Lemma 5.2.6. Let 퐵 be a subring of a ring 퐴, and let 푎 ∈ 퐴. The following are equivalent:

(i) The element 푎 is integral over 퐵.

(ii) The 퐵-subalgebra 퐵[푎] generated by 푎 is a finite 퐵-algebra.

(iii) The 퐵-subalgebra 퐵[푎] is contained in a finite 퐵-subalgebra 퐶.

(iv) There is a faithful 퐵[푎]-module 푀 which is finitely generated over 퐵.

Proof. (i) ⇒ (ii). Clear. (ii) ⇒ (iii). Take 퐶 = 퐵[푎]. (iii) ⇒ (iv). Take 푀 = 퐶.

(iv) ⇒ (i). Suppose 푀 = 퐵⟨푚1, … , 푚푟⟩. Then, 푎 ⋅ 푚푖 = ∑푗 푏푖,푗 ⋅ 푚푗 for some in . The matrix | satisfies 푏푖,1, … , 푏푖,푟 퐵 푋 = (푎훿푖,푗 − 푏푖,푗 | 1 ≤ 푖 ≤ 푟, 1 ≤ 푗 ≤ 푟)

푟 ∑ (푎훿푖,푗 − 푏푖,푗) ⋅ 푚푗 = 0 푗=1 but (adj 푋) 푋 = (det 푋) 퐼, so det (푋) = 0, since 푀 is a faithful 퐵[푎]-module. Thus 푎 is integral over 퐵.

Lemma 5.2.7. If 퐴 is a finite 퐵-algebra and 퐵 is a finite 퐶-algebra, then 퐴 is a finite 퐶-algebra.

46 2. Integral extensions

Proof. Exercise.

Corollary 5.2.8. Let 퐵 be a subring of a ring 퐴. If 푎1, … , 푎푛 are all integral over 퐵, then the 퐵-subalgebra 퐵[푎1, … , 푎푛] is a finite 퐵-algebra. Proof. Use induction on 푛.

Corollary 5.2.9. If 퐵 is a subring of 퐴 and 퐶 = {푎 ∈ 퐴 | 푎 is integral over 퐵} then 퐶 is a 퐵-subalgebra of 퐴.

Proof. Clearly, 퐵 is contained in 퐶, and if 푥 and 푦 are in 퐶, we must show that 푥 + 푦 ∈ 퐶, −푥 ∈ 퐶, and 푥푦 ∈ 퐶; but by the previous corollary, the 퐵-subalgebra 퐵[푥, 푦] is a finite 퐵-algebra, so 푥 + 푦, −푥, and 푥푦 are all integral over 퐵 by lemma 5.2.6.

Definition 5.2.10. The integral closure of 퐵 in 퐴 is the 퐵-subalgebra 퐶 defined above. We say 퐵 is integrally closed in 퐴 just if 퐶 = 퐵. An integral extension of 퐵 is a ring 퐴 containing 퐵 such that the integral closure of 퐵 in 퐴 is 퐴 itself. A normal domain is an integral domain 퐵 such that 퐵 is integrally closed in Frac 퐵.

Example 5.2.11. The ring ℤ is a normal domain, as is any unique factorisation [1] domain. In particular, the polynomial ring 푘[푥1, … , 푥푛] over a field 푘 is a normal domain. We will also see that the ring of integers 풪퐾 is a normal domain.

Definition 5.2.12. If 푋 and 푌 are affine schemes, a morphism 푓 ∶ 푋 → 푌 is integral morphism of schemes just if 풪(푋) is an integral extension of the image of 풪(푌) under 푓♯. We also say 푋 is integral over 푌 in this case.

Remark 5.2.13. A morphism of (affine) schemes is finite if and only if itisof finite type and integral.

Corollary 5.2.14. Let 푋, 푌, 푍 be affine schemes. If 푓 ∶ 푋 → 푌 and 푔 ∶ 푌 → 푍 are integral morphisms of schemes, then so is 푔 ∘ 푓 ∶ 푋 → 푍.

Proof. Let 퐴 = 풪(푋), 퐵 = 풪(푌), 퐶 = 풪(푍), and let 푎 ∈ 퐴. There are 푏0, … , 푏푛−1 in 퐵 such that

♯ ♯ ♯ 푛−1 푛 푓 (푏0) + 푓 (푏1)푎 + ⋯ + 푓 (푏푛−1)푎 + 푎 = 0 [1] See question 3 on examples sheet 3.

47 V. Finiteness conditions

but the 퐶-subalgebra of 퐵 generated by 푏0, … , 푏푛−1 is a finite 퐵-algebra, so ♯ ♯ the 퐶-subalgebra of 퐴 generated by 푓 (푏0), … , 푓 (푏푛−1), 푎 is a finite 퐶-algebra, hence integral over 퐶.

Corollary 5.2.15. Let 퐵 be a subring of a ring 퐴. If 퐶 is the integral closure of 퐵 in 퐴, then 퐶 is integrally closed in 퐴.

Proof. If 푥 is integral over 퐶, it is also integral over 퐵.

Exercise 5.2.16. Let 퐴 be an integral extension of 퐵. Prove the following claims:

(i) If 퐼 is an ideal of 퐴 and 퐽 = 퐼 ∩ 퐵, then 퐵 ∕ 퐽 is an integral extension of 퐴 ∕ 퐼.

−1 (ii) If 푆 is a multiplicatively closed subset of 퐵, then 퐴[푆 ] is an integral −1 extension of 퐵[푆 ]. Lemma 5.2.17. Let 퐶 be the integral closure of a subring 퐵 in a ring 퐴. If 푆 is a −1 −1 multiplicatively closed subset of 퐵, then 퐶[푆 ] is the integral closure of 퐵[푆 ] −1 in 퐴[푆 ]. −1 −1 −1 Proof. We know 퐶[푆 ] is integral over 퐵[푆 ], and if 푎 ∕ 푠 ∈ 퐴[푆 ] and 푎 ∕ 푠 −1 is integral over 퐵[푆 ], then

푛−1 푛 푏0 푏1 푎 푏푛−1 푎 푎 + + ⋯ + 푛−1 + 푛 = 0 푠0 푠1 푠 푠푛−1 푠 푠

−1 for some 푏푖/푠푖 in 퐵[푆 ]. Let 푡 = 푠0 ⋯ 푠푛−1 and clear denominators:

푛 푛−1 (푠푡) (푠푡) 푠푡 푛−1 푛 −1 푏0 + 푏1 (푡푎) + ⋯ + 푏푛−1 (푡푎) + (푡푎) = 0 ∈ 퐴[푆 ] 푠0 푠1 푠푛−1 Hence, for some 푢 in 푆,

푛 푛−1 (푠푡푢) (푠푡푢) 푠푡푢 푛−1 푛 푏0 + 푏1 (푢푡푎) + ⋯ + 푏푛−1 (푢푡푎) + (푢푡푎) = 0 ∈ 퐴 푠0 푠1 푠푛−1

−1 and thus 푢푡푎 is integral over 퐵. But 푢푡 ∈ 푆, so 푎 ∕ 푠 ∈ 퐶[푆 ]. Lemma 5.2.18. Normality is a local property: if 퐴 is an integral domain, the following are equivalent:

(i) 퐴 is a normal domain.

48 2. Integral extensions

(ii) 퐴픭 is normal for all primes ideals 픭 of 퐴.

(iii) 퐴픪 is normal for all maximal ideals 픪 of 퐴.

Proof. Observe that we may embed each 퐴픭 into 퐾 = Frac 퐴. Let 퐶 be the integral closure of 퐴 in 퐾. 퐴 is normal if and only if the inclusion 퐴 ↪ 퐶 is an isomorphism (of 퐴-modules), but by lemma 3.2.10, this can be checked locally, and we already saw that 퐴 being a normal domain implies each 퐴픭 is a normal domain.

The normality of 퐴 is related to singularities of Spec 퐴. To be precise, a normal affine variety has singular locus of codimension at least 2.

Example 5.2.19. For any field 푘, the map 푥 ↦ 푥2 defines a finite morphism 1 1 푓 ∶ 픸푘 → 픸푘. Indeed, writing 푘[푥] for the ring of the domain and 푘[푦] for the ring of the codomain, the homomorphism 푓♯ ∶ 푘[푦] → 푘[푥] satisfies 푓♯(푦) = 푥2; hence 푓♯ makes 푘[푦] into a finite 푘[푥]-algebra, and in particular 푓 is an integral morphism.

Lemma 5.2.20. Let 퐵 be a subring of a ring 퐴, and suppose 퐴 is an integral extension of 퐵. Let 픭 ∈ Spec 퐴, 픮 = 픭 ∩ 퐵. Then, 픭 is a maximal ideal of 퐴 if and only if 픮 is a maximal ideal of 퐵.

Proof. We know 퐴 ∕ 픭 is integral over 퐵 ∕ 픮, so it suffices to show that 퐴 is a field if and only if 퐵 is a field, when 퐴 is an integral domain.

Suppose 퐵 is a field. Let 푥 ∈ 퐴. If 푥 ≠ 0, then there are 푏0, … , 푏푛−1 in 퐵 such that 푛−1 푛 푏0 + 푏1푥 + ⋯ + 푏푛−1푥 + 푥 = 0

Choose 푏0, … , 푏푛 so that 푛 is minimal with this property. Since 퐴 is an integral domain, 푏0 ≠ 0, thus,

푏 + 푏 푥 + ⋯ + 푥푛−1 푥 ⋅ 1 2 = −1 푏0 and hence 푥 is invertible in 퐴. Thus 퐴 is a field. Conversely, suppose 퐴 is a field. Let 푢 ∈ 퐵. If 푢 ≠ 0, then 1 ∕ 푢 exists in 퐴 and is integral over 퐵, say

푏1 푏푛−1 1 푏 + + ⋯ + + = 0 0 푢 푢푛−1 푢푛 49 V. Finiteness conditions

for some 푏0, … , 푏푛−1. But then

1 푏 푢푛−1 + 푏 푢푛−2 + ⋯ + 푏 = − 0 1 푛−1 푢 so 1 ∕ 푢 ∈ 퐵. Thus 퐵 is a field.

Corollary 5.2.21. Let 퐴 be an integral extension of 퐵. Let 픭 and 픭′ be prime ideals of 퐴. If 픭 ⊆ 픭′ and 픭 ∩ 퐵 = 픭′ ∩ 퐵, then 픭 = 픭′.

′ −1 Proof. Let 픮 = 픭 ∩ 퐵 = 픭 ∩ 퐵, and let 푆 = 퐵 ⧵ 픮. We showed that 퐴[푆 ] is −1 integral over 퐵픮 = 퐵[푆 ]. Let 픪 be the maximal ideal of 퐵픮, and let 픓 and ′ ′ −1 픓 be the extensions of 픭 and 픭 (respectively) in 퐴[푆 ]. Since 픭 ∩ 푆 = ∅ and 픭′ ∩ 푆 = ∅, we know from theorem 3.1.11 that 픓 and 픓′ are prime ideals −1 ′ ′ of 퐴[푆 ], and 픓 ∩ 퐵픮 = 픓 ∩ 퐵픮 = 픪, so 픓 and 픓 must both be maximal ideals, by the preceding lemma. But 픓 ⊆ 픓′, so 픓 = 픓′, and it follows that 픭 = 픭′.

Theorem 5.2.22 (Going-up theorem). Let 퐴 be an integral extension of 퐵. If 픮 ∈ Spec 퐵, then there is at least one 픭 in Spec 퐴 such that 픭 ∩ 퐵 = 픮.

−1 Proof. Let 푆 = 퐵 ⧵ 픮. We know that the localisation 퐴[푆 ] is an integral −1 −1 −1 extension of 퐵[푆 ], and 퐵픮 = 퐵[푆 ] ≠ 0, so 퐴[푆 ] ≠ 0. By theorem 1.3.10, −1 there is a maximal ideal 픪 in 퐴[푆 ], and by lemma 5.2.20, 픪∩퐵픮 is a maximal ideal. By pulling back 픪 to 퐴, we obtain the required prime 픭 of 퐴.

3 Noether normalisation and Nullstellensätze

In this section we only consider commutative rings.

Lemma 5.3.1 (Preparation lemma). Let 푘 be a field, and let 푓 ∈ 푘[푥1, … , 푥푛]. If 푓 ≠ 0, then there is an isomorphism 푘[푥1, … , 푥푛] ≅ 푘[푦1, … , 푦푛] such that 푓 is a constant multiplied by a polynomial monic in 푦푛, i.e.

푑−1 푑 푖 푓 = 푎푦푛 + ∑ 푎푖(푦1, … , 푦푛−1)푦푛 푖=0

× for some 푎 ∈ 푘 and 푎푖(푦1, … , 푦푛−1) ∈ 푘[푦1, … , 푦푛−1].

50 3. Noether normalisation and Nullstellensätze

푛 퐼 푖1 푖푛 Proof. Let 퐼 = (푖1, … , 푖푛) ∈ ℕ , and write 퐱 for 푥1 ⋯ 푥푛 . Lexicographic- ally order ℕ푛, and let 퐼 be the greatest multi-index such that 퐱퐼 has a non-zero coefficient in 푓. Then, for 푚1 ≫ 푚2 ≫ ⋯ ≫ 푚푛−1 ≫ 1,

푚1 푚푛−1 푚1푖1+⋯+푚푛−1푖푛−1+푖푛 푓(푦1 + 푦푛 , … , 푦푛−1 + 푦푛 , 푦푛) = 푎퐼푦푛 + lower order terms

Thus, the homomorphism 푘[푥1, … , 푥푛] → 푘[푦1, … , 푦푛] defined by

푚1 푚푛−1 푥1 ↦ 푦1 + 푦푛 , … , 푥푛−1 ↦ 푦푛−1 + 푦푛 , 푥푛 ↦ 푦푛 is an isomorphism with the required properties.

Lemma 5.3.2 (Noether normalisation). Let 푘 be a field. If 푅 is a non-trivial 푘-algebra of finite type, then 푅 is a finite extension of a polynomial ring (in finitely many variables) over 푘, i.e. there is a natural number 푛 and an injective homomorphism 푘[푦1, … , 푦푛] ↣ 푅 making 푅 into a finite 푘[푦1, … , 푦푛]-algebra.

Proof. Let 푓 ∶ 푘[푥1, … , 푥푁] ↠ 푅 be a surjective homomorphism: such a homomorphism exists because 푅 is finitely generated over 푘. The claim is obvious for 푁 = 0, and we proceed by induction on 푁. Let 퐼 = ker 푓; if 퐼 = 0 then we are done. Otherwise, let 푓 ∈ 퐼, 푓 ≠ 0. By the preparation lemma above, we may assume 푓 is monic in 푥푁. Thus, we have a relation of the form

푑−1 푑 푖 푥푁 + ∑ 푎푖(푥1,…, 푥푁−1)푥푁 = 0 ∈ 푅 푖=0 where 푥푗 = 푓(푥푗), so 푥푁 is integral over the 푘-subalgebra 푆 of 푅 generated by 푥1,…, 푥푁−1; hence 푅 is a finite 푆-algebra. By the induction hypothesis, 푆 is a finite 푘[푦1, … , 푦푛]-algebra, so 푅 is finite over 푘[푦1, … , 푦푛] as well.

Corollary 5.3.3 (Weak Nullstellensatz). Let 푘 be a field, and let 푅 be a 푘-algebra of finite type. If 푅 is a field, then 푅 is a finite extension of 푘.

Proof. We know there are 푦1, … , 푦푛 in 푅 such that the 푘-subalgebra 푘[푦1, … , 푦푛] is a polynomial ring and 푅 is finite over 푘[푦1, … , 푦푛]. If 푛 = 0, then we are done. If 푛 > 0, then 1 ∕ 푦1 ∈ 푅, but then 1 ∕ 푦1 must be integral over 푘[푦1, … , 푦푛] — a contradiction. (Polynomial rings are unique factorisation domains and hence normal.)

51 V. Finiteness conditions

Corollary 5.3.4 (Easy Nullstellensatz). Let 푘 be an algebraically closed field. If 푓1, … , 푓푟 are in 푘[푥1, … , 푥푛], then either

(i) there are 푔1, … , 푔푟 such that 푔1푓1 + ⋯ + 푔푟푓푟 = 1, or

(ii) the polynomials 푓1, … , 푓푟 have a common zero over 푘.

Proof. Let 푅 = 푘[푥1, … , 푥푛]/⟨푓1, … , 푓푟⟩. If (i) is false, then 푅 is a non-trivial 푘-algebra of finite type. By theorem 1.3.10, 푅 has a maximal ideal 픪, and 푅∕픪 is field which is also a 푘-algebra of finite type. The preceding result implies 푅∕픪 must be a finite field extension of 푘, but 푘 is algebraically closed, so 푅 ∕ 픪 ≅ 푘.

Thus, the polynomials 푓1, … , 푓푟 have a common zero over 푘.

Definition 5.3.5. The Jacobson radical of a ring is the intersection of all its maximal ideals.

Lemma 5.3.6. Let 푘 be a field. If 푅 is a 푘-algebra of finite type, then the nilradical and Jacobson radical of 푅 coincide.

Proof. By theorem 1.3.17, the nilradical is contained in the Jacobson radical. Suppose 푓 is in the Jacobson radical of 푅. We need to show that 푓 is also contained in every prime ideal 픭 of 푅. Dividing by 픭, it is enough to prove the claim when 푅 is an integral domain of finite type over 푘. Suppose 푓 ≠ 0. Then, 푅[1 ∕ 푓] is also an integral domain of finite type over 푘 and contains a maximal ideal 픪. Let 픮 = 픪 ∩ 푅. Now, 푅/픮 is an integral domain containing 푘 and contained in 푅[1 ∕ 푓]/픪, but 푅[1 ∕ 푓]/픪 is a finite field extension of 푘 by the weak Nullstellensatz, so 푅/픮 must also be a finite field extension of 푘. Thus 픮 is a maximal ideal of 푅, and 푓 ∉ 픮. So 푓 is not in the Jacobson radical of 푅.

Theorem 5.3.7 (Hilbert’s Nullstellensatz). Let 푘 be an algebraically closed field, and let 퐼 be an ideal of 푘[푥1, … , 푥푛]. If we define

푛 | 풵(퐼) = {(푎1, … , 푎푛) ∈ 푘 | ∀푓 ∈ 퐼. 푓(푎1, … , 푎푛) = 0}

| ℐ(푌) = {푓 ∈ 푘[푥1, … , 푥푛] | ∀ (푎1, … , 푎푛) ∈ 푌. 푓(푎1, … , 푎푛) = 0} then ℐ(풵(퐼)) = √퐼.

52 3. Noether normalisation and Nullstellensätze

Proof. Let . Certainly . Let ; it 퐽 = ℐ(풵(퐼)) 퐼 ⊆ √퐼 ⊆ 퐽 푅 = 푘[푥1, … , 푥푛]/√퐼 is a reduced 푘-algebra of finite type, so by the above lemma it has trivial Jacobson radical. Suppose 푓 ∉ √퐼: then the image of 푓 in 푅 is non-zero and there is a maximal ideal 픪 of 푅 not containing 푓; but 픪 corresponds to a point (푎1, … , 푎푛) of 풵(퐼) by the weak Nullstellensatz, and 푓(푎1, … , 푎푛) ≠ 0. Thus, 푓 ∉ 퐽, and so √퐼 = 퐽.

VI Dimension theory

In this chapter we only consider commutative rings.

1 Artinian rings

Definition 6.1.1. Let 푅 be a ring. A chain of prime ideals of length 푟 is a strictly ascending chain of the form

픭0 ⊊ 픭1 ⊊ ⋯ ⊊ 픭푟

where each 픭푖 is a prime ideal of 푅. The Krull dimension of 푅 is the supremum of the lengths of all such chains.

Examples 6.1.2. • Any field has Krull dimension 0.

• The ring ℤ has Krull dimension 1.

Definition 6.1.3. Let 푋 be a topological space. The irreducible dimension of 푋 is the supremum of the lengths of chains of irreducible closed subsets of 푋.

Clearly, if 푋 is an affine scheme, then the irreducible dimension 푋 is equal to the Krull dimension of 풪(푋). Lemma 6.1.4. Any prime ideal of an artinian ring is a maximal ideal.

Proof. Let 픭 be a prime ideal of an artinian ring 푅. Then, 푅 ∕ 픭 is an artinian integral domain. Let 푥 ∈ 푅 ∕ 픭. Then, we have a descending chain of ideals:

2 3 ⟨푥⟩ ⊇ ⟨푥 ⟩ ⊇ ⟨푥 ⟩ ⊇ ⋯

55 VI. Dimension theory

Thus, for some positive integer 푛, there is an 푎 in 푅 ∕ 픭 such that 푥푛 = 푎푥푛+1. If 푥 ≠ 0, then 푎푥 = 1. Thus, 푅 ∕ 픭 is a field, and 픭 is a maximal ideal. Lemma 6.1.5. An artinian ring has only finitely many distinct prime ideals.

Proof. Let 푅 be a ring. If 픪1, 픪2, 픪3,… are distinct maximal ideals, then we have an infinite strictly descending chain of ideals:

픪1 ⊋ 픪1 ∩ 픪2 ⊋ 픪1 ∩ 픪2 ∩ 픪3 ⊋ ⋯

(If 픪1 ∩ ⋯ ∩ 픪푛 = 픪1 ∩ ⋯ ∩ 픪푚 and 푛 ≤ 푚, then 픪푚 must contain some 픪푖 where 푖 ≤ 푛, and so we must have 푛 = 푚.) Lemma 6.1.6. The nilradical of an artinian ring is nilpotent, i.e. there is a positive integer 푛 such that 푥푛 = 0 for every nilpotent element 푥.

Proof. Let 픯 be the nilradical of an artinian ring 푅. Then, we have a descending chain of ideals: 픯 ⊇ 픯2 ⊇ 픯3 ⊇ ⋯ so there is a 푁 such that 픯푁 = 픯푁+1 = ⋯. Let 퐼 = 픯푁. Consider the poset Σ of all ideals 퐽 of 푅 such that 퐼퐽 ≠ ⟨0⟩. Suppose 퐼 ≠ 0; then Σ is non-empty since ⟨1⟩ ∈ Σ. Since 푅 is artinian, Σ must have a minimal member 퐽, and 퐽 ≠ 0 since 퐼퐽 ≠ 0. Let 푥 ∈ 퐽. If 푥 ≠ 0 then ⟨푥⟩ ∈ Σ, and ⟨푥⟩ ⊆ 퐽, so 퐽 = ⟨푥⟩ by minimality; but 퐼 (퐼퐽) = 퐼2퐽 = 퐼퐽 ≠ ⟨0⟩, so 퐼퐽 ∈ Σ as well, and 퐼퐽 ⊆ 퐽, so 퐼퐽 = 퐽, hence 푥 = 푎푥 for some 푎 in 퐼. But then 푥 = 푎푥 = 푎2푥 = ⋯ = 0, since 푎 is nilpotent — a contradiction. Thus 퐼 = 0, i.e. the nilradical is nilpotent. Theorem 6.1.7. A ring is artinian if and only if it is either

• noetherian and of Krull dimension 0, or • the trivial ring.

Proof. Let 푅 be a non-trivial artinian ring. Clearly, its Krull dimension is 0, since all prime ideals are maximal by lemma 6.1.4. We need to show that 푅 is noetherian. By lemma 6.1.5, 푅 has only finitely many maximal ideals, say

픪1, … , 픪푛. Let 픯 be the nilradical of 푅. By theorem 1.3.17, 픯 = 픪1 ∩ ⋯ ∩ 픪푛, and by lemma 6.1.6, there is 푁 ≫ 1 such that 픯푁 = ℐ(0). Consider the filtration below:

2 푅 ⊃ 픪1 ⊃ 픪1픪2 ⊃ ⋯ ⊃ 픪1 ⋯ 픪푛 ⊃ 픪1 픪2 ⋯ 픪푛 ⊃ ⋯ ⊇ ⟨0⟩

56 2. Discrete valuations and Dedekind domains

Each quotient is of the form 퐼 ∕ 픪퐼 for some maximal ideal 픪 and some ideal 퐼, and this is naturally an artinian (푅 ∕ 픪)-vector space. But any artinian vector space has finite dimension and so is also noetherian, and the filtration isfinite, so 푅 must also be noetherian. Conversely, let 푅 be a noetherian ring. If 푅 has Krull dimension 0, then every prime ideal is maximal, and so the nilradical 픯 is the intersection of finitely many maximal ideals. By lemma 5.1.15, there is an integer 푁 such that 픯푁 ⊆ ⟨0⟩ ⊆ 픯. By considering the filtration above, we may conclude that 푅 is artinian.

2 Discrete valuations and Dedekind domains

Definition 6.2.1. A discrete valuation on a field 퐾 is a map 휈 ∶ 퐾 → ℤ∪{+∞} satisfying the following axioms:

• 휈(푥) = +∞ if and only if 푥 = 0.

• 휈(푥푦) = 휈(푥) + 휈(푦) for all 푥 and 푦 in 퐾.

• 휈(푥 + 푦) ≥ min {휈(푥), 휈(푦)} for all 푥 and 푦 in 퐾.

The trivial valuation is the discrete valuation 휈 such that 휈(푥) = 0 for all 푥 ≠ 0.

Exercise 6.2.2. Let 휈 be a discrete valuation on a field 퐾.

(i) Show that 휈(1) = 0 and 휈(푥) = 휈(−푥).

(ii) Show that the set

풪퐾,휈 = {푥 ∈ 퐾 | 휈(푥) ≥ 0} is a subring of 퐾. It is called the valuation ring of 퐾 at 휈.

Example 6.2.3. Let 푝 be a prime number. The 푝-adic valuation on ℚ is the

discrete valuation 휈푝 defined as follows: if 푎 and 푏 are integers not divisible by 푝, 푛 then 휈푝(푝 푎 ∕ 푏) = 푛 for all integers 푛. The valuation ring of ℚ at 휈푝 is the local ring ℤ⟨푝⟩.

Example 6.2.4. Let 푘 be a field, and let 푓 be irreducible in 푘[푥1, … , 푥푛]. We can define a discrete valuation 휈푓 on the field 푘(푥1, … , 푥푛) in a similar fashion: 푛 if 푎 and 푏 are elements of 푘[푥1, … , 푥푛], then 휈푓(푓 푎 ∕ 푏) = 푛 for all integers 푛. The valuation ring is the local ring . 푘[푥1, … , 푥푛]⟨푓⟩

57 VI. Dimension theory

Definition 6.2.5. A discrete valuation ring is an integral domain 푅 such that there is a discrete valuation 휈 on 퐾 = Frac 푅 such that 푅 = 풪퐾,휈.

Proposition 6.2.6. Let 푅 = 풪퐾,휈.

(i) If 픪 = {푓 ∈ 푅 | 휈(푓) > 0}, then 픪 is the unique maximal ideal of 푅.

(ii) If 휈(푓) = 푛, then the ideal of 푅 generated by 푓 is {푔 ∈ 푅 | 휈(푔) ≥ 푛}.

(iii) If 퐼 is an ideal of 푅, then 퐼 = ⟨푓⟩ for some 푓 in 퐼.

In particular, 푅 is a local principal ideal domain. If 푅 is not a field, then 푅 is a noetherian local domain of Krull dimension 1.

Proof. (i). If 푓 is invertible in 푅, then 휈(푓) = 0, so 픪 is the set of non-units. It is an ideal since 휈 is a valuation, and the claim follows from lemma 3.1.9. (ii). Certainly, ⟨푓⟩ ⊆ {푔 ∈ 푅 | 휈(푔) ≥ 푛}. If 휈(푔) ≥ 푛, then 휈(푔 ∕ 푓) ≥ 0, so 푔 ∕ 푓 ∈ 푅, and 푔 ∈ ⟨푓⟩ as required. (iii). If 퐼 = ⟨0⟩, then we are done. Otherwise, since ℕ is well-ordered, there is an element 푓 of 퐼 of minimal valuation, and it is clear that 퐼 = ⟨푓⟩.

Theorem 6.2.7. Let 푅 be a local noetherian domain of Krull dimension 1. The following are equivalent:

(i) 푅 is a discrete valuation ring.

(ii) 푅 is a normal domain.

(iii) The unique maximal ideal of 푅 is a principal ideal.

(iv) If 픪 is the unique maximal ideal of 푅 and 푘 is the residue field, then 2 dim푘 픪/픪 = 1. Proof. (i) ⇒ (ii). Let 퐾 = Frac 푅 and let 푥 ∈ 퐾. Suppose

푛−1 푛 푎0 + 푎1푥 + ⋯ + 푎푛−1푥 + 푥 = 0 for some 푎0, … , 푎푛−1 in 푅. If 푅 = 풪퐾,휈 and 휈(푥) < 0, then

푛 푛−1 휈(푥 ) = 휈(푎0 + ⋯ + 푎푛−1푥 )

≥ min {휈(푎0), 휈(푎1) + 휈(푥), … , 휈(푎푛−1) + (푛 − 1)(푥)} ≥ (푛 − 1) 휈(푥)

58 2. Discrete valuations and Dedekind domains

But 휈(푥푛) = 푛휈(푥푛) < (푛 − 1) 휈(푥) — a contradiction. So 휈(푥) ≥ 0, and 푥 ∈ 푅. Thus 푅 is a normal domain. (ii) ⇒ (iii). Omitted: see [Eisenbud, 1995, § 11.2]. (iii) ⇔ (iv). Use corollary 3.2.13. (iii) ⇒ (i). Since 푅 is an integral domain, ⟨0⟩ is prime, and the only prime ideals of 푅 must be ⟨0⟩ and 픪. Thus, if 퐼 is any proper ideal of 푅, √퐼 = 픪. Since 푅 is noetherian, lemma 5.1.15 implies 픪푁 ⊆ 퐼 ⊆ 픪 for some positive integer 푁. Suppose 픪 = ⟨푡⟩. Then, 픪푛 = ⟨푡푛⟩ for all natural numbers 푛, and

픪 ⊊ 픪2 ⊊ 픪3 ⊊ ⋯ since 푡푛 = 푎푡푛+1 would imply 푎푡 = 1 (a contradiction). Now, for 푦 ∈ 푅, if 푦 ≠ 0, there is a greatest 푛 such that 푦 ∈ 픪푛, since 픪푁 ⊆ ⟨푦⟩ ⊆ 픪 for all 푛 푛+1 푛 푁 ≫ 1. Set 휈(푦) = 푛. The image of 푦 in 픪 /픪 is non-zero, so 픪 = ⟨푦⟩ by corollary 3.2.13, hence 푦 = 푢푡푛 for some 푢 ∈ 푅×. Thus, we may extend 휈 to 퐾 to obtain a discrete valuation on 퐾 such that 풪퐾,휈 = 푅.

Definition 6.2.8. A Dedekind domain is a noetherian normal domain of Krull dimension 1.

Proposition 6.2.9. Let 푅 be a Dedekind domain.

(i) If 픪 is a maximal ideal of 푅, then 푅픪 is a discrete valuation ring.

(ii) Let 퐾 = Frac 푅. There is a distinct discrete valuation on 퐾 for each maximal ideal of 푅.

Proof. (i). By the properties of localisation, 푅픪 is a local noetherian normal domain of Krull dimension 1, so is a discrete valuation ring by the preceding theorem. (ii). Clear.

Examples 6.2.10. • Any principal ideal domain which is not a field is a Dedekind domain.

• Let 퐾 be a number field. The ring of integers 풪퐾 is a Dedekind domain, but not necessarily a principal ideal domain. The failure of 풪퐾 to be a principal ideal domain is measured by the ideal class group of 퐾, also

called the Picard group of 풪퐾.

59 VI. Dimension theory

• Let 푋 be a smooth affine algebraic curve over a field 푘. Then, 풪(푋) is a Dedekind domain. It is a principal ideal domain if and only if 푘 is algebraically closed and the completion 푋 has genus 0.

3 Noetherian local rings

Definition 6.3.1. Let 푅 be a ring, and let 픭 ∈ Spec 푅. The codimension of 픭 is the supremum of the lengths of chains of prime ideals contained in 픭.

Recall theorem 3.1.11: there is an inclusion-preserving bijection between

the primes of 푅픭 and the primes of 푅 contained in 픭, so the codimension of 픭 is equal to the Krull dimension of 푅픭.

Theorem 6.3.2 (Krull’s Hauptidealsatz). Let 푅 be a noetherian ring. If 푎 is not an invertible element of 푅, then every minimal prime containing 푎 has codimension at most 1.

Proof. Let 픭 be a minimal prime containing 푎. We wish to show that dim 푅픭 ≤ 1. Notice that the ideal 푅픭⟨푎 ∕ 1⟩ has 픭픭 minimal over it, but 픭픭 is maximal in 푅픭, so 픭픭 must be the only prime ideal of 푅픭 containing 푎 ∕ 1. We may now assume 픭 is a maximal ideal without loss of generality. Let 픮 ∈ Spec 푅. If 픮 ≠ 픪 then 푎 ∉ 픮. Let 픮(푖) be the 푖-th symbolic power of 픮, i.e. 푖 the inverse image of 푅픮픮 in 푅. Consider the chain of ideals

⟨푎⟩ + 픮(1) ⊇ ⟨푎⟩ + 픮(2) ⊇ ⟨푎⟩ + 픮(3) ⊇ ⋯

Since 픭 is the only prime of 푅 containing 푎, the residue ring 푅/⟨푎⟩ has a unique prime ideal; hence 푅/⟨푎⟩ is a noetherian local ring of Krull dimension 0 and is therefore an artinian local ring by theorem 6.1.7. Thus, for 푛 ≫ 1, we must have ⟨푎⟩ + 픮(푛) = ⟨푎⟩ + 픮(푛+1). Since ⟨푎⟩ ⊆ 픭, we must have

픮(푛) ⟨푎⟩ 픮(푛) ⟨푎⟩ + 픮(푛) 푅/픭 ⊗푅 = 푅/픭 ⊗푅 + = 푅/픭 ⊗푅 = 0 픮(푛+1) (⟨푎⟩ 픮(푛+1) ) ⟨푎⟩ + 픮(푛+1)

(푛) (푛+1) 푛 푛+1 thus by Nakayama’s lemma (3.2.12), 픮 = 픮 , and so 푅픮픮 = 푅픮픮 . Na- 푛 kayama’s lemma then implies 푅픮픮 = 0, so 픮푅픮 is nilpotent. Thus, 푅픮 must be an artinian local ring, so dim 푅픮 = 0. Thus, dim 푅 ≤ 1.

60 3. Noetherian local rings

Corollary 6.3.3. Let 푅 be a noetherian ring. If 푥1, … , 푥푐 are elements of 푅, then every minimal prime of 푅 containing ⟨푥1, … , 푥푐⟩ has codimension at most 푐.

Proof. We proceed by induction on 푐. Let 픭 be a minimal prime containing

⟨푥1, … , 푥푐⟩. By localising, we may assume without loss of generality that 픭 is maximal, and it suffices to show that dim 푅 ≤ 푐 in this case. As before, 픭 is the unique prime of 푅 containing ⟨푥1, … , 푥푐⟩, and 푅/⟨푥1, … , 푥푐⟩ is an artinian ring. Thus, 픭/⟨푥1, … , 푥푐⟩ is nilpotent in 푅/⟨푥1, … , 푥푐⟩. Suppose

픭0 ⊊ 픭1 ⊊ ⋯ ⊊ 픭푟 is a strictly ascending chain of prime ideals in 푅; we need to show 푟 ≤ 푐. Since 픭 is the unique maximal ideal of 푅, we may assume 픭푟 = 픭. Since 푅 is noetherian, we may also assume 픭푟−1 is maximal amongst the primes strictly contained in 픭. Clearly, it is enough to show that 픭푟−1 is minimal over an ideal generated by 푐 − 1 elements. Since 픭푟−1 ≠ 픭, {푥1, … , 푥푐} ⊈ 픭푟−1; say 푥1 ∉ 픭푟−1. Then, 픭 is a minimal prime over 픭푟−1 + ⟨푥1⟩, so 푅/(픭푟−1 + ⟨푥1⟩) is artinian. Thus, for 2 ≤ 푖 ≤ 푐 and 푛 푛 ≫ 0, 푥푖 = 푎푖푥1 + 푦푖 for some 푎푖 in 푅 and 푦푖 in 픭푟−1. Note that ⟨푥1, 푦2, … , 푦푐⟩ contains a power of 픭; but 푦2, … , 푦푐 are in 픭푟−1, and the image of 픭 is a minimal prime containing 푥1 in 푅/⟨푦2, … , 푦푐⟩, so 픭/⟨푦2, … , 푦푐⟩ has codimension at most 1 in 푅/⟨푦2, … , 푦푐⟩, by Krull’s Hauptidealsatz. Since 픭푟−1 ⊊ 픭푟, this implies 픭푟−1 is a minimal prime containing ⟨푦2, … , 푦푐⟩ as required. Corollary 6.3.4. Any noetherian local ring has finite Krull dimension.

Proof. Let 픪 be the maximal ideal of a noetherian local ring 푅. Since 푅 is noetherian, 픪 = ⟨푥1, … , 푥푐⟩ for some 푥1, … , 푥푐. But dim 푅 = codim 픪 and codim 픪 ≤ 푐 by the above result.

Remark 6.3.5. If 푅 is a noetherian local ring with maximal ideal 픪 and residue field 푘, then 2 dim 푅 ≤ dim푘 픪/픪 This implies that any prime ideal in any noetherian ring has finite codimension. Remark 6.3.6. Nagata [1962] showed that there is a noetherian ring 푅 with infinite Krull dimension. Of course, the local ringsof 푅 must have unboundedly large finite dimension.

61 VI. Dimension theory

4 Affine varieties

Lemma 6.4.1. Let 푘 be a field, and let 푛 be a natural number. If 픪 is a maximal ideal of 푘[푥1, … , 푥푛], then 픪 can be generated by 푛 elements.

Proof. By the weak Nullstellensatz, 푘[푥1, … , 푥푛]/픪 is a finite field extension of 푘. For 0 ≤ 푖 ≤ 푛, let 퐹푖 be the image of the subalgebra 푘[푥1, … , 푥푖] in 푘[푥1, … , 푥푛]/픪. Each 퐹푖 is a finite 푘-algebra and an integral domain, so must also be a finite field extension of 푘. We obtain a filtration

푘 = 퐹0 ⊆ 퐹1 ⊆ ⋯ ⊆ 퐹푛

and for 1 ≤ 푖 ≤ 푛, there is a maximal ideal 픪푖 of 퐹푖−1[푥푖] such that there is a canonical isomorphism 퐹푖 ≅ 퐹푖−1[푥푖]/픪푖 . But 퐹푖−1[푥푖] is a principal ideal domain, so 픪푖 = ⟨푓푖⟩ for some irreducible element 푓푖 in 퐹푖−1[푥푖]. We may lift each 푓푖 to 푘[푥1, … , 푥푖], and this yields a generating set of 푛 elements for 픪.

Corollary 6.4.2. The Krull dimension of 푘[푥1, … , 푥푛] is 푛.

Proof. Let 푅 = 푘[푥1, … , 푥푛]. It is clear that dim 푅 ≥ 푛, but we just showed that dim 푅픪 ≤ 푛 for all maximal ideals 픪 of 푅, so dim 푅 ≤ 푛. Lemma 6.4.3. Let 푅 be an integral domain of finite type over a field 푘. Then,

dim 푅 ≤ tr deg(Frac 푅 | 푘)

Proof. By the Noether normalisation lemma (5.3.2), 푅 contains a polynomial ring 푆 = 푘[푥1, … , 푥푛] and 푅 is finite over 푆. Thus, Frac 푅 is a finite extension of Frac 푆, and tr deg(Frac 푅 | 푘) = tr deg(Frac 푆 | 푘) = 푛. Now, if we have a strictly ascending chain of primes

픭0 ⊊ 픭1 ⊊ ⋯ 픭푟

in 푅, by the going-up theorem (5.2.22), we obtain a strictly ascending chain of primes

픭0 ∩ 푆 ⊊ 픭1 ∩ 푆 ⊊ ⋯ 픭푟 ∩ 푆 in 푆, hence 푟 ≤ dim 푆 = 푛.

Lemma 6.4.4. Let 푋 be an affine variety over a field 푘, i.e. 풪(푋) is an integral domain of finite type over 푘. If tr deg(Frac 풪(푋) | 푘) = 푛, 푔 ∈ 풪(푋), and 푔 ∉ 푘, then any irreducible component of 풱(⟨푔⟩) has transcendence degree exactly 푛−1.

62 4. Affine varieties

Proof. By replacing 푋 with 풟(ℎ) for a suitable ℎ in 풪(푋), we may assume 푌 = 풱(⟨푔⟩) is irreducible. We wish to show tr deg(Frac 풪(푌) | 푘) = 푛 − 1. 푛 First, assume 푋 = 픸푘. By the preparation lemma (5.3.1), we may assume 푔 is of the form 푒 푔 = 푐푥푛 + lower order terms × 푛−1 for some 푐 in 푘 and positive integer 푒. Thus, we get a finite morphism 푌 → 픸푘 and tr deg(Frac 풪(푌) | 푘) = 푛 − 1 as required. 푛 In general, there is only a finite morphism 푓 ∶ 푋 → 픸푘. Let ℎ be the 푛 1 morphism ⟨푓, 푔⟩ ∶ 푋 → 픸푘 ×푘 픸푘. The morphism ℎ is finite because 푓 is finite and 푔 is integral over 푘[푥1, … , 푥푛]. Let Φ be the monic polynomial that 푔 satisfies. Since 풪(푋) is an integral domain, we may assume Φ is irreducible. 푛+1 This implies 푋 is mapped into an irreducible hypersurface 푍 = 풱(⟨Φ⟩) in 픸푘 . 푛 We thus obtain a factorisation of 푓 ∶ 푋 → 픸푘 as a dominant morphism 푋 → 푍 푛 followed by a finite morphism 푍 → 픸푘. The hypersurface 푌 in 푋 is the inverse 푛+1 푑 푑−1 image of 푆 = 풱(⟨푥푛+1⟩) ∩ 푍 ⊂ 픸푘 . If Φ = 푥푛+1 + 푎푑−1푥푛+1 + ⋯ + 푎0, then | , but . So is isomorphic to a hypersurface 푎0|푆 = 0 푎0 ≠ 0 ∈ 푘[푥1, … , 푥푛] 푆 푛 in 픸푘, and as we showed above, tr deg(Frac 풪(푆) | 푘) = 푛 − 1. Theorem 6.4.5. Let 푋 be an affine variety over a field 푘. The following numbers are equal:

(i) The irreducible dimension of 푋.

(ii) The Krull dimension of 풪(푋).

(iii) The transcendence degree of Frac 풪(푋) over 푘.

(iv) The Krull dimension of 풪(푋)픪 for any maximal ideal 픪 of 풪(푋). Proof. We previously showed that

dim 풪(푋)픪 ≤ dim 풪(푋) ≤ tr deg(Frac 풪(푋) | 푘)

Fix a maximal ideal 픪. To prove the opposite inequality, we need to find a strictly increasing chain of irreducible closed subsets of 푋

{푚} = 푌0 ⊊ 푌1 ⊊ ⋯ ⊊ 푌푛 with 푛 = tr deg(Frac 풪(푋) | 푘). If 푛 = 0 then we are done. We proceed by induction on 푛. If 푌0 = 푋, then 풪(푋) would be a field and tr deg(풪(푋) | 푘) = 0; so if

63 VI. Dimension theory

푛 > 0 then 푌0 ≠ 푋. Thus we have a chain of length 1: 푌0 ⊊ 푋. By the preceding lemma, there is a irreducible closed subscheme 푌 with tr deg(Frac 풪(푌) | 푘) =

푛 − 1, and we can arrange for 푌0 ⊊ 푌 ⊊ 푋. The claim follows. Definition 6.4.6. A catenary ring is a ring 푅 with the following property: for any two prime ideals 픭 and 픮, every strictly ascending maximal chain of ideals starting at 픭 and ending at 픮 have the same length.

Example 6.4.7. If 푅 is a catenary noetherian local domain,[1] then

codim 픭 + dim 푅/픭 = dim 푅 for all prime ideals 픭 of 푅. We showed that every noetherian local ring has finite Krull dimension, but not every noetherian local ring is catenary: see [Nagata, 1962, p. 203] or [Reid, 1995, p. 148].

Theorem 6.4.8. (i) Any localisation of a catenary ring is catenary.

(ii) All algebras of finite type over a field 푘 are catenary. Proof. (i). Recall theorem 3.1.11: there is an inclusion-preserving bijective cor- −1 respondence between prime ideals in a localisation 푅[푆 ] and prime ideals of −1 푅 not meeting 푆. It is therefore clear that 푅[푆 ] is catenary if 푅 is catenary. (ii). Let 푅 be a 푘-algebra of finite type. Fix two prime ideals 픭 and 픮 of 푅, and suppose 픭 ⊆ 픮. We need to show that if dim 푅/픭 − dim 푅/픮 > 1, then ′ ′ there is a prime ideal 픭 such that 픭 ⊊ 픭 ⊊ 픮. Choose 푔 ∈ 푅/픭 such that 푔 ≠ 0 and 픮/픭 ⊆ ⟨푔⟩: we know every irreducible component of 풱(⟨푔⟩) has dimension ′ dim 푅/픭 − 1, so this yields the required 픭 . Noting that every ideal is contained in a prime ideal, induction then proves the claim.

5 Regular local rings

We previously saw that a noetherian local ring 푅 with unique maximal ideal 픪 and residue field 푘 has 2 dim 푅 ≤ dim푘 픪/픪 [1] More generally, one could assume that 푅 is a Cohen–Macaulay noetherian local ring: see [Eisenbud, 1995, Cor. 18.11].

64 5. Regular local rings

We now make the following definition:

Definition 6.5.1. A regular local ring is a local ring 푅 with unique maximal 2 ideal 픪 and residue field 푘 such that dim 푅 = dim푘 픪/픪 . The 푘-vector space 2 픪/픪 is called the Zariski cotangent space.

2 Example 6.5.2. Let 푅 = 푘 푥1, … , 푥푛 . An element of 픪 픪 can be [ ]⟨푥1,…,푥푛⟩ / 푛 thought of as the germ of a differential 1-form at the origin of 픸푘. In particular, 2 dim푘 픪/픪 = 푛 = dim 푅, so 푅 is a regular local ring.

Now, let 푋 be an affine scheme of finite type over a field 푘. We can embed 푋 푛 as a closed subscheme of 픸푘 for 푛 ≫ 0. Algebraically, we have a surjective homomorphism 푘[푥1, … , 푥푛] ↠ 풪(푋), thus there is an ideal ℐ(푋) of 푘[푥1, … , 푥푛] such that 풪(푋) ≅ 푘[푥1, … , 푥푛]/ℐ(푋).

Definition 6.5.3. Let 푋 be a scheme over a field 푘.A 푘-rational point of 푋 is a morphism of 푘-schemes Spec 푘 → 푋. We write 푋(푘) for the set of 푘-rational points of 푋.

When is the local ring of 푋 at a 푘-rational point 푃 a regular local ring?

Definition 6.5.4. Let 푋 be an affine scheme of finite type over a field 푘. We say 푋 is smooth of dimension 푛 just if 푋 has the following properties:

• All the irreducible components of 푋 have dimension 푛.

푛+푚 • Fixing a closed 푘-embedding 푋 ↪ 픸푘 , if the ideal ℐ(푋) is generated by , then the Jacobian matrix | 푓1, … , 푓푟 (휕푓푖/휕푥푗 | 1 ≤ 푖 ≤ 푟, 1 ≤ 푗 ≤ 푛 + 푚) has rank 푚 at each (not necessary 푘-rational!) point of 푋.

More explicitly, a matrix has rank at least 푚 if there is some 푚 × 푚 minor which is invertible; so we are demanding that the zero locus of all the 푚 × 푚 minors of the Jacobian matrix does not meet 푋, considered as subschemes of 푛+푚 픸푘 .

2 Example 6.5.5. Let 푋 = 풱(푥푦) ⊂ 픸푘. All the irreducible components of 푋 have dimension 1 by Krull’s Hauptidealsatz (theorem 6.3.2), so 푋 is smooth of dimension 1 if and only if the matrix

(푦 푥)

65 VI. Dimension theory has rank 1 everywhere on 푋, so if and only if 풱(푥, 푦) ∩ 푋 = ∅. But it is clear that 풱(푥, 푦) ∩ 푋 = {(0, 0)}, so 푋 is not smooth.

Lemma 6.5.6. Let 푋 be an affine scheme of finite type over 푘, and let 푃 be a

푘-rational point of 푋. Then, the local ring 풪푋,푃 is a regular local ring if and only if 푃 is in the smooth locus of 푋.

푛+푚 Proof. Let 푛 = dim 푋 and fix a closed 푘-embedding 휑 ∶ 푋 ↪ 픸푘 . Let 픪 be the maximal ideal of 풪(푋) corresponding to 푃. There is a natural isomorph- 2 2 ism of 푘-vector spaces 픪푃/픪푃 ≅ 픪/픪 , where 픪푃 is the unique maximal ♯ ideal of 풪푋,푃. Now, 픪 must be the image under 휑 ∶ 푘[푥1, … , 푥푛+푚] ↠ 풪(푋) 2 of the ideal 픫 = ⟨푥1 − 푎1, … , 푥푛+푚 − 푎푛+푚⟩ for some 푎1, … , 푎푛+푚 in 푘, and 픪 must be the image of 픫2. Let ℐ(푋) = ker 휑♯. The inverse image of 픪2 un- ♯ 2 2 2 2 ⊕(푛+푚) der 휑 is 픫 + ℐ(푋), hence 픫/(픫 + ℐ(푋)) ≅ 픪/픪 . But 픫/픫 ≅ 푘 , 2 2 so 픪/픪 is isomorphic to 픫/픫 modulo the subspace spanned by the vectors | 휕푓 휕푥 (푃) | 1 ≤ 푗 ≤ 푛 + 푚 1 ≤ 푖 ≤ 푟 , where ℐ(푋) = ⟨푓 , … , 푓 ⟩. Thus, {( 푖/ 푗 | ) | } 1 푟 풪푋,푃 is a regular local ring if and only if the matrix (휕푓푖/휕푥푗 ) (푃) has rank 푚, so if and only if 푋 is smooth of dimension 푛 at 푃.

1 What is 픸푘 if 푘 is not algebraically closed? It is immediate from the definition that we have a generic point ⟨0⟩ and closed points ⟨푓⟩ for each irreducible monic polynomial 푓.

Definition 6.5.7. A polynomial 푓 in one variable over a field is separable just if 푓 and its derivative 푓′ are coprime, or equivalently just if 푓 factors into distinct linear factors over the algebraic closure 푘.

Definition 6.5.8. A perfect field is a field 푘 such that either

• char 푘 = 0, or

• char 푘 = 푝 > 0 and every element is a 푝-th power.

Remark 6.5.9. If 푘 is perfect then every irreducible polynomial over 푘 is separable. Now, assume 푘 is perfect. Let 푓(푥) be an irreducible monic polynomial over 푘. Then, 푓(푥) = (푥 − 푎1) ⋯ (푥 − 푎푛) for some distinct 푎1, … , 푎푛 in 푘. By Galois theory, Gal(푘 | 푘) acts transitively on {푎1, … , 푎푛}, so the set is an orbit of 푘 under

66 5. Regular local rings

1 Gal(푘 | 푘). Thus, the set of closed points of 픸푘 are in natural bijection with the set of Galois orbits of 푘. Example 6.5.10. The monic irreducible polynomials over ℝ are either linear or quadratic, so

1 픸ℝ = {⟨0⟩} ∪ ℝ ∪ {푎 + 푏푖 | 푎 ∈ ℝ, 푏 ∈ ℝ, 푏 > 0} We may regard it as the closed upper-half plane.

Examples 6.5.11. For each prime number 푝, the local ring ℤ⟨푝⟩ is a regular local ring (and moreover a discrete valuation ring). The ring of 푝-adic integers ℤ푝 is also a regular local ring.

Similarly, the local ring 푘 푥1, … , 푥푛 is regular and is of dimension [ ]⟨푥1,…,푥푛⟩ 푛, as is the power series 푘 푥 , … , 푥 . J 1 푛K Theorem 6.5.12. Let 푋 be an affine scheme of finite type over a field 푘. If 푋 is smooth of dimension 푛 over 푘, then every local ring of 푋 is a noetherian regular local ring.

Notice that, for 푋 = Spec 푅 and 픭 ∈ Spec 푅, dim 푅픭 is equal to codim 픭 and may not be equal to dim 푅. What we mean here is that 푅픭 is a regular local ring of dimension codim 픭. Geometrically, this means that if 푋 is a smooth variety of dimension 푛 and 푌 is a closed subvariety of 푋, then a dense open subset of 푌 is a complete intersection. Theorem 6.5.13 (Auslander–Buchsbaum). Every noetherian regular local ring is a unique factorisation domain. In particular, it is a normal domain.

Proof. See [Eisenbud, 1995, Ch. 19, Ch. 20, Ex. 20.15]. Corollary 6.5.14. Let 푋 be smooth affine variety of dimension 푛 over 푘. If 푌 is a closed subvariety of codimension 1, then the local ring 풪푋,푌 is a discrete valuation ring. Lemma 6.5.15. If 푅 is a unique factorisation domain, then every prime ideal of codimension 1 is a principal ideal. Proof. Let 픭 be a prime ideal of codimension 1. The zero ideal is prime, so ⟨0⟩ ⊊ 픭 and there is no intermediate prime ideal. Let 푓 ∈ 픭. Suppose 푓 ≠ 0, and let

푓1 be an irreducible factor of 푓. Since ⟨푓1⟩ is a prime ideal and ⟨0⟩ ⊊ ⟨푓1⟩ ⊆ 픭, we must have ⟨푓1⟩ = 픭.

67 VI. Dimension theory

Corollary 6.5.16. Let 푋 be a smooth affine variety over a field 푘. If 푌 is a closed subvariety of codimension 1, then ideal of 푌 in 풪(푋) is locally principal. (It is the invertible sheaf 풪(−푌).)

Proof. Use theorem 6.5.12, theorem 6.5.13, and lemma 6.5.15.

68 Bibliography

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Freyd, Peter [1964] Abelian categories. An introduction to the theory of functors. Harper’s Series in Modern Mathematics. New York: Harper & Row Publishers, 1964. xi+164.

Lang, Serge [2002] Algebra. Third. Graduate Texts in Mathematics 211. New York: Springer-Verlag, 2002. xvi+914. isbn: 0-387-95385-X.

Nagata, Masayoshi [1962] Local rings. Interscience Tracts in Pure and Applied Mathematics 13. New York: Interscience Publishers, a division of John Wiley & Sons, 1962. xiii+234. isbn: 0-470-62865-0.

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