Lecture Notes Were Prepared for the Graduate Course Algebra III (ID: 011M4002Y) in Spring 2016, University of the Chinese Academy of Sciences

YANQI LAKE LECTURESON ALGEBRA

/ Wen-Wei Li

Part 3 Commutative rings Completions Dimension theory Yanqi Lake Lectures on Algebra

Part 3

Wen-Wei Li Chinese Academy of Sciences Version: 2019-06-14

The cover page uses the fonts Bebas Neue and League Gothique, both licensed under the SIL Open Font License.

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. To view a copy of this license, visit http:// creativecommons.org/licenses/by-nc/4.0/. Contents

Introduction 1

1 Warming up 3 1.1 Review on ring theory ...... 3 1.2 Localization of rings and modules ...... 5 1.3 Radicals and Nakayama’s lemma ...... 9 1.4 Noetherian and Artinian rings ...... 11 1.5 What is commutative algebra? ...... 12

2 Primary decompositions 15 2.1 The support of a module ...... 15 2.2 Associated primes ...... 16 2.3 Primary and coprimary modules ...... 18 2.4 Primary decomposition: the main theorem ...... 19 2.5 Examples and remarks ...... 21

3 Integral dependence, Nullstellensatz and flatness 25 3.1 Integral extensions ...... 25 3.2 Nullstellensatz ...... 28 3.3 Flatness: the first glance ...... 30 3.4 Structure of flat modules ...... 33 3.5 Faithful flatness and surjectivity ...... 36

4 Going-up, going-down, gradings and filtrations 39 4.1 Going-up and going-down ...... 39 4.2 Subsets in the spectrum ...... 42 4.3 Graded rings and modules ...... 45 4.4 Filtrations ...... 46 4.5 Theorems of Artin–Rees and Krull ...... 48

5 From completions to dimensions 51 5.1 Completions ...... 51 5.2 Further properties of completion ...... 55 5.3 Hilbert–Samuel polynomials ...... 56 5.4 Definition of Krull dimension ...... 58 5.5 Krull’s theorems and regularity ...... 63 ⋅ iv ⋅ CONTENTS

6 Dimension of finitely generated algebras 65 6.1 Dimensions in fibers ...... 65 6.2 Calculation for polynomial algebras ...... 67 6.3 Noether normalization and its consequences ...... 68

7 Serre’s criterion for normality and depth 73 7.1 Review of discrete valuation rings ...... 73 7.2 Auxiliary results on the total fraction ring ...... 74 7.3 On normality ...... 76 7.4 Serre’s criterion ...... 77 7.5 Introduction to depth ...... 78

8 Some aspects of Koszul complexes 83 8.1 Preparations in homological algebra ...... 83 8.2 Auxiliary results on depth ...... 85 8.3 Koszul complexes and depth ...... 86

Bibliography 91

Index 93 Introduction

In the beginning, these lecture notes were prepared for the graduate course Algebra III (ID: 011M4002Y) in Spring 2016, University of the Chinese Academy of Sciences. For some reasons, it took place in the Yuquanlu campus instead of the Yanqi Lake campus as it should be. The course is a sequel to Algebra I (fields, modules and representations) and II (homological algebra). The topic of this Part III is commutative algebra, or more precisely commutative ring theory. Each “Lecture” in these notes took roughly one week, say ap- proximately four hours of lecture, but the materials were only partially covered. My initial intention was to give a traditional course on commutative algebra as proposed by the syllabus prescribed by UCAS. For various reasons, my plan failed. For example, there are too few discussions on depth, regular sequences and Cohen–Macaulay modules, too few applications of completions, and the computational aspects haven’t been touched. Moreover, the homological aspect of commutative algebra is almost non- existent in these notes, namely the Auslander–Buchsbaum Formula, the properties of regular local rings, etc. Last but not least, the exercises herein are scarcely sufficient. These notes were also used for the Enhanced Program for Graduate Study held at the Beijing International Center of Mathematical Research, Peking University, during Spring 2019 (course ID: 00102057). As the title suggests, some backgrounds from the Part I are presumed, namely the basic notions of rings, modules and their chain conditions, as well as familiarity with tensor products and some Galois theory. We occasionally presume some basic knowl- 푖 edge of homological algebra, such as the functors Tor푖 and Ext . Sometimes I made free use of the language of derived categories. This was indeed covered in the preceding course Algebra II, and it should be common sense for the future generations. As the reader might have observed, these notes were prepared in a rush; certain paragraphs have not been proofread yet and many proofs are silly. I am very grateful to the students for various corrections and improvements, and I will try to polish these notes in the future.

Conventions

Throughout these lectures, we consider only associative rings with unit 1, and the rings and algebras are assumed to be commutative and nonzero unless otherwise specified. The ideal generated by elements 푥1, 푥2,… in a ring 푅 is denoted by (푥1, 푥2, …) or sometimes ⟨푥1, 푥2, …⟩; the 푅-algebra of polynomials in variables 푋, 푌, … with coefficients in ⋅ 2 ⋅ Introduction

푅 is denoted by 푅[푋, 푌, …]. We write 푅× for the group of invertible elements in a ring 푅. A ring without zero-divisors except 0 is called an integral domain, or simply a domain. The localization of a ring 푅 with respect to a multiplicative subset 푆 will be written as 푅[푆−1]. For any sets 퐸, 퐹, let 퐸 ∖ 퐹 ∶= {푥 ∈ 퐸 ∶ 푥 ∉ 퐹}. The cardinality of 퐸 is denoted by |퐸|. The usual logical connectives such as ∃, ∀, ∧, ∨ and so forth will occasionally be used. Writing 퐴 ∶= 퐵 means that the expression 퐴 is defined to be 퐵. We will use the standard notations ℤ, ℚ, ℝ, ℂ to denote the set of integers, of rational numbers, etc. Sans serif fonts are reserved for categories, such as Ab (abelian groups) and Ring. When denoting morphisms in a category by arrows, monomorphisms (resp. epi- morphisms, isomorphisms) will be indicated ↪ (resp. ↠, →∼ ).

Possible references

The reader is expected to have basic familiarity with groups, rings and modules, as covered in my lecture notes on Algebra I. We will make use of some really elementary homological algebra as our course proceeds — so keep calm. Our main references will be [11] and [8]. The Bourbaki volumes [5, 3] serve as our ultimate source. The readers are also encouraged to consult the relevant materials in Stacks Project.

《教授生涯》, 李桦, 木刻版画, 年. Lecture 1 Warming up

The reader might be familiar with most of the materials in this lecture. Our goal is to fix notation and present the basic structural results on Noetherian and Artinian rings or modules, including the celebrated Nakayama’s Lemma which will be used repeatedly.

1.1 Review on ring theory

Let 푅 be a ring, supposed to be commutative with unit 1 ≠ 0 as customary. Recall that an ideal 퐼 ⊊ 푅 is called ⋄ prime, if 푎푏 ∈ 퐼 ⟺ (푎 ∈ 퐼) ∨ (푏 ∈ 퐼); ⋄ maximal, if 퐼 is maximal among the proper ideals of 푅 with respect to inclusion. Recall the following standard facts

⋄ 퐼 is prime if and only if 푅/퐼 is an integral domain, i.e. has no zero divisors except 0;

⋄ 퐼 is maximal if and only if 푅/퐼 is a field; in particular, maximal ideals are prime;

⋄ every proper ideal 퐼 is contained in a maximal ideal (an application of Zorn’s Lemma).

Definition 1.1.1 (Local rings). The ring 푅 is called local if it has a unique maximal ideal, semi-local if it has only finitely many maximal ideals. Let 픪 be the maximal ideal of a local ring 푅. We call 푅/픪 the residue field of 푅.A local homomorphism between local rings 휑 ∶ 푅1 → 푅2 is a ring homomorphism such that 휑(픪1) ⊂ 픪2. Consequently, local homomorphisms induce embeddings on the level of residue fields.

Sometimes we denote a local ring by the pair (푅, 픪). Remark 1.1.2. Let 푅 be a local ring with maximal ideal 픪, then 푅× = 푅 ∖ 픪. The is easily seen as follows. An element 푥 ∈ 푅 is invertible if and only if 푅푥 = 푅. Note that 푅푥 = 푅 is equivalent to that 푥 is not contained in any maximal ideal, and the only maximal ideal is 픪. ⋅ 4 ⋅ Warming up

Throughout these lectures, we shall write

Spec(푅) ∶= {prime ideals of 푅}, MaxSpec(푅) ∶= {maximal ideals of 푅}.

They are called the spectrum and the maximal spectrum of 푅, respectively. The upshot is that Spec(푅) comes with a natural topology.

Definition 1.1.3 (Zariski topology). For any ideal 픞 ⊂ 푅, set 푉(픞) ∶= {픭 ∈ Spec(푅) ∶ 픭 ⊃ 픞}. Then there is a topology on Spec(푅), called the Zariski topology, whose closed subset are precisely 푉(픞), for various ideals 픞.

Indeed, we only have to prove the family of subsets {푉(픞) ∶ 픞 ⊂ 푅} is closed under finite union and arbitrary intersections. It boils down to the easy observation that 푉(픞) ∪ 푉(픟) = 푉(픞픟) (check this!) and ⋂픞∈풜 푉(픞) = 푉 (∑픞∈풜 픞), where 풜 is any family of ideals. −1 Given a ring homomorphism 휑 ∶ 푅1 → 푅2, if 퐼 ⊂ 푅2 is an ideal, then 휑 (퐼) ⊂ 푅1 is also an ideal.

Proposition 1.1.4. Given 휑 as above, it induces a continuous map

♯ 휑 ∶ Spec(푅2) ⟶ Spec(푅1) 픭 ⟼ 휑−1(픭) with respect to the Zariski topologies on spectra.

Proof. Clearly, 푎푏 ∈ 휑−1(픭) is equivalent to 휑(푎)휑(푏) ∈ 픭, which is in turn equivalent to (휑(푎) ∈ 픭) ∨ (휑(푏) ∈ 픭) when 픭 is prime. ♯ To show the continuity of 휑 , observe that for any ideal 픞 ⊂ 푅1 and 픭 ∈ Spec(푅2), −1 we have 휑 (픭) ⊃ 픞 if and only if 픭 ⊃ 휑(픞), i.e. 픭 ∈ 푉(휑(픞)푅2). Hence the preimage of closed subsets are still closed. More operations on spectra:

⋄ Take 푅1 to be a subring of 푅2 and 휑 be the inclusion map, the map above becomes 픭 ↦ 픭 ∩ 푅1. ⋄ Take 휑 ∶ 푅 ↠ 푅/퐼 to be a quotient homomorphism, then 휑−1 is the usual bijection from Spec(푅/퐼) onto 푉(퐼).

−1 ⋄ In general, 휑 does not induce MaxSpec(푅2) → MaxSpec(푅1), as illustrated in the case 휑 ∶ ℤ ↪ ℚ.

At this stage, we can prove a handy result concerning prime ideals.

Proposition 1.1.5 (Prime avoidance). Let 퐼 and 픭1, … , 픭푛 be ideals of 푅 such that 퐼 ⊂ 푛 ⋃푖=1 픭푖. Suppose that ⋄ either 푅 contains an infinite field, or ⋄ at most two of the ideals 픭1, … , 픭푛 are non-prime, then there exists 1 ≤ 푖 ≤ 푛 such that 퐼 ⊂ 픭푖. §1.2 Localization of rings and modules ⋅ 5 ⋅

Proof. If 푅 contains an infinite field 퐹, the ideals are automatically 퐹-vector subspaces 푟 of 푅. Since 퐼 = ⋃푖=1 퐼 ∩ 픭푖 whereas an 퐹-vector space cannot be covered by finitely many proper subspaces, there must exist some 푖 with 퐼 ∩ 픭푖 = 퐼. Under the second assumption, let us argue by induction on 푛 that ∀푖 퐼 ⊄ 픭푖 implies 푛 퐼 ⊄ ⋃푖=1 픭푖. The case 푛 = 1 is trivial. When 푛 ≥ 2, by induction we may choose, for 푛 each 푖, an element 푥푖 ∈ 퐼 ∖ ⋃푗≠푖 픭푗. Suppose on the contrary that 퐼 ⊂ ⋃푗=1 픭푗, then we would have 푥푖 ∈ 픭푖, for all 푖 = 1, … , 푛. When 푛 = 2 we have 푥1 +푥2 ∉ 픭1 ∪픭2 and 푥1 +푥2 ∈ 퐼, a contradiction. When 푛 > 2, we may assume 픭1 is prime, therefore

푛 푛 푥1 + ∏ 푥푗 ∉ ⋃ 픭푖, 푗=2 푖=1

again a contradiction.

Exercise 1.1.6. The following construction from [8, Exercise 3.17] shows that the assumptions of Proposition 1.1.5 cannot be weakened. Take 푅 = (ℤ/2ℤ)[푋, 푌]/(푋, 푌)2, which has a basis {1, 푋, 푌} (modulo (푋, 푌)2) as a ℤ/2ℤ-vector space. Show that the image 픪 of (푋, 푌) in 푅 is the unique prime ideal, and can be expressed as a union of three ideals properly contained in 픪.

1.2 Localization of rings and modules

Let 푆 be a multiplicative subset of 푅, which means that (a) 1 ∈ 푆, (b) 푆 is closed under multiplication, and (c) 0 ∉ 푆. The localization of 푅 with respect to 푆 is the ring 푅[푆−1] formed by classes [푟, 푠] with 푟 ∈ 푅, 푠 ∈ 푆, modulo the equivalence relation

[푟, 푠] = [푟′, 푠′] ⟺ ∃푡 ∈ 푆, (푟푠′ − 푟′푠)푡 = 0.

You should regard [푟, 푠] as a token for 푟/푠; the ring structure of 푅[푆−1] is therefore evident. In brief, localization amounts to formally inverting the elements of 푆, whence the notation 푅[푆−1]. Note that condition (c) guarantees 푅[푆−1] ≠ {0}.

Exercise 1.2.1. Given 푅 and 푆, show that 푟 ↦ 푟/1 yields a natural homomorphism 푅 → 푅[푆−1] and show that its kernel equals {푟 ∶ ∃푠 ∈ 푆, 푠푟 = 0}.

The universal property of 푅 → 푅[푆−1] can be stated using commutative diagrams as follows.

⎧ 푅 푅[푆−1] { 휑 ∶ 푅 → 푅′ ∶ ring homomorphism ∀ ⎨ , ∃! { ′ × 휑 ⎩ s.t. 휑(푆) ⊂ (푅 ) 푅′ ⋅ 6 ⋅ Warming up

Consequently, if 푆 ⊂ 푅× then 푅 ≃ 푅[푆−1] canonically. Furthermore, the homomorphism 푅 → 푅[푆−1] induces a bijection

Spec(푅[푆−1]) 1∶1 {픭 ∈ Spec(푅) ∶ 픭 ∩ 푆 = ∅} ⊂ Spec(푅)

픭푅[푆−1] = {푟/푠 ∶ 푟 ∈ 픭, 푠 ∈ 푆} 픭 (1–1)

픮 its preimage.

Exercise 1.2.2. Check the properties above.

Let us review some important instances of the localization procedure.

1. Take 푆 to be the subset of non zero-divisors of 푅. This is easily seen to be a multiplicative subset (check it!) and 퐾(푅) ∶= 푅[푆−1] is called the total fraction ring of 푅. The reader is invited to check that 푅 → 퐾(푅) is the “biggest localization” such that the natural homomorphism 푅 → 푅[푆−1] is injective. Hint: state this in terms of universal properties. When 푅 is an integral domain, we shall take 푆 ∶= 푅 ∖ {0}; in this case the total fraction ring Frac(푅) ∶= 퐾(푅) is the well-known field of fractions of 푅.

2. Take any 픭 ∈ Spec(푅) and 푆 ∶= 푅 ∖ 픭. From the definition of prime ideals, one infers that 푆 is a multiplicative subset of 푅. The corresponding localization is −1 denoted by 푅 → 푅픭 ∶= 푅[푆 ]. We see from (1–1) that

MaxSpec(푅픭) = {픭푅픭};

in particular, 푅픭 is a local ring with maximal ideal 픭푅픭. This is the standard way to produce local rings; we say that 푅픭 is the localization of 푅 at the prime 픭. 3. Suppose 푓 ∈ 푅 is not nilpotent, that is, 푓 푛 ≠ 0 for every 푛. Take 푆 ∶= {푓 푛 ∶ 푛 ≥ 0}. The corresponding localization is denoted by the self-explanatory notation 푅 → 푅[푓 −1].

Exercise 1.2.3. Describe the following localizations explicitly.

(a) 푅 = ℤ, and we localize at the prime ideal (푝) where 푝 is a prime number.

(b) 푅 = ℂ[푋1, … , 푋푛] and we localize at the maximal ideal generated by 푋1, … , 푋푛. (c) 푅 = ℂJ푋K (the ring of formal power series) and 푆 ∶= {푋푛 ∶ 푛 ≥ 0}.

Exercise 1.2.4. Prove that 푅[푋]/(푓 푋 − 1) →∼ 푅[푓 −1] by mapping 푋 ↦ 푓 −1. Hint: use the universal property.

Always let 푆 ⊂ 푅 be a multiplicative subset. The localization 푀[푆−1] of an 푅- module 푀 can be defined in the manner above, namely as the set of equivalence classes [푚, 푠] with 푚 ∈ 푀 and 푠 ∈ 푆, such that

[푚, 푠] = [푚′, 푠′] ⟺ ∃푡 ∈ 푆, 푡(푠′푚 − 푠푚′) = 0. §1.2 Localization of rings and modules ⋅ 7 ⋅

As in the case of rings, we shall write 푚/푠 instead of [푚, 푠]. It is an 푅[푆−1]-module, equipped with a natural homomorphism 푀 → 푀[푆−1] of 푅-modules. This yields a functor: for any homomorphism 푓 ∶ 푀 → 푁 we have a natural 푓 [푆−1] ∶ 푀[푆−1] → 푁[푆−1], mapping 푚/푠 to 푓 (푚)/푠; furthermore 푓 [푆−1] ∘ 푔[푆−1] = (푓 ∘ 푔)[푆−1] whenever composition makes sense. A slicker interpretation is to use the natural isomorphism 푀[푆−1] →∼ 푅[푆−1] ⊗ 푀 푅 which maps 푚/푠 to (1/푠) ⊗ 푚. Hereafter, we shall identify 푀[푆−1] and 푅[푆−1] ⊗ 푀 푅 without further comments. −1 In the same vein, we may define 푀[푓 ] and 푀픭, for non-nilpotent 푓 ∈ 푅 and 픭 ∈ Spec(푅) respectively. Localization “commutes” with several standard operation on modules, which we sketch below. The details are left to the reader.

⋄ For 푅-modules 푀, 푁, we have a natural isomorphism of 푅[푆−1]-modules

(푀 ⊗ 푁)[푆−1] ≃ 푀[푆−1] ⊗ 푁[푆−1], 푅 푅[푆−1] (푀 ⊕ 푁)[푆−1] ≃ 푀[푆−1] ⊕ 푁[푆−1].

−1 Same for arbitrary direct sums. This is easily seen by viewing 푀[푆 ] as 푀 ⊗푅 푅[푆−1].

⋄ Note that Hom푅(푀, 푁) is also an 푅-module: simply set (푟푓 )(푚) = 푟 ⋅ 푓 (푚) for any 푓 ∈ Hom푅(푀, 푁). There is a natural homomorphism

−1 −1 −1 Hom푅(푀, 푁)[푆 ] → Hom푅[푆−1] (푀[푆 ], 푁[푆 ]) (1–2)

sending 푠−1 ⊗ 휑 to 푠−1휑 ∶ 푚/푡 ↦ 휑(푚)/푠푡. It is clearly an isomorphism for 푀 = 푅, 푎 thus also for 푀 = 푅 where 푎 ∈ ℤ≥0, but this is not the case in general, as there is no uniform bound for the “denominators” for any given 휓 ∶ 푀[푆−1] → 푁[푆−1] — some finiteness condition is needed. Let us assume 푀 to be finitely presented, i.e. there is an exact sequence

푎 푏 푅 → 푅 → 푀 → 0, 푎, 푏 ∈ ℤ≥0.

In this case (1–2) is an isomorphism, as easily seen from the commutative diagram with exact rows:

0 Hom(푀, 푁)[푆−1] Hom(푅푏, 푁)[푆−1] Hom(푅푎, 푁)[푆−1]

≃ ≃

0 Hom(푀[푆−1], 푁[푆−1]) Hom(푅[푆−1]푏, 푁[푆−1]) Hom(푅[푆−1]푎, 푁[푆−1])

Here we used the fact that localization preserves exactness: see the Proposition 1.2.5 below.

⋄ Let 휑 ∶ 푅 → 푅′ be a ring homomorphism and 푆 ⊂ 푅 be a multiplicative subset, so that 푆′ ∶= 휑(푆) ⊂ 푅′ is also multiplicative. View 푅′ as an 푅-module, then we ⋅ 8 ⋅ Warming up

have ∼ ⎵⎵⎵⎵⎵푅′[(푆′)−1] ⎵⎵⎵푅′[푆−1] 푅[푆−1] ⊗ 푅′ 푅 as ring as module

푟′/휑(푠) (1/푠) ⊗ 푟′

푟′휑(푟)/휑(푠) (푟/푠) ⊗ 푟′

⋄ As a special case, take 휑 to be a quotient homomorphism 푅 → 푅/퐼, we get a natural isomorphism

푅[푆−1] (푅/퐼)[(푆′)−1] ≃ 푅[푆−1] ⊗ (푅/퐼) = 푅 퐼[푆−1]

from the right exactness of ⊗.

We prove an easy yet fundamental property of localizations, namely they are exact functors.

Proposition 1.2.5 (Exactness of localization). Let 푆 be any multiplicative subset of 푅. If

푓푖 ⋯ → 푀푖 푀푖+1 → ⋯ is an exact sequence of 푅-modules, then

−1 푓푖[푆 ] −1 −1 ⋯ → 푀푖[푆 ] 푀푖+1[푆 ] → ⋯ is also exact.

Proof. By homological common sense, we are reduced to the exact sequences (i) 0 → 푀′ → 푀 → 푀″ (i.e. left exactness), (ii) 푀′ → 푀 → 푀″ → 0 (i.e. right exactness). The case (ii) is known for tensor products in general. As for (i), note that for every homomorphism 푔 ∶ 푀 → 푀″,

푦 ∵푦/푠=푡푦/푡푠 푦 ker (푔[푆−1]) = { ∈ 푀[푆−1] ∶ ∃푡 ∈ 푆, 푡푔(푦) = 0} { ∶ 푦 ∈ ker(푔)} 푠 푠 = im [ker(푔)[푆−1] → 푀[푆−1]] .

Thus it remains to show that if 푓 ∶ 푀′ ↪ 푀, then 푓 [푆−1] ∶ 푀′[푆−1] → 푀[푆−1] is injective as well. If 푥/푠 ↦ 푓 (푥)/푠 = 0 under 푓 [푆−1], there exists 푡 ∈ 푆 such that 푡푓 (푥) = 푓 (푡푥) = 0 in 푀, therefore 푡푥 = 0 in 푀′, but the latter condition implies 푥/푠 = 0 in 푀′[푆−1].

Lemma 1.2.6. Let 푀 be an 푅-module. The localizations 푀 → 푀픪 for various maximal ideals 픪 assemble into an injection

푀 ↪ ∏ 푀픪. 픪∈MaxSpec(푅) §1.3 Radicals and Nakayama’s lemma ⋅ 9 ⋅

Proof. Let 푚 ∈ 푀 be such that 푚 ↦ 0 ∈ 푀픪 for all 픪. This means that for all 픪 there exists 푠 ∈ 푅 ∖ 픪 such that 푠푚 = 0. Hence the annihilator ideal ann푅(푚) ∶= {푟 ∈ 푅 ∶ 푟푚 = 0} is not contained in any maximal ideal, thus ann푅(푚) = 푅. There is an analogue for rings. Observe that when 푅 is an integral domain, all the localizations 푅[푆−1] can be regarded as subrings of the field of fractions Frac(푅).

Lemma 1.2.7. Let 푅 be an integral domain, then 푅 = ⋂픪∈MaxSpec(푅) 푅픪 as subrings of Frac(푅). Proof. Only the inclusion ⊃ requires proof. Let 푥 ∈ Frac(푅) and define 퐷 ∶= {푟 ∈ 푅 ∶ 푟푥 ∈ 푅} (the ideal of denominators). Suppose 푥 ∈ 푅픪 for all maximal 픪, then 퐷 ⊄ 픪 for all maximal 픪. The same reasoning as above leads to 퐷 = 푅. It will be important to gain finer control on the ideals 픪 in the assertion above, say by using some prime ideals “lower” than the maximal ones. We will return to this issue later.

1.3 Radicals and Nakayama’s lemma

We begin with two important notions of radicals. The first version is defined as follows. Given an ideal 퐼 of 푅, the nilpotent radical √퐼 is defined to be

√퐼 ∶= {푟 ∈ 푅 ∶ ∃푛, 푟푛 ∈ 퐼}.

2푛 2푛 2푛 푘 2푛−푘 It is readily seen to be an ideal from the binomial identity (푎+푏) = ∑푘=0 ( 푘 )푎 푏 . It should also be clear that √퐼 ⊂ 푅 equals the preimage of √0 ⊂ 푅/퐼.

Exercise 1.3.1. Show that √√퐼 = 퐼 for all ideal 퐼. Proposition 1.3.2. Let 퐼 be a proper ideal of 푅. We have

√퐼 = ⋂ 픭. 픭∈Spec(푅) 픭⊃퐼

Proof. By replacing 푅 by 푅/퐼, this is easily reduced to the case 퐼 = {0}. If 푟 is nilpotent and 픭 ∈ Spec(푅), then 푟푛 = 0 ∈ 픭 implies 푟 ∈ 픭. Conversely, suppose that 푟 is not nilpotent. There exists a prime ideal in 푅[푟−1], which comes from some 픭 ∈ Spec(푅) with 푟 ∈ 푅 ∖ 픭 by (1–1). Hence 푟 ∉ 픭.

Remark 1.3.3. A ring 푅 is called reduced if √0 = {0}. In any case, 푅red ∶= 푅/√0 is a reduced ring. Furthermore, any reduced quotient of 푅 factors through 푅red. The second radical is probably familiar to the readers. The Jacobson radical rad(푅) of the ring 푅 is the intersection of all maximal ideals. The previous proposition implies rad(푅) ⊃ √(0). Note that

푎 ∈ rad(푅) ⟹ (1 + 푎) ∈ 푅×. ⋅ 10 ⋅ Warming up

Indeed, 1+푎 cannot be contained in any maximal ideal 픪, for otherwise 1 = (1+푎)−푎 ∈ 픪, which is absurd. To prove the celebrated Nakayama’s Lemma, let us recall an easy variant of the Cayley–Hamilton theorem from linear algebra.

Lemma 1.3.4. Suppose that 퐼 ⊂ 푅 is an ideal, 푀 is an 푅-module with generators 푥1, … , 푥푛 and 푛 푛−1 휑 ∈ End푅(푀) satisfies 휑(푀) ⊂ 퐼푀, then there exists a polynomial 푃(푋) = 푋 +푎푛−1푋 + 푛 푛−1 ⋯ + 푎0 ∈ 푅[푋] with 푎푖 ∈ 퐼, such that 푃(휑) = 휑 + 푎푛−1휑 + ⋯ + 푎0 = 0. 푛 Proof. Write 휑(푥푖) = ∑푗=1 푎푖푗푥푗 where 푎푖푗 ∈ 퐼. Set 퐴 ∶= (푎푖푗)1≤푖,푗≤푛 ∈ Mat푛(푅). Regard 푀 as an 푅[푋]-module by letting 푋 act as 휑. Then we have the matrix equation over 푅[푋]

⎛푥1⎞ ⎛0⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (푋 ⋅ id푛×푛 − 퐴) ⎜ ⋮ ⎟ = ⎜⋮⎟ ⎜ ⎟ ⎜ ⎟ ⎝푥푛⎠ ⎝0⎠ ∨ Multiplying by the cofactor matrix (푋 ⋅ id푀 − 퐴) on the left, we see that 푃(푋) ∶= det(푋 ⋅ id푛×푛 − 퐴) ∈ 푅[푋] acts as 0 on each 푥푖, thus on the whole 푀. This is the required polynomial. Theorem 1.3.5 (Nakayama’s Lemma). Suppose that 푀 is a finitely generated 푅-module and 퐼 is an ideal of 푅 such that 퐼푀 = 푀. Then there exists 푎 ∈ 퐼 such that (1 + 푎)푀 = 0. If 퐼 ⊂ rad(푅), then we have 푀 = {0} under these assumptions.

Proof. Write 푀 = 푅푥1 + ⋯ + 푅푥푛. Plug 휑 = id푀 into Lemma 1.3.4 to deduce that 푃(id푀) = 1 + 푎⎵⎵⎵⎵⎵⎵⎵푛−1 + ⋯ + 푎0 acts as 0 on 푀. This proves the first part. Assume further- =∶푎∈퐼 more that 퐼 ⊂ rad(푅), then 1 + 푎 ∈ 푅× so that 푀 = (1 + 푎)푀 = 0.

Nakayama’s Lemma is named after Tadashi Nakayama (1912— 1964). Picture borrowed from [14].

Corollary 1.3.6. Let 푀 be a finitely generated 푅-module, and let 퐼 ⊂ rad(푅) be an ideal of 푅. If the images of 푥1, … , 푥푛 ∈ 푀 in 푀/퐼푀 form a set of generators, then 푥1, … , 푥푛 generate 푀. §1.4 Noetherian and Artinian rings ⋅ 11 ⋅

Proof. Apply Theorem 1.3.5 to 푁 ∶= 푀/(푅푥1 + ⋯ + 푅푥푛); our assumption 푀 = 퐼푀 + 푅푥1 + ⋯ + 푅푥푛 entails that 퐼푁 = 푁, thus 푁 = 0. We record another amusing consequence of Theorem 1.3.5.

Proposition 1.3.7. Let 푀 be a finitely generated 푅-module and 휓 ∈ End푅(푀). If 휓 is surjective then 휓 is an automorphism. Proof. Introduce a variable 푌. Make 푀 into an 푅[푌]-module by letting 푌 act as 휓. Put 퐼 ∶= (푌) so that 퐼푀 = 푀. Theorem 1.3.5 yields some 푄(푌) ∈ 푅[푌] satisfying (1 − 푄(푌)푌)푀 = 0, that is, 푄(휓)휓 = id푀.

1.4 Noetherian and Artinian rings

An 푅-module 푀 is called Noetherian (resp. Artinian) if every ascending (resp. descending) chain of submodules eventually stabilizes. Recall that in a short exact sequence 0 → 푀′ → 푀 → 푀″ → 0, we have 푀 is Noetherian (resp. Artinian) if and only if 푀′ and 푀″ are. Being both Noetherian and Artinian is equivalent to being a module of finite length, that is, a module admitting composition series. If we take 푀 ∶= 푅 on which 푅 acts by multiplication, then the submodules are precisely the ideals of 푅. We say that 푅 is a Noetherian (resp. Artinian) ring if 푅 as an 푅-module is Noetherian (resp. Artinian); this translates into the corresponding chain conditions on the ideals. Both chain conditions are preserved under passing to quotients and localizations. Finitely generated modules over a Noetherian ring are Noetherian. The following result ought to be known to the readers. Theorem 1.4.1 (Hilbert’s Basis Theorem). If 푅 is Noetherian, then so is the polynomial algebra 푅[푋1, … , 푋푛] for any 푛 ∈ ℤ≥1. Joint with the foregoing remarks, we infer that finitely generated algebras over Noetherian rings are still Noetherian. On the other hand, being Artinian is a rather stringent condition on rings. Theorem 1.4.2. A ring 푅 is Artinian if and only if 푅 is of finite length as an 푅-module. Such rings are semi-local. Proof. As noticed before, having finite length implies that 푅 is Noetherian as well as Artinian. Assume conversely that 푅 is an Artinian ring. First we claim that MaxSpec(푅) is finite, i.e. 푅 is semi-local. If there were an infinite sequence of distinct maximal ideals 픪1, 픪2,…, we would have an infinite chain

픪1 ⊃ 픪1픪2 ⊃ 픪1픪2픪3 ⊃ ⋯ .

This chain is strictly descending, since 픪1 ⋯ 픪푖 = 픪1 ⋯ 픪푖+1 would imply 픪푖+1 ⊃ 픪1 ⋯ 픪푖, hence 픪푖+1 ⊃ 픪푗 for some 1 ≤ 푗 ≤ 푖 because maximal ideals are prime. From the Artinian property we conclude that there are only finitely many maximal ideals 픪1, … , 픪푛 of 푅. 푘 푘+1 Set 픞 ∶= 픪1 ⋯ 픪푛. Since 푅 is Artinian we must have 픞 = 픞 for some 푘 > 0. We claim that 픞푘 = 0. ⋅ 12 ⋅ Warming up

Put 픟 ∶= {푟 ∈ 푅 ∶ 푟픞푘 = 0}, we have to show 픟 = 푅. If not, let 픟′ be a minimal ideal lying strictly over 픟. Thus 픟′ = 푅푥 + 픟 for any 푥 ∈ 픟′ ∖ 픟. We must have 픞푥 + 픟 ⊊ 픟′, for otherwise 푀 ∶= 픟′/픟 is finitely generated (say by 푥) and satisfies 푀 = 픞푀, then Theorem 1.3.5 plus 픞 ⊂ rad(푅) would imply 푀 = {0}, which is absurd. By minimality we have 픟 = 픞푥 + 픟. It follows that 픞푥 ⊂ 픟, i.e. 픞푘+1푥 = 0; from 픞푘 = 픞푘+1 we infer 푥 ∈ 픟. Contradiction. All in all, we obtain a descending chain of ideals

푅 ⊃ 픪1 ⊃ 픪1픪2 ⊃ ⋯ ⊃ 픪1 ⋯ 픪푛 = 픞 2 ⊃ 픞픪1 ⊃ 픞픪1픪2 ⊃ ⋯ ⊃ 픞픪1 ⋯ 픪푛 = 픞 ⊃ ⋯ ⊃ 픞푘 = {0}.

Each subquotient thereof, which is a priori an 푅-module, is actually an 푅/픪푖-vector space for some 1 ≤ 푖 ≤ 푛. Such a vector space must also satisfy the descending chain condition on vector subspaces, otherwise pulling-back will contradict the Artinian assumption on 푅. Artinian vector spaces must be finite-dimensional. It follows that all these subquotients are of finite length, hence so is 푅 itself.

Corollary 1.4.3. A ring 푅 is Artinian if and only if it is Noetherian and every prime ideal of 푅 is maximal.

Proof. If 푅 is Artinian, then 푅 is of finite length, hence is Noetherian as well. For every 푘 prime ideal 픭, we have 픭 ⊃ {0} = (픪1 ⋯ 픪푛) in the notations of the proof above, therefore 픭 ⊃ 픪푖 for some 푖, so 픭 = 픪푖 is maximal. Conversely, if 푅 is Noetherian but of infinite length, then the nonempty set of ideals

풮 ∶= {ideals 퐼 ⊊ 푅 ∶ 푅/퐼 has infinite length}

contains a maximal element 픭. We contend that 픭 is prime. If 푥푦 ∈ 픭 with 푥, 푦 ∉ 픭, then 푅/(픭 + 푅푥) has finite length by the choice of 픭; on the other hand, 픞 ∶= {푟 ∈ 푅 ∶ 푟푥 ∈ 픭} ⊋ 픭 (as 푦 ∈ 픞), hence 푅/픞 has finite length as well. From the short exact sequence 푥 0 → 푅/픞 푅/픭 → 푅/(픭 + 푅푥) → 0 we see 푅/픭 has finite length, contradiction. If we assume moreover that every prime ideal is maximal, then for the 픭 chosen above, 푅/픭 will be a field, thus of finite length. This is impossible.

Rings whose prime ideals are all maximal are said to have dimension zero, in the sense of Krull dimensions; we shall return to this point in §5.4.

1.5 What is commutative algebra?

In broad terms, commutative algebra is the study of commutative rings. Despite its in- trinsic beauty, we prefer to motivate from an external point of view. See also [8]. (A) Algebraic geometry. To simplify matters, we consider affine algebraic varieties over an algebraically closed field 한. Roughly speaking, such a variety is the zero 푛 푛 locus 풳 = {푓1 = ⋯ = 푓푟 = 0} in 픸 = 한 of 푓푖 ∈ 한[푋1, … , 푋푛]. The choice of equations is of course non-unique: what matters is the ideal 퐼(풳) ∶= {푓 ∈ 한[푋1, … , 푋푛] ∶ 푓 |풳 = 0}. §1.5 What is commutative algebra? ⋅ 13 ⋅

Conversely, every ideal 픞 defines a subset 푉(픞) ∶= {푥 ∈ 한푛 ∶ ∀푓 ∈ 픞, 푓 (푥) = 0}. As consequences of Hilbert’s Nullstellensatz, which we will discuss later, we have

푉 ∘ 퐼(풳) = 풳, 퐼 ∘ 푉(픞) = √픞.

One can deduce from this that the (closed) subvarieties of 픸푛 are in bijection with ideals 픞 satisfying √픞 = 픞. Furthermore, 한[풳] ∶= 한[푋1, … , 푋푛]/퐼(풳) may be regarded as the ring of “regular functions” (i.e. functions definable by means of polynomials) on 풳, and MaxSpec(한[풳]) is in bijection with the points of 풳: to 푥 = (푥1, … , 푥푛) ∈ 풳 we attach

픪푥 = (푋1 − 푥1, … , 푋푛 − 푥푛) ⊃ 퐼(풳).

By passing to the ring 한[풳], we somehow obtain a description of 풳 that is independent of embeddings into affine spaces. Moreover, 풳 inherits the Zariski topology from that on Spec(한[풳]). Naively speaking, the geometric properties of 풳 transcribe in ring-theoretic terms to the reduced Noetherian 한-algebra 한[풳]. For example, assume that 푓 ∈ 한[풳] is not nilpotent, then the formation of 한[풳][푓 −1] corresponds to taking the Zariski-open subset 풳푓 = {푥 ∈ 풳 ∶ 푓 (푥) ≠ 0} of 풳. This may be explained as follows:

풳푓 ≃ {(푥1, … , 푥푛, 푦) ∶ 푓1(푥1, … , 푥푛) = ⋯ = 푓푟(푥1, … , 푥푛) = 0, 푓 (푥1, … , 푥푛)푦 = 1}

푛+1 which is also an affine algebraic variety in 한 , and one may verify that 한[풳푓 ] = 한[풳][푓 −1]. (B) Invariant theory. Let 퐺 be a group acting on a finite-dimensional 한-vector space 푉 from the right, and let 한[푉] be the 한-algebra of polynomials on 푉. Thus 한[푉] carries a left 퐺-action by 푔푓 (푣) = 푓 (푣푔). For “reasonable” groups 퐺, say finite or 한- algebraic ones, the classical invariant theory seeks to describe the subalgebra 한[푉]퐺 of invariants1 in terms of generators and relations. In particular, one has to know when is the algebra 한[푉]퐺 finitely generated. This is actually the source of many results in commutative algebra, such as the Basis Theorem and Nullstellensatz of Hilbert. For example, let the symmetric group 퐺 = 픖푛 act on 푉 = 한푛 in the standard manner, then our question is completely answered by the 퐺 following classical result: 한[푉] equals the polynomial algebra 한[푒1, … , 푒푛], where 푒푖 stands for the 푖-th elementary symmetric function in 푛 variables. The same questions may be posed for any affine algebraic variety 푉. From the geometric point of view, if 한[푉]퐺 is finitely generated, it will consist of regular functions of some kind of quotient variety 푉//퐺. The study of quotients in this sense naturally leads to geometric invariant theory, for which we refer to [16] for details.

1More generally, we are also interested in the algebra of invariant differential operators with polynomial coefficients. ⋅ 14 ⋅ Warming up Lecture 2

Primary decompositions

We shall follow [8, §3] and [11, §8] closely.

2.1 The support of a module

Fix a ring 푅. For any 푅-module 푀 and 푥 ∈ 푀, we define the annihilator ann푅(푥) ∶= {푟 ∈ 푅 ∶ 푟푥 = 0}; it is an ideal of 푅. Also define ann푅(푀) ∶= {푟 ∈ 푅 ∶ 푟푀 = 0} = ⋂푥∈푀 ann푅(푥). Definition 2.1.1. The support of an 푅-module 푀 is

Supp(푀) ∶= {픭 ∈ Spec(푅) ∶ 푀픭 ≠ 0} .

Let us unwind the definition: 픭 ∉ Supp(푀) means that every 푥 ∈ 푀 maps to 0 ∈ 푀픭, equivalently ann푅(푥) ⊄ 픭. To test whether 픭 ∉ Supp(푀), we only need to check the foregoing condition for 푥 ranging over a generating set of 푀.

Proposition 2.1.2. If 푀 is finitely generated then Supp(푀) = 푉(ann푅(푀)); in particular it is Zariski-closed in Spec(푅).

푛 Proof. Suppose 푀 = 푅푥1 + ⋯ + 푅푥푛. Set 퐼푖 ∶= ann푅(푥푖) and observe that ⋂푖=1 퐼푖 = 푛 ann푅(푀). Then 픭 ∈ Supp(푀) if and only if 픭 ⊃ 퐼푖 for some 푖, that is, 픭 ∈ ⋃푖=1 푉(퐼푖). To conclude, note that 푉(퐼) ∪ 푉(퐽) = 푉(퐼퐽) = 푉(퐼 ∩ 퐽) for any ideals 퐼, 퐽 ⊂ 푅 (an easy exercise).

Finite generation is needed in the result above. Consider the ℤ-module 푀 ∶= 푎 ⨁푎≥1 ℤ/푝 ℤ. Its support is {푝} but ann(푀) = {0}.

Proposition 2.1.3. For an exact sequence 0 → 푀′ → 푀 → 푀″ → 0 we have Supp(푀) = ′ ″ Supp(푀 ) ∪ Supp(푀 ). For arbitrary direct sums we have Supp(⨁푖 푀푖) = ⋃푖 Supp(푀푖). Proof. Again, we use the exactness of localization for the first assertion. For 픭 ∈ Spec(푅) ′ ″ we have an exact 0 → 푀픭 → 푀픭 → 푀픭 → 0, hence 푀픭 ≠ 0 if and only if 픭 ∈ Supp(푀′) ∪ Supp(푀″). The second assertion is obvious. ⋅ 16 ⋅ Primary decompositions

From an 푅-module 푀, one can build a “field of modules” over Spec(푅) by assigning to each 픭 the 푅픭-module 푀픭, and Supp(푀) is precisely the subset of 푀픭 over which the field is non-vanishing. This is how the support arises in algebraic geometry. Amore precise description will require the notion of quasi-coherent sheaves on schemes.

Exercise 2.1.4. Let 푀, 푁 be finitely generated 푅-modules. Show that Supp(푀 ⊗ 푁) = 푅 Supp(푀) ∩ Supp(푁). Hint: since localization commutes with ⊗, it suffices to prove that 푀 ⊗ 푁 ≠ {0} when 푀, 푁 are both nonzero finitely generated modules over a local 푅 ring 푅. Nakayama’s Lemma implies that 푀 ⊗ 한 and 한 ⊗ 푁 are both nonzero where 한 푅 푅 is the residue field of 푅. Now

(푀 ⊗ 한) ⊗ (한 ⊗ 푁) ≃ 푀 ⊗ (한 ⊗ 한) ⊗ 푁 ≃ (푀 ⊗ 푁) ⊗ 한 푅 한 푅 푅 한 푅 푅 푅 is nonzero.

2.2 Associated primes

Throughout this section, 푅 will be a Noetherian ring.

Definition 2.2.1. A prime ideal 픭 is said to be an associated prime of an 푅-module 푀 if ann푅(푥) = 픭 for some 푥 ∈ 푀; equivalently, 푅/픭 embeds into 푀. Denote the set of associated primes of 푀 by Ass(푀).

Example 2.2.2. For the ℤ-module 푀 ∶= ℤ/푛ℤ with 푛 ∈ ℤ>1, one easily checks that Ass(푀) is the set of prime factors of 푛.

Lemma 2.2.3. Consider the set 풮 ∶= {ann푅(푥) ∶ 푥 ∈ 푀, 푥 ≠ 0} of ideals, partially ordered by inclusion. Every maximal element in 풮 is prime.

Proof. Let 픭 = ann푅(푥) be a maximal element of 풮 and suppose 푎푏 ∈ 픭. If 푏 ∉ 픭, then

푏푥 ≠ 0, 푎푏푥 = 0, 푅 ≠ ann푅(푏푥) ⊃ ann푅(푥) = 픭.

Hence 푎 ∈ ann푅(푏푥) = 픭 by the maximality of 픭 in 풮. Definition 2.2.4. Call 푟 ∈ 푅 a zero divisor on 푀 if 푟푥 = 0 for some 푥 ∈ 푀 ∖ {0}.

Theorem 2.2.5. Let 푀 be an 푅-module.

(i) We have 푀 = {0} if and only if Ass(푀) = ∅.

(ii) The union of all 픭 ∈ Ass(푀) equals the set of zero divisors on 푀.

(iii) For any multiplicative subset 푆 ⊂ 푅, we have

Ass(푀[푆−1]) = {픭[푆−1] ∶ 픭 ∈ Ass(푀), 픭 ∩ 푆 = ∅} .

(iv) If 0 → 푀′ → 푀 → 푀″ is exact, then Ass(푀′) ⊂ Ass(푀) ⊂ Ass(푀′) ∪ Ass(푀″). §2.2 Associated primes ⋅ 17 ⋅

Proof. (i) Clearly Ass({0}) = ∅. If 푀 ≠ 0, the set 풮 in Lemma 2.2.3 is then nonempty, hence contains a maximal element 픭 because 푅 is Noetherian; this yields 픭 ∈ Ass(푀). (ii) Elements of any 픭 ∈ Ass(푀) are all zero divisors by the very definition of associated primes. Conversely, if 푟 ∈ ann푅(푥) for some 푥 ∈ 푀 ∖ {0}, there must exist some maximal element 픭 of 풮 with 픭 ⊃ ann푅(푥) as 푅 is Noetherian; so 픭 is the required associated prime containing 푟. (iii) If 픭 ∈ Spec(푅), 픭 ∩ 푆 = ∅ and there is some 푅/픭 ↪ 푀, then

{0} ≠ 푅[푆−1]/픭[푆−1] ≃ (푅/픭)[푆−1] ↪ 푀[푆−1]

by the exactness of localizations, hence 픭[푆−1] ∈ Ass(푀[푆−1]). Conversely, every element of Ass(푀[푆−1]) has the form 픭[푆−1] for some 픭 ∈ Spec(푅) disjoint from 푆. Also recall that 픭 equals the preimage of 픭[푆−1] under 푅 → 푅[푆−1]. There exist 푥 ∈ 푀 and −1 푠 ∈ 푆 such that 픭[푆 ] = ann푅[푆−1](푥/푠) = ann푅[푆−1](푥). Ideals in a Noetherian ring being finitely generated, we infer that ∃푡 ∈ 푆 with 픭 ⊂ ann푅(푡푥). It remains to show −1 픭 = ann푅(푡푥). If 푟푡푥 = 0 for some 푟 ∈ 푅, then 푟 maps into 푟/1 ∈ 픭[푆 ] = ann푅[푆−1](푥); thus 푟 ∈ 픭. (iv) It suffices to treat the second ⊂. Suppose that 픭 ∈ Ass(푀) and 푅/픭 ≃ 푁 for some submodule 푁 ⊂ 푀. Identify 푀′ with ker(푀 → 푀″). If 푀′ ∩ 푁 = {0} then 푅/픭 ≃ 푁 ↪ 푀″, so 픭 ∈ Ass(푀″). If there exists 푥 ∈ 푀′ ∩ 푁 ⊂ 푁 with 푥 ≠ 0, then we ′ have ann푅(푥) = 픭 since 푁 ≃ 푅/픭 and 픭 is prime; in this case 픭 ∈ Ass(푀 ).

We remark that 0 → 푀′ → 푀 → 푀″ → 0 being exact does not imply Ass(푀) = 푝 Ass(푀′) ∪ Ass(푀″). To see this, consider 푅 = ℤ and 0 → ℤ ℤ → ℤ/푝ℤ → 0 for some prime number 푝.

Exercise 2.2.6. Show that Ass(푀1 ⊕ 푀2) = Ass(푀1) ∪ Ass(푀2).

Theorem 2.2.7. For every 푅-module 푀 we have Supp(푀) = ⋃픭∈Ass(푀) 푉(픭), in particular Ass(푀) ⊂ Supp(푀). Furthermore, every minimal element of Supp(푀) with respect to inclusion is actually a minimal element of Ass(푀).

Proof. For any prime 픮 we have 푀픮 ≠ 0 ⟺ Ass(푀픮) ≠ ∅, and the latter condition holds precisely when there exists 픭 ∈ Ass(푀) with 픭 ∩ (푅 ∖ 픮) = ∅, i.e. 픮 ⊃ 픭. This proves the first assertion. The second assertion is a direct consequence.

Observe that if 픭 ∈ Supp(푀), then 픮 ⊃ 픭 ⟹ 픮 ∈ Supp(푀): the reason is that

푀 = (푀 ) 픭 픮 픭푅픮 (2–1)

thus the occurrence of non-minimal elements in Supp(푀) is unsurprising. In contrast, the non-minimal elements in Ass(푀) are somehow mysterious. These non-minimal associated primes are called embedded primes.

Exercise 2.2.8. Prove the formula (2–1) of “localization in stages”.

Hereafter we impose finite generation on 푀. This implies 푀 is Noetherian. ⋅ 18 ⋅ Primary decompositions

Theorem 2.2.9. Let 푀 be a finitely generated 푅-module. There exists a chain 푀 = 푀푛 ⊃ 푀푛−1 ⊃ ⋯ ⊃ 푀0 = {0} of submodules such that for every 0 < 푖 ≤ 푛, the subquotient 푀푖/푀푖−1 is isomorphic to 푅/픭푖 for some prime ideal 픭푖. Furthermore we have Ass(푀) ⊂ {픭1, … , 픭푛}; in particular Ass(푀) is a finite set.

Proof. Assume 푀 ⊋ 푀0 ∶= {0}. There exists 픭1 ∈ Ass(푀) together with a submodule 푀1 ⊂ 푀 isomorphic to 푅/픭1. Furthermore Ass(푀1) = Ass(푅/픭1) = {픭1} as easily seen. Hence the Theorem 2.2.5 entails Ass(푀) ⊂ {픭1} ∪ Ass(푀/푀1). If 푀1 = 푀 we are done. Otherwise we start over with 푀/푀1, finding 푀1 ⊂ 푀2 ⊂ 푀 with 푀2/푀1 ≃ 푅/픭2 where 픭2 ∈ Ass푅(푀/푀1), and so forth. This procedure termi- nates in finite steps since 푀 is Noetherian.

2.3 Primary and coprimary modules

The classical framework of primary decompositions concerns ideals, but it is advanta- geous to allow modules here. As before, the ring 푅 is Noetherian.

Definition 2.3.1. An 푅-module 푀 is called coprimary if Ass(푀) is a singleton. A submodule 푁 ⊊ 푀 is called a 픭-primary submodule if 푀/푁 is coprimary with associated prime 픭 ∈ Spec(푅).

Proposition 2.3.2. The following are equivalent for a nonzero 푅-module 푀.

(i) 푀 is coprimary;

(ii) for every zero divisor 푟 ∈ 푅 for 푀 and every 푥 ∈ 푀, there exists 푛 ≥ 1 such that 푟푛푥 = 0.

The condition (ii) is usually called the local-nilpotency of 푟 on 푀. When 푀 = 푅/퐼 for some ideal 퐼, it translates into: all zero divisors of the ring 푅/퐼 are nilpotent. Proof. (i) ⟹ (ii): Suppose Ass(푀) = {픭} and 푥 ∈ 푀 ∖ {0}. From ∅ ≠ Ass(푅푥) ⊂ Ass(푀) we infer that Ass(푅푥) = {픭}, thus by Theorem 2.2.7 we see 푉(픭) = Supp(푅푥) = 푉(ann(푅푥)). Hence 픭 = √ann(푅푥). This implies (ii) by the definition of √ . (ii) ⟹ (i): It is routine to check that

픭 ∶= {푟 ∈ 푅 ∶ ∀푥 ∈ 푀 ∃푛 ≥ 1, 푟푛푥 = 0}

is an ideal of 푅. For every 픮 ∈ Ass(푀) there exists 푥 ∈ 푀 with ann푅(푥) = 픮. Every 푟 ∈ 픭 has some power falling in 픮, thus 픭 ⊂ 픮. Conversely, (ii) and Theorem 2.2.5 imply 픮 = ann푅(푥) ⊂ 픭. From 픮 = 픭 we conclude 푀 is coprimary with the unique associated prime 픭.

Exercise 2.3.3 (Classical definition of primary ideals). Let 퐼 be a proper ideal of 푅. Show that 푅/퐼 is coprimary if and only if the following holds:

∀푎, 푏 ∈ 푅, (푎푏 ∈ 퐼) ∧ (푎 ∉ 퐼) ⟹ ∃푛 ≥ 1, 푏푛 ∈ 퐼.

In this case we also say 퐼 is a primary ideal of 푅. Show that {√퐼} = Ass(푅/퐼) if 퐼 is a primary ideal. Hint: apply Proposition 2.3.2. §2.4 Primary decomposition: the main theorem ⋅ 19 ⋅

Exercise 2.3.4. Let 픪 be a maximal ideal of 푅. Show that every ideal 퐼 ⊊ 푅 containing some power of 픪 is primary, and Ass(푅/퐼) = {픪}. Hint: show that 픪 is the only prime ideal containing 퐼 = ann푅(푅/퐼).

Lemma 2.3.5. Let 픭 ∈ Spec(푅) and 푁1, 푁2 ⊂ 푀 are 픭-primary submodules. Then 푁1 ∩ 푁2 is a 픭-primary submodule of 푀.

Proof. We have 푀/푁1 ∩ 푁2 ↪ 푀/푁1 ⊕ 푀/푁2. Since 푁1 ∩ 푁2 ≠ 푀, we have

∅ ≠ Ass(푀/푁1 ∩ 푁2) ⊂ Ass(푀/푁1) ∪ Ass(푀/푁2) = {픭} by Theorem 2.2.5.

2.4 Primary decomposition: the main theorem

We still assume 푅 Noetherian and fix a finitely generated 푅-module 푀. Theorem 2.4.1 (Lasker–Noether). Let 푁 ⊊ 푀 be an 푅-submodule. Then we can express 푁 as 푁 = 푀1 ∩ ⋯ ∩ 푀푛, for some 푛 ≥ 1 and primary 푅-submodules 푀푖, say with Ass(푀/푀푖) = {픭푖} for 푖 = 1, … , 푛. Such a decomposition is called a primary decomposition of 푁. We say it is irredundant if none of the 푀푖 can be dropped, and minimal if there is no such decomposition with fewer terms.

(i) We have Ass(푀/푁) ⊂ {픭1, … , 픭푛}, equality holds when the decomposition is irredundant. (ii) If the decomposition is minimal, then for every 픭 ∈ Ass(푀/푁) there exists a unique 1 ≤ 푖 ≤ 푛 with 픭 = 픭푖; consequently 푛 = |Ass(푀/푁)|. (iii) Consider a primary decomposition of 푁. Let 푆 ⊂ 푅 be any multiplicative subset, and assume without loss of generality that 픭1, … , 픭푚 are the primes among {픭1, … , 픭푛} which are disjoint from 푆, then 푚 ≥ 1 ⟺ 푁[푆−1] ⊊ 푀[푆−1] and

−1 −1 −1 푁[푆 ] = 푀1[푆 ] ∩ ⋯ ∩ 푀푚[푆 ] is a primary decomposition of the 푅[푆−1]-submodule 푁[푆−1] ⊊ 푀[푆−1]; this decomposition of 푁[푆−1] is minimal if the one for 푁 is. Note that the “irredundant” condition in [11, (8.D)] corresponds to minimality here. The rings whose ideals all have primary decompositions are called Laskerian rings, thus part (i) of the Theorem says Noetherian implies Laskerian, but there exist non- Noetherian examples. Proof. Establish the existence of primary decompositions first. Replacing 푀 by 푀/푁, we may assume 푁 = {0} from the outset. We claim that ⎧ { 푄(픭) is 픭-primary, ∀픭 ∈ Ass(푀), ∃푄(픭) ⊂ 푀, ⎨ (2–2) { ⎩ Ass(푄(픭)) = Ass(푀) ∖ {픭}. ⋅ 20 ⋅ Primary decompositions

Granting this, 푄 ∶= ⋂픭∈Ass(푀) 푄(픭) yields the required decomposition since Ass(푀) is finite and Ass(푄) = ∅. Establish (2–2) as follows. Put 훹 ∶= {픭}. By Zorn’s Lemma we get a maximal element 푄(픮) from the set

∅ ≠ {푄 ⊂ 푀 ∶ submodule, Ass(푄) ⊂ Ass(푀) ∖ 훹}

which is partially ordered by inclusion (details omitted, and you may also use the Noetherian property of 푀). Since

Ass(푀) ⊂ Ass(푀/푄(픭)) ∪ Ass(푄(픭)),

it suffices to show Ass(푀/푄(픭)) ⊂ 훹. Let 픮 ∈ Ass(푀/푄(픭)) so that there exists 푀 ⊃ 푄′ ⊃ 푄(픭) with 푄′/푄(픭) ≃ 푅/픮. Since Ass(푄′) ⊂ Ass(푄(픭)) ∪ {픮}, maximality forces 픮 ∈ 훹 (otherwise Ass(푄′) ⊂ Ass(푀) ∖ 훹), whence (2–2). Now we turn to the properties (i) — (iii). (i) The obvious embedding

푛 푀/푁 ↪ ⨁ 푀/푀푖 푖=1

푛 together with Theorem 2.2.5 yield Ass(푀/푁) ⊂ ⋃푖=1 Ass(푀/푀푖) = {픭1, … , 픭푛}. Now assume the given primary decomposition is irredundant, we have

푀 ∩ ⋯ ∩ 푀 푀 ∩ ⋯ ∩ 푀 {0} ≠ 2 푛 = 2 푛 푁 푀1 ∩ (푀2 ∩ ⋯ ∩ 푀푛) 푀1 + 푀2 ∩ ⋯ ∩ 푀푛 ≃ ↪ 푀/푀1. 푀1

Thus Ass(푀/푁) contains Ass((푀2 ∩ ⋯ ∩ 푀푛)/푁) = {픭1}. Same for 픭2, … , 픭푛. (ii) Suppose 푁 = 푀1 ∩ ⋯ ∩ 푀푛 is an irredundant primary decomposition, so that Ass(푀/푁) = {픭1, … , 픭푛}. If 픭푖 = 픭푗 for some 1 ≤ 푖 ≠ 푗 ≤ 푛, Lemma 2.3.5 will imply that 푀푖 ∩ 푀푗 is primary, leading to a shorter primary decomposition. This is impossible when the primary decomposition is minimal. (iii) Suppose 푆 ∩ 픭푖 = ∅ (equivalently, 푖 ≤ 푚). By Theorem 2.2.5 and the exactness −1 −1 −1 of localization, 푀푖[푆 ] ⊂ 푀[푆 ] will be 픭푖[푆 ]-primary. On the other hand 푆 ∩ 픭푖 ≠ −1 −1 −1 ∅ implies Ass((푀/푀푖)[푆 ]) = ∅ by Theorem 2.2.5, thus 푀[푆 ]/푀푖[푆 ] = 0. Since localization respects intersections, we obtain

푚 −1 −1 푁[푆 ] = ⋂ 푀푖[푆 ]. 푖=1

In particular 푁[푆−1] is proper if and only if 푚 ≥ 1. It remains to show Ass(푀[푆−1]/푁[푆−1]) has 푚 elements if the original primary decomposition is minimal. Indeed, that set is just {픭[푆−1] ∶ 픭 ∈ Ass(푀/푁), 픭 ∩ 푆 = ∅} −1 −1 by Theorem 2.2.5, which equals {픭1[푆 ], … , 픭푚[푆 ]} (distinct) by (ii). §2.5 Examples and remarks ⋅ 21 ⋅

A natural question arises: to what extent are minimal primary decompositions unique? For those 푀푖 whose associated primes are minimal in Ass(푀/푁), the answer turns out to be positive.

Corollary 2.4.2. Let 푁 = 푀1 ∩ ⋯ ∩ 푀푛 be a minimal primary decomposition of 푁 ⊊ 푀. Suppose that 푀1 is 픭-primary where 픭 ∶= 픭1 is a minimal element in Ass(푀/푁), then 푀1 equals the preimage of 푁픭 under 푀 → 푀픭. Call it the 픭-primary component of 푁. Proof. Recall the proof of Theorem 2.4.1, especially the part (iii); here we localize with respect to 푆 ∶= 푅 ∖ 픭. The minimality assumption entails

푁픭 = 푀1,픭 ⊂ 푀픭.

It remains to show that the preimage of 푀1,픭 under 푀 → 푀픭 equals 푀1, in other words the injectivity of the natural map 푀/푀1 → (푀/푀1)픭 = 푀픭/푀1,픭. Indeed, ̄푥∈ 푀/푀1 maps to 0 if and only if there exists 푠 ∉ 픭 with 푠 ̄푥= 0, but Theorem 2.2.5 implies that the zero divisors of 푀/푀1 must lie in 픭. It follows that the non-uniqueness of minimal primary decompositions can only arise from embedded primes in Ass(푀/푁).

2.5 Examples and remarks

Primary decompositions are most often applied in the case 푀 = 푅 and 푁 = 퐼 is a proper ideal. The goal is to express 퐼 as an intersection of primary ideals. To begin with, let us take 푅 = ℤ. Observations: ⋄ The primary ideals of 푅 take the form (푝)푛, where 푛 ≥ 1 and 푝 is a prime number or zero. This may be deduced from Exercise 2.3.3 or directly from definitions.

⋄ The irredundant primary decompositions of ℤ/푛ℤ, for 푛 > 1, corresponds to the factorization of 푛 into prime-powers. There are no embedded primes in this case; the irredundant primary decomposition is unique and automatically minimal. Exercise 2.5.1. Justify the foregoing assertions. Therefore one can regard primary decompositions as a generalization of factorization of integers, now performed on the level of ideals. The most important case is 푅 = 한[푋1, … , 푋푛] (fix some field 한), as it is naturally connected to classical problems in algebraic geometry. Let us consider a simple yet non-trivial example from [8, §3]. Example 2.5.2. Take 푅 = 한[푋, 푌] and 퐼 = (푋2, 푋푌). The reader is invited to check that

퐼 = (푋) ∩ (푋2, 푋푌, 푌2) = (푋) ∩ (푋2, 푌).

Claim: this gives two minimal primary decompositions of 퐼. The ideal (푋) is prime, hence primary. In fact, (푋2, 푋푌, 푌2) = (푋, 푌)2 and (푋2, 푌) are both primary ideals associated to the maximal ideal (푋, 푌). This follows either by direct arguments or by Exercise 2.3.4, noting that (푋, 푌)2 = (푋2, 푋푌, 푌2) is contained in (푋2, 푌). The embedded prime (푋, 푌) is seen to be responsible non-uniqueness of primary decompositions. ⋅ 22 ⋅ Primary decompositions

Emanuel Lasker (1868–1941) first obtained the primary decomposition for finitely generated 한-algebras and the algebras of conver- gent power series in 1905. His method involves techniques from elimination theory. His result is then generalized and rewritten by Emmy Noether in 1921, in which the ascending chain condition plays a pivotal role. Lasker is best known for being the World Chess Champion from 1894 to 1921. (Picture taken from Wikime- dia Commons)

To see the geometry behind, recall that 푉(픞) ∪ 푉(픟) = 푉(픞픟) = 푉(픞 ∩ 픟) for any ideals 픞, 픟, thus expressing 퐼 as an intersection means breaking the corresponding geometric object into a union of simpler pieces. Also recall that for an ideal 퐼 ⊂ 한[푋, 푌], the points in ⋂푓 ∈퐼{푓 = 0} are in bijection with the maximal ideals lying over 퐼, at least for 한 algebraically closed (Nullstellensatz). Thus we may interpret these primary decompositions as equalities among “geometric objects” embedded in 한2: ⎧ 2 2 2 {{푋 = 0} ∪ {푋 = 푋푌 = 푌 = 0} {푋 = 0, 푋푌 = 0} = ⎨ 2 ⎩{{푋 = 0} ∪ {푋 = 푌 = 0} . 1. The geometric object defined by 푋 = 0 inside 한2 is certainly the 푌-axis: the regular functions living on this space form the 한-algebra 한[푋, 푌]/(푋) = 한[푌]. 2. The geometric object defined by 푋2 = 푋푌 = 푌2 = 0 looks “physically” like the origin (0, 0), but the 한-algebra of “regular functions” (in an extended sense) living on it equals 한[푋, 푌]/(푋2, 푋푌, 푌2): by restricting a polynomial function 휕푓 푓 (푋, 푌) to this “thickened point”, we see not only 푓 (0, 0) but also 휕푥 (0, 0) and 휕푓 2 2 휕푦 (0, 0). In other words, we shall view 푋 = 푋푌 = 푌 = 0 as the first-order infinitesimal neighborhood of (0, 0) ∈ 한2. 3. In a similar vein, 푋2 = 푌 = 0 physically defines (0, 0), but by restricting 푓 to that 휕푓 thickened point, we retrieve 푓 (0, 0) as well as 휕푥 (0, 0). Therefore we obtain the first-order infinitesimal neighborhood of 0 inside the 푋-axis. Both decomposition says that we obtain the 푌-axis together with first-order infinitesimal information at the origin (0, 0). This is also a nice illustration of the use §2.5 Examples and remarks ⋅ 23 ⋅

of nilpotent elements in scheme theory.

Example 2.5.3 (Symbolic powers). Let 픭 be a prime ideal in a Noetherian ring 푅 and fix 푛 ≥ 1. Observe that 픭 is the unique minimal element in Supp(푅/픭푛) = 푉(픭푛). By Theorem 2.2.7 픭 ∈ Ass(푅/픭푛), so it makes sense to denote by 픭(푛) the 픭-primary component (Corollary 2.4.2) of 픭푛, called the 푛-th symbolic power of 픭. In general 픭(푛) ⊋ 픭푛. For a nice geometric interpretation of symbolic powers due to Nagata and Zariski, we refer to [8, §3.9].

Getting primary decomposition of ideals in polynomial algebras is a non-trivial task. Thanks to the pioneers in computational commutative algebra, this can now achieved on your own computer, eg. by the open-source SageMath system. ⋅ 24 ⋅ Primary decompositions Lecture 3 Integral dependence, Nullstellensatz and flatness

The materials below largely come from [8, §4], [1, V.4]and [11, §§5–6]. In what follows, any ring 푅 and the 푅-algebras are assumed to be commutative. For elements 푎, 푏, … in an 푅-algebra, denote by 푅[푎, 푏, …] the 푅-subalgebra generated by them.

3.1 Integral extensions

Consider an 푅-algebra 퐴. Recall that to give an 푅-algebra 퐴 is the same as to give a ring homomorphism 휑 ∶ 푅 → 퐴. We shall switch freely between algebra and homomorphisms, omitting 휑 if necessary. In many concrete circumstances 푅 is simply a subring of 퐴. Definition 3.1.1. An element 푥 ∈ 퐴 is said to be integral over 푅 if there exists a monic 푛 푛−1 polynomial 푃(푋) = 푋 + 푎푛−1푋 + ⋯ + 푎0 ∈ 푅[푋] (with 푛 ≥ 1) such that 푃(푥) = 0. If every 푥 ∈ 퐴 is integral over 푅, we say 퐴 is integral over 푅. Note that elements from 푅 are trivially integral: take 푃(푋) with 푛 = 1. In the case 퐴 = ℂ and 푅 = ℤ, we recover the notion of algebraic integers. When 푅 is a field, we usually say algebraic instead of integral. The monic assumption on 푃 is crucial when 푅 is not a field, as illustrated by the following proof. Proposition 3.1.2. An element 푥 ∈ 퐴 is integral over 푅 if and only if there exists an 푅- submodule 푀 ⊂ 퐴 such that ⋄ 푀 is a finitely generated 푅-module; ⋄ 푥푀 ⊂ 푀, thus 푀 is an 푅[푥]-module; ⋄ 푀 is a faithful 푅[푥]-module i.e. ann푅[푥](푀) = {0}, and If 푥 is integral, 푀 ∶= 푅[푥] satisfies the conditions listed above. Proof. This is a familiar application of Cayley-Hamilton theorem, which we recall be- 푛 푛−1 low. If 푥 is integral, say 푥 + 푎푛−1푥 + ⋯ + 푎0 = 0, a straightforward induction shows that 푛−1 푅[푥] = ∑ 푅푥푖. 푖=0 ⋅ 26 ⋅ Integral dependence, Nullstellensatz and flatness

In particular we may take 푀 ∶= 푅[푥] to be the required submodule, which is faithful as 1 ∈ 푅[푥]. Conversely, given a submodule 푀 as above, with generators 푥1, … , 푥푚, we make 푀 into an 푅[푋]-module by letting the variable 푋 act as multiplication by 푥. 푚 Writing 푥 ⋅ 푥푖 = ∑푗=1 푎푖푗푥푗, there is the matrix equation

⎛푥1 ⎞ ⎜ ⎟ ⎜ ⎟ (푋 ⋅ 1푚×푚 − 퐴) ⎜ ⋮ ⎟ = 0, 퐴 = (푎푖푗)1≤푖,푗≤푚 ∈ Mat푚×푚(푅) ⎜ ⎟ ⎝푥푚⎠

∨ over 푅[푋]. Now multiply both sides by the cofactor matrix (푋 ⋅ 1푚×푚 − 퐴) , we get 푃(푋)푥푖 = 0 for all 푖, where 푃 ∈ 푅[푋] is the characteristic polynomial of 퐴, i.e. 푃(푥)푀 = {0}. Since 푀 is faithful as an 푅[푥]-module, we get 푃(푥) = 0. Corollary 3.1.3. The integral elements in an 푅-subalgebra 퐴 form a subalgebra. In particular, 퐴 is integral over 푅 if and only if it has a set of integral generators. Proof. Let 푎, 푏 ∈ 퐴 be integral elements. One readily checks that ⋄ 푅[푎, 푏] is finitely generated as an 푅-module (say by certain monomials 푎푖푏푗);

⋄ 푅[푎, 푏] is faithful (as an 푅[푎, 푏]-module) because it contains 1. Thus 푅[푎, 푏] witnesses the integrality of 푎+푏 and 푎푏, since they both stabilize 푅[푎, 푏]. Proposition 3.1.4. Consider ring homomorphisms 푅 → 퐴 → 퐵 such that 퐴 is integral over 푅. If 푦 ∈ 퐵 is integral over 퐴, then it is integral over 푅.

푛 푛−1 Proof. Assume 푦 + 푎푛−1푦 + ⋯ + 푎0 = 0 with 푎0, … , 푎푛−1 ∈ 퐴 integral over 푅. The 푅[푎0, … , 푎푛−1]-module 푅[푎0, … , 푎푛−1][푦] is also finitely generated over 푅, faithful and preserved by 푦, hence it witnesses the integrality of 푦 over 푅. This subalgebra of integral elements in 퐴 is called the integral closure of 푅 in 퐴. If the integral closure equals the image of 푅 in 퐴, we say 푅 is integrally closed in 퐴. The integral closure is automatically integrally closed by virtue of Proposition 3.1.4. Definition 3.1.5. Let 푅 be an integral domain and denote by 퐾 its field of fractions. The integral closure of 푅 in 퐾 is called the normalization of 푅. The domain 푅 is said to be normal if 푅 is integrally closed in 퐾. The first examples of normal domains come from unique factorization domains (UFD) that you have seen in undergraduate algebra, including ℤ, ℚ[푋, 푌], etc. Proposition 3.1.6. Unique factorization domains are normal. Proof. Given 푥 = 푟/푠 ∈ 퐾 with coprime 푟, 푠 ∈ 푅. If there is an integral dependence 푛 푛−1 푛 푛−1 푛 relation 푥 + 푎푛−1푥 + ⋯ + 푎0 = 0, we will have 푟 + 푎푛−1푟 푠 + ⋯ + 푎0푠 = 0, hence 푠 ∣ 푟푛. As 푟 is coprime to 푠, we see 푠 ∈ 푅× and 푥 ∈ 푅. Let us show that taking integral closure commutes with localizations. Geometri- cally, this means that taking integral closure is a local operation on Spec(푅). §3.1 Integral extensions ⋅ 27 ⋅

Lemma 3.1.7. Let 퐴 be an 푅-algebra, 푆 be a multiplicative subset of 푅. Denote by 푅̃ the integral closure of 푅 in 퐴. Then the integral closure of 푅[푆−1] in 퐴[푆−1] equals 푅[푆̃ −1]. 푛 푛−1 푖 Proof. Suppose that 푥 ∈ 퐴 satisfies 푥 + ∑푖=0 푏푖푥 = 0 with 푏0, … , 푏푛−1 ∈ 푅. For all 푛 푛−1 푖 푠 ∈ 푆 we deduce ( 푥 ) + ∑ 푏푖 ⋅ ( 푥 ) = 0 in 퐴[푆−1]. Therefore 푅[푆̃ −1] is in the 푠 푖=0 푠푛−푖 푠 integral closure of 푅[푆−1] in 퐴[푆−1]. Conversely, suppose that 푥/푠 ∈ 퐴[푆−1] satisfies 푛 푛−1 푖 푥 푎푖 푥 ( ) + ∑ ( ) = 0, 푎푖 ∈ 퐴, 푠푖 ∈ 푆. 푠 푖=0 푠푖 푠 Cleaning denominators, we see that 푥푠 ⋯ 푠 is integral over 푅, hence 푥 = 푥푠1⋯푠푛−1 ∈ 1 푛−1 푠1⋯푠푛 푅[푆̃ −1]. In particular, localizations of a normal domain are still normal. We have the following converse.

Exercise 3.1.8. Let 푅 be a domain. If for every maximal ideal 픪 of 푅 we have 푅픪 is normal, then so is 푅 itself. Hint: recall that 푅 = ⋂픪 푅픪 inside the fraction field 퐾. Exercise 3.1.9. Let 푅 ∶= ℂ[푋, 푌]/(푌2 − 푋3). Show that (i) 푅 is a domain, or equivalently, 푌2 − 푋3 is an irreducible polynomial; (ii) the element 푥 ∶= 푌/̄ 푋̄ ∈ Frac(푅) does not lie in 푅, where 푋,̄ 푌̄ denote the images of 푋, 푌 ∈ ℂ[푋, 푌]; (iii) show that 푥3 ∈ 푅, therefore 푅 is not a normal ring. Remark 3.1.10. The rationale of the previous Exercise comes from singularity. Consider the cuspidal curve 퐶 ∶= {(푥, 푦) ∈ ℂ2 ∶ 푦2 = 푥3}. It has an isolated singularity at (0, 0) since ∇(푦2 − 푥3) = (0, 0) at the origin. On the other hand, it admits a (non-injective) parametrization from ℂ by 푡 ↦ (푥 = 푡2, 푦 = 푡3). Following Milnor [12], a nice way to understand the “cusp” at (0, 0) is to cut 퐶 by a 3-sphere 푆 ∶= {(푥, 푦) ∈ ℂ2 ∶ |푥|2 + |푦|2 = 휖} enclosing the origin, with 0 < 휖 ≪ 1. Writing the parametrization above as 푡 = 푟푒푖휃 in polar coordinates, we get the equation 푟4 + 푟6 = 휖 which has a unique positive root 휌. Hence 푆 ∩ 퐶 is described by the closed curve 휃 ↦ (휌2푒2푖휃, 휌3푒3푖휃) in 푆 ≃ 핊3. We call 푆 ∩ 퐶 the link of singularity at (0, 0); in this case it is equivalent to the (2, 3)-torus knot, i.e. the trefoil shown below. ⋅ 28 ⋅ Integral dependence, Nullstellensatz and flatness

If we started from a non-singular curve in ℂ2, the result would then be an unknotted 핊1 inside 핊3. In this sense the link provides a measure for the singularity. Our study of normality has to be paused here. We refer to [13, §§8–9] for a beautiful discussion on the algebro-geometric content of normality, as well as important consequences such as Zariski’s Main Theorem. Roughly speaking, normality means there is only one branch through each point of the corresponding variety.

3.2 Nullstellensatz

Our aim is to present a generalization of the celebrated Nullstellensatz, which is one of the cornerstones of algebraic geometry. We shall also write the nilpotent radical of a ring 푅 as nil(푅) ∶= √0푅. For an ideal 픞 ⊂ 푅, we write 픞[푋] for the ideal of 푅[푋] formed by polynomials with all coefficients lying in 픞. Definition 3.2.1. A ring 푅 is called a Jacobson ring if every prime ideal 픭 satisfies

픭 = ⋂ 픪. (3–1) 픪∶maximal ideal ⊃픭

Equivalently, we require that the Jacobson radical rad(푅/픭) = nil(푅/픭) = {0} for all 픭. Note that ⊃ always holds. Observations: ⋄ Quotients of Jacobson rings are still Jacobson. ⋄ Fields are trivially Jacobson. Exercise 3.2.2. Prove that every principal ideal domain (a domain in which every ideal is generated by one element) with infinitely many maximal ideals is Jacobson. Theorem 3.2.3 (E. Snapper). For any 푅, the polynomial algebra 푅[푋] satisfies rad(푅[푋]) = nil(푅[푋]).

푖 Proof. To show that nil(푅[푋]) ⊃ rad(푅[푋]), let 푓 (푋) = ∑푖 푎푖푋 ∈ rad(푅[푋]), then 푖+1 × 1 + 푋푓 (푋) = 1 + ∑푖 푎푖푋 ∈ 푅[푋] . By looking at the reduction modulo 픭 of 푓 (푋) for every prime ideal 픭 of 푅, we see that 푎푖 ∈ ⋂ 픭 = nil(푅) for all 푖. Hence 푓 (푋) ∈ nil(푅)[푋] ⊂ nil(푅[푋]). Lemma 3.2.4. Let 푅 ⊂ 퐴 be integral domains such that 퐴 is a finitely generated 푅-algebra. If rad(푅) = {0}, then rad(퐴) = {0}. Proof. We may assume that 퐴 is generated by a single element 푎 ∈ 퐴 over 푅. If 푎 is transcendental over 퐾 ∶= Frac(푅), Theorem 3.2.3 above can be applied as nil(푅[푋]) = 푛 푖 {0}. Let us assume that 푎 satisfies 푓 (푎) = 0 for some 푓 (푋) = ∑푖=0 푟푖푋 ∈ 푅[푋] with 푟푛 ≠ 0. Let 푏 ∈ rad(퐴) and suppose 푏 ≠ 0. Hereafter we embed everything into the 퐾-algebra Frac(퐴). Since 푎 is algebraic over 퐾, so is every element from 푅[푎] or even 퐾[푎] ⊂ Frac(퐴). Hence 푏 is integral over 퐾 as well. By cleaning denominators, we §3.2 Nullstellensatz ⋅ 29 ⋅

푚 푖 arrive at 푔(푏) = 0 for some 푔(푋) = ∑푖=0 푠푖푋 ∈ 푅[푋] with the smallest possible degree 푚. Since 퐴 is a domain, we have 푠0 ≠ 0. Using rad(푅) = {0}, there exists a maximal ideal 픪 of 푅 such that 푟푛푠0 ∉ 픪. Taking ′ ′ localization at 픪, we get the subring 퐴 ∶= 퐴 ⊗푅 푅픪 ⊂ Frac(퐴) containing 푅픪, and 퐴 is also a finitely generated 푅픪-module since we inverted 푟푛. Nakayama’s Lemma for ′ ′ 푅픪-modules implies 픪퐴 ⊊ 퐴 , thus 픪퐴 ⊊ 퐴. Now choose a maximal ideal 픪퐴 of 퐴 over 픪. We must have 픪퐴 ∩푅 = 픪, which entails 푠0 ∉ 픪퐴. This is a contradiction since 푚 푖 푠0 = − ∑푖=1 푠푖푏 ∈ rad(퐴). Theorem 3.2.5 (Nullstellensatz). Let 퐴 be a finitely generated 푅-algebra. Assume that 푅 is a Jacobson ring, then the following statements hold. (i) 퐴 is a Jacobson ring.

(ii) Let 픫 ∈ Spec(퐴) be maximal, then its image 픪 ∈ Spec(푅) is maximal as well, and 퐴/픫 is a finite extension of the field 푅/픪. Proof. We start with (i). One may assume 푅 ⊂ 퐴 from the outset. Condition (3–1) for 퐴 amounts to rad(퐴/픭) = 0 for every 픭 ∈ Spec(퐴). Apply the previous Lemma to the integral domains 푅/푅 ∩ 픭 ⊂ 퐴/픭 to prove (i). Now turn to (ii). Using (i) and induction, we may assume that 퐴 = 푅[푎] for some 푎 ∈ 퐴. By considering the homomorphism 푅/픪 ↪ 퐴/픫 between Jacobson rings, we may further reduce to the case 픫 = {0} and 픪 = {0}, so that 푅 is a domain embedded in the field 퐴. In particular 푎 cannot be transcendental (as 푅[푋] is not a field) and must 푛 푖 satisfy ∑푖=0 푐푖푎 = 0 for some 푐0, … , 푐푛 ∈ 푅 with 푐푛 ≠ 0. Let 픨 be any maximal ideal of 푅 not containing 푐푛, which exists since rad(푅) = {0}. As in the proof of the previous Lemma, 푎 becomes integral over 푅픨 and Nakayama’s Lemma for 푅픨-modules entails 픨퐴 ≠ 퐴, hence 픨 = 0 because 퐴 is a field. This implies that 푅 is a field and 퐴 is a finite extension of 푅.

Corollary 3.2.6. Let 한 be an algebraically closed field, and 퐴 ∶= 한[푋1, … , 푋푛]. The maximal 푛 푛 ideals of 퐴 are in bijection with 한 by attaching to each 푥 ∶= (푥1, … , 푥푛) ∈ 한 the ideal

픪푥 = {푓 ∈ 퐴 ∶ 푓 (푥) = 0} = (푋1 − 푥1, … , 푋푛 − 푥푛). ∼ Proof. Since 퐴/픪푥 → 한 by evaluation at 푥, we see 픪푥 is indeed maximal. It is routine to show that 푥 = 푦 ⟺ 픪푥 = 픪푦. It remains to show that every maximal ideal 픫 contains some 픪푥. Indeed, Theorem 3.2.5 implies the field 퐴/픫 is algebraic over 한, ∼ hence 퐴/픫 ≃ 한 as 한-algebras. Let 푥푖 be the image of 푋푖 under 퐴 ↠ 퐴/픫 → 한 and set 푥 ∶= (푥1, … , 푥푛), then 픫 ⊃ 픪푥 as required. Corollary 3.2.7. Keep the notations above and set

푍(픞) ∶= {푥 ∈ 한푛 ∶ ∀푓 ∈ 픞, 푓 (푥) = 0}, 퐼(풳) ∶= {푓 ∈ 퐴 ∶ ∀푥 ∈ 푋, 푓 (푥) = 0}

for ideals 픞 ⊂ 퐴 and subsets 풳 ⊂ 한푛, then we have 퐼푍(픞) = √픞 for all 픞. If we identify 한푛 with MaxSpec(퐴) by the previous Corollary, then 푍(픞) is just the intersection of 푉(픞) ⊂ Spec(퐴) and MaxSpec(퐴). Details are left to the readers. ⋅ 30 ⋅ Integral dependence, Nullstellensatz and flatness

Proof. If 푓 ∈ 퐴 and 푓 푛 ∈ 픞 for some 푛, the vanishing of 푓 푛 on 푍(픞) will entail that of 푓 , hence the inclusion ⊃ holds. Assume conversely that 푓 ∈ 퐴 vanishes on 푍(픞). This means: for every maximal ideal 픪푥 we have

픪푥 ⊃ 픞 ⟺ 푥 ∈ 푍(픞) ⟹ 푓 (푥) = 0 ⟺ 푓 ∈ 픪푥. Hence 푓 ∈ ⋂ 픪푥 = √픞, 픪푥⊃픞 the last equality being based on Definition 3.2.1 since 퐴/픞 is a Jacobson ring. This concludes the ⊂. Remark 3.2.8. How about 푍퐼(풳)? Unwinding definitions, it is seen to equal the setof points that “satisfy the algebraic equations that 풳 satisfies.” The Zariski topology on 한푛 is defined by stipulating the subsets {푥 ∈ 한푛 ∶ 푓 (푥) = 0} to be closed, for all 푓 ∈ 퐴, so we obtain 푍퐼(풳) = 풳̄, the Zariski-closure of 풳. The reader is invited to verify that by identifying 한푛 with MaxSpec(퐴), the foregoing topology is induced from the Zariski topology on the prime spectrum Spec(퐴). 푛 An (algebraic, closed) subvariety of 한 is the vanishing locus 푓1 = ⋯ = 푓푚 = 0 for some 푓1, … , 푓푚 ∈ 한[푋1, … , 푋푛]; it is determined by the ideal 픞 = (푓1, … , 푓푚), in fact it depends only on √픞. An ideal 픞 is called radical if √픞 = 픞. To recap, we obtain a dictionary:

푛 Subvariety 풳 in 한 Radical ideal 픞 in 한[푋1, … , 푋푛]

Points of 풳 MaxSpec(퐴), 퐴 = 퐴풳 ∶= 한[푋1, … , 푋푛]/픞 Union of two varieties product or intersection of two ideals Intersection of varieties sum of ideals

Polynomial map 풳 → 풴 한-homomorphism 퐴풴 → 퐴풳 ⋮ ⋮

If one allows arbitrary rings 퐴 instead of just 한[푋1, … , 푋푛]/픞, and consider Spec(퐴) instead of MaxSpec(퐴) (the latter is well-behaved only for Jacobson rings), the result is the category of affine schemes. A proper treatment of these ideas should be left to the Algebraic Geometry course, if it exists......

3.3 Flatness: the first glance

To begin with, we consider a ring 퐴 and a module 푁. It has been observed that the additive functor 푁 ⊗− ∶ 퐴-Mod → 퐴-Mod is right exact, namely it preserves the exactness 퐴 of sequences like • → • → • → 0. If we consider exactness of sequences like 0 → • → • → •, the corresponding notion is left exactness. The same applies to any additive functor 퐹 instead of 푁 ⊗ −. Being both 퐴 §3.3 Flatness: the first glance ⋅ 31 ⋅

left and right exact is equivalent to that 퐹 preserves all exact sequences; in this case we say 퐹 is an exact functor. Definition 3.3.1. We say 푁 is a flat 퐴-module if 푁 ⊗ − is exact. We say 푁 is faithfully flat 퐴 if for every sequence 푀• = [⋯ → 푀푖 → 푀푖−1 → ⋯] of 푅-modules, we have 푀• ⊗ 푁 is 푅 exact if and only if 푀• is. Remark 3.3.2. In view of the right-exactness of ⊗, to assure flatness of 푁 it suffices that 푁 ⊗ − preserves kernels. 퐴 Now we consider a ring homomorphism 퐴 → 퐵, which makes 퐵 into an 퐴-algebra. Tensor product now gives an additive functor, often called the base change: 퐵 ⊗ − ∶ 퐴-Mod → 퐵-Mod. 퐴 Thus we can also talk about flatness and faithful flatness of 퐵 over 퐴. Since 퐵 is naturally an 퐴-module, this notion is compatible with the previous one. Example 3.3.3. Let 푆 be a multiplicative subset of 퐴, then 퐴[푆−1] is flat over 퐴. It is not faithfully flat in general, however; see Theorem 3.5.6. (푖) Example 3.3.4. A routine fact is that for any family (푀• ) of complexes of 퐴-modules, 푖∈퐼 we have (푖) (푖) ∀푖 ∈ 퐼, 푀• is exact ⟺ ⨁ 푀• is exact. 푖∈퐼 Recall that ⊗ preserves direct sums. It follows that a direct sum of modules is flat if and only if each summand is flat. From this we deduce the flatness of free modules since 퐴 ⊗ 푀 ≃ 푀 functorially for each 푀. Furthermore, projective modules are flat as 퐴 they are direct summands of free modules. Exercise 3.3.5. Show that ℤ/푛ℤ is not flat over ℤ for 푛 > 1. We list some basic properties below. ⊳ Tensor products Let 푀, 푁 be flat (resp. faithfully flat) 푅-modules, then so is 푀 ⊗ 푅 푁. This follows the associativity constraint of tensor products: (− ⊗ 푀) ⊗ 푁 ≃ − ⊗ (푀 ⊗ 푁). ⊳ Transitivity Given ring homomorphisms 퐴 → 퐵 → 퐶, if 퐵 is flat (resp. faithfully flat) over 퐴 and 퐶 is flat (resp. faithfully flat) over 퐵, then 퐶 is also flat (resp. faithfully flat) over 퐴. This follows from the transitivity of base change, namely there is a isomorphism of functors 퐴-Mod → 퐶-Mod. (− ⊗ 퐵) ⊗ 퐶 →∼ − ⊗ 퐶. 퐴 퐵 퐴

⊳ Base change Suppose 푁 is a flat (resp. faithfully flat) 퐴-module and 퐵 is any 퐴- algebra, then 푁 ⊗퐵 is a flat (resp. faithfully flat) 퐵-module. Again, any 퐵-module 퐴 푀 can be viewed as an 퐴-module, and there is a functorial isomorphism (푁 ⊗ 퐵) ⊗ 푀 →∼ 푁 ⊗ 푀. 퐴 퐵 퐴 ⋅ 32 ⋅ Integral dependence, Nullstellensatz and flatness

푑푖 Remark 3.3.6. A sequence [⋯ → 푀푖 푀푖−1 → ⋯] of 푅-modules is a complex (resp. exact) if and only if so is its localization at 픪, for every maximal ideal 픪. Indeed: ⋄ (푀•, 푑•) is a complex if and only if im(푑푖−1푑푖) = 0 for all 푖. Since localization is an exact functor, it preserves images and we know a module 푁 is zero if and only if 푁픪 = 0 for all 픪. ⋄ a complex (푀•, 푑•) is exact if and only if H푖(푀•) = 0 for all 푖. The same reasoning applies since localization preserves 퐻푖. Proposition 3.3.7. The following are equivalent for an 푅-module 푁. (i) 푁 is flat over 푅, (ii) 푁픭 is flat over 푅픭 for all prime ideal 픭, (iii) 푁픪 is flat over 푅픪 for all maximal ideal 픪. Proof. Direct consequence of the exactness of localization and Remark 3.3.6.

Lemma 3.3.8. Let 휑 ∶ 푅 → 푅′ be a ring homomorphism, 픭′ ∈ Spec(푅′) maps to 픭 ∈ Spec(푅) ♯ ′ under 휑 . If 휑 is flat, so is the induced homomorphism 푅픭 → 푅픭′ . ′ ′ Proof. Set 푆 = 푅 ∖ 픭 so that 휑(푆) ⊂ 푅 ∖ 픭 . Factorize 푅픭 → 푅픭′ as

′ −1 ′ 푅픭 → 푅⎵⎵⎵⎵⎵[휑(푆) ] → 푅픭′ . as a ring

The first arrow is also the base-change to 푅픭 of 휑 (as a homomorphism of 푅-modules), whereas the second one is a localization of 푅′. Their composite is therefore flat.

Proposition 3.3.9. Let 휑 ∶ 푅 → 푅′ be a ring homomorphism. The following are equivalent:

(i) 푅′ is flat over 푅,

′ ′ ′ ♯ ′ (ii) 푅픭′ is flat over 푅픭 for all 픭 ∈ Spec(푅 ) with 픭 = 휑 (픭 ); (iii) Idem, but for 픭′ ∈ MaxSpec(푅′).

′ −1 −1 Proof. (i) ⟹ (ii): Set 푆 ∶= 푅∖픭. Base change implies 푅 [푆 ] is flat over 푅[푆 ] = 푅픭. ′ −1 Since 푅픭′ is a localization of 푅[푆 ] (exercise), we conclude by transitivity. (ii) ⟹ (iii): Trivial. (iii) ⟹ (i): By Remark 3.3.6 applied to complexes of 푅′-modules, it suffices to ′ ′ ′ show the exactness of the functor − ⊗ 푅픭′ for all 픭 ∈ MaxSpec(푅 ). In view of Lemma 푅 ′ ′ 3.3.8, the factorization of 푅 → 푅픭′ into 푅 → 푅픭 → 푅픭′ and the flatness of 푅 → 푅픭 show ′ that 푅픭′ is indeed flat over 푅. Lemma 3.3.10 (Equational criterion of flatness). An 푅-module 푁 is flat if and only if for 푟 all 푟 ≥ 1, 푎1, … , 푎푟 ∈ 푅, 푥1, … , 푥푟 ∈ 푁 verifying ∑푖=1 푎푖푥푖 = 0, there exist 푠 ∈ ℤ≥1, an 푅-valued matrix 퐵 = (푏푖푗)1≤푖≤푟 and 푦1, … , 푦푠 ∈ 푁 such that 1≤푗≤푠

⎛푥1⎞ ⎛푦1⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⋮ ⎟ = 퐵 ⎜ ⋮ ⎟ ,(푎1 ⋯ 푎푟) 퐵 = 0. ⎜ ⎟ ⎜ ⎟ ⎝푥푟⎠ ⎝푦푠⎠ §3.4 Structure of flat modules ⋅ 33 ⋅

푓 Proof. Suppose 푀 is flat and consider the exact sequence 0 → ker(푓 ) → 푅⊕푟 푅 where 푓 (푡1, … , 푡푟) = ∑푖 푎푖푡푖. We obtain an exact

(푥푖)푖↦∑푖 푎푖푥푖 0 → ker(푓 ) ⊗ 푀 → 푀⊕푟 푀. 퐴

푠 Thus if (푥1, … , 푥푟) ↦ 0 by the arrow above, we can express it as ∑푗=1(푏1푗, … , 푏푟푗) ⊗ 푦푗 ∈ ker(푓 ) ⊗ 푀 as required. 퐴 To show the converse, we invoke the fact that flatness is equivalent to the injectivity of 픞 ⊗ 푁 → 픞푁 for all finitely generated ideal 픞. See Proposition 3.4.1.

It will be useful to rephrase the condition in Lemma 3.3.10 as follows: for all ⊕푟 ⋄ homomorphism 푥 ∶ 푅 → 푀, where 푟 ∈ ℤ≥1, and ⋄ 퐾 ⊂ ker(푥): submodule generated by one element, there exist some 푠 ∈ ℤ≥1 and a commutative diagram

푅⊕푟 퐵 푅⊕푠 s.t. 퐾 ⊂ ker(퐵). 푥 푦 푀

Indeed, 푥 (resp. 푦) corresponds to (푥1, … , 푥푟) via 푥 ∶ (푡1, … , 푡푟) ↦ ∑푖 푡푖푥푖 (resp. 푦 ∶ (푢1, … , 푢푠) ↦ ∑푗 푢푗, 푦푗), and 퐾 ⊂ ker(푥) corresponds to 푅 ⋅ (푎1, … , 푎푟). The homomorphism 퐵 corresponds to the matrix (푏푖푗)푖,푗. Remark 3.3.11. For flat 푁, the equational criterion so rephrased is applicable to any 푦 finitely generated 퐾 ⊂ ker(푥). Indeed, one may iterate the construction for 푅⊕푠 푀, etc. to make every generator of 퐾 mapped to 0.

3.4 Structure of flat modules

Recall from homological algebra that the right exact functor 푁 ⊗ − ∶ 푅-Mod → 푅-Mod 퐴 푅 has (Tor푖 (푁, −))푖≥0 as its left derived functors.

Proposition 3.4.1. The following are equivalent for an 푅-module 푁:

(i) 푁 is flat,

푅 (ii) Tor푖 (푁, −) = 0 for all 푖 > 0,

푅 (iii) Tor1 (푁, −) = 0,

푅 (iv) Tor1 (푁, 푅/픞) = 0 for all finitely generated ideal 픞, or equivalently 픞 ⊗ 푁 → 푁 is 푅 injective. ⋅ 34 ⋅ Integral dependence, Nullstellensatz and flatness

Proof. First, the equivalence mentioned in (iv) is a consequence of the exact sequence 푅 푅 ⎵⎵⎵⎵⎵⎵Tor1 (푁, 푅) → Tor1 (푁, 푅/픞) → 푁 ⊗ 픞 → 푁 → 푁 ⊗ (푅/픞) → 0 푅 푅 =0 deduced from 0 → 픞 → 푅 → 푅/픞 → 0. Clearly (i) ⟹ (ii) ⟹ (iii) ⟹ (iv). To show (iii) ⟹ (i), note that if ′ ″ 푅 ″ ′ ″ 0 → 푀 → 푀 → 푀 → 0 is exact, then Tor1 (푁, 푀 ) → 푁 ⊗푀 → 푁 ⊗푀 → 푁 ⊗푀 → 0 is exact. 푅 We show (iv) ⟹ (i), (ii) or (iii) as follows: Tor1 (푁, 푀) = 0 can be tested for 푅 finitely generated 푀 only, since Tor푖 (푁, −) preserves filtered inductive limits such as 푀 = lim {f.g. submodules}. −−→ We do induction on the minimal number 푛 of generators of 푀. If 푀 = 푅푥1 + ⋯ + 푅푥푛, ′ ′ ″ put 푀 ∶= ∑푖<푛 푅푥푖 so that we have a short exact sequence 0 → 푀 → 푀 → 푀 → ″ 0 where 푀 is generated by the image of 푥푛, hence isomorphic to some 푅/픞. Since 푅 ′ 푅 푅 ″ Tor1 (푁, 푀 ) → Tor1 (푁, 푀) → Tor1 (푁, 푀 ) is exact, we are reduced to the 푛 = 1 case, i.e. 푀 = 푅/픞. It boils down to assure 픞 ⊗ 푁 ↪ 푁. Again, using the exactness of filtered lim and the fact that ⊗ respects lim, it suffices to test this on finitely generated 픞. −−→ −−→ For a down-to-earth approach, see [8, §6.3]. Corollary 3.4.2. If 푟 ∈ 푅 is not a zero divisor, then 푟 is not a zero divisor for any flat 푅-module 푁. Proof. Take 픞 ∶= 푅푟, which is ≃ 푅, and contemplate on 푁 ≃ 픞 ⊗ 푁 ↪ 푁. 푅 Exercise 3.4.3. Suppose 푅 is a principal ideal domain. Show that 푁 is flat if and only if 푁 has no zero divisors except 0. Hint: the ideals take the form 픞 = (푡), so the condition (iv) amounts to 푡푥 = 0 ⟺ 푥 = 0. Exercise 3.4.4. For all field 한, show that (i) 한[푋, 푌]/(푋푌 − 푋) is not flat over 한[푋], (ii) 한J푡K[푌, 푍]/(푌푍 − 푡) is flat over 한J푡K.

Lemma 3.4.5. Suppose 퐴 → 퐵 is flat. Write 푀퐵 ∶= 퐵 ⊗ 푀 for any 퐴-module 퐵. Then there 퐴 퐵 퐴 are natural isomorphisms Tor푖 (푀퐵, 푁퐵) ≃ 퐵 ⊗ Tor푖 (푀, 푁) for all 푖. 퐴

Proof. Take a projective resolution 0 ← 푀 ← 푃• of 퐴-modules. Since 퐵 is flat over 퐴, its base-change 0 ← 푀퐵 ← 푃•,퐵 to 퐵 is still a projective resolution; we are using the fact 퐵 that base-change preserves projectivity. Hence H푖(푃•,퐵 ⊗푁퐵) computes Tor푖 (푀퐵, 푁퐵). 퐵 On the other hand, by flatness the homology groups are equal to H푖(푃• ⊗ 푁) ⊗ 퐵, 퐴 퐴 퐴 that is, Tor푖 (푀, 푁) ⊗ 퐵. We leave it to the reader to convince him- or herself that the 퐴 isomorphism so constructed is natural. Another way is to use to associativity and commutativity of tensor products on the derived level, namely the flatness of 퐴 → 퐵 implies

L L L L L 푀퐵 ⊗퐵 푁퐵 ≃ (푀 ⊗퐴 퐵) ⊗퐵 (푁 ⊗퐴 퐵) ≃ (푀 ⊗퐴 푁) ⊗퐴 퐵 in the derived categories, and it remains to take H푖. §3.4 Structure of flat modules ⋅ 35 ⋅

In particular, let 푆 ⊂ 푅 be a multiplicative subset. By Example 3.3.3 we infer

−1 푅[푆 ] −1 −1 푅 −1 Tor푖 (푀[푆 ], 푁[푆 ]) ≃ Tor푖 (푀, 푁)[푆 ]. Theorem 3.4.6. Let 푅 be a local ring with maximal ideal 픪. Let 푀 be a finitely generated 푅-module. The following are equivalent: ⋄ 푀 is free, ⋄ 푀 is projective. If we assume moreover that 푀 is finitely presented, both conditions are equivalent to the flatness of 푀. Proof. Free modules are known to be projective. Now let 푀 be finitely generated projective, and take a basis 1̄푥 , …푛 ̄푥 of the 푅/픪-vector space 푀/픪푀, together with liftings 푀 ∋ 푥푖 ↦푖 ̄푥 . Nakayama’s Lemma then implies the surjectivity of 훷 ∶ 푅⊕푛 ⟶ 푀

(푎1, … , 푎푛) ⟼ 푎1푥1 + ⋯ + 푎푛푥푛. As 푀 is projective, 훷 admits a section so that we may identify 푀 with a direct summand ⊕푛 ⊕푛 of 푅 , namely 푀 ⊕ 푁 = 푅 for some 푁. Taking − ⊗푅 푅/픪 leads to (푅/픪)⊕푛 = 푀/픪푀 ⊕ 푁/픪푁 as vector spaces over 푅/픪,

and by comparing dimensions we see 푁/픪푁 = {0}, which in turn gives 푁 = {0} by Nakayama’s Lemma (푁 is finitely generated since 푅⊕푛 ↠ 푁). Hence 푀 ≃ 푅⊕푛 is free. Now turn to the second assertion. Projective modules are flat since they are direct summands of free modules, and it remains to show that every flat 푀 with finite pre- 푥 sentation 푅⊕푞 → 푅⊕푟 푀 → 0 is a direct summand of a free module. Let’s plug 푥 ∶ 푅⊕푟 ↠ 푀 and 퐾 ∶= ker(푥) into the equational criterion of flatness (Lemma 3.3.10), rephrased as in Remark 3.3.11. Let 푁 be the image of 퐵 ∶ 푅⊕푟 → 푅⊕푠. One readily sees that 푦 induces 푁 →∼ 푀. This furnishes a section 푠 ∶ 푀 → 푅⊕푠 for 푦. We deduce the following result characterizing finitely presented projective modules: in geometric language, they correspond to vector bundles over the affine scheme Spec(푅).

Corollary 3.4.7. Let 푀 be a finitely presented 푅-module. Then 푀 is projective if and only 푀픪 is free for every maximal ideal 픪.

Proof. In view of Theorem 3.4.6, it suffices to show 푀 is projective if and only if 푀픪 is for all maximal ideal 픪. One direction is easy: if 푀 is a direct summand of a free module, then so is 푀픪. Conversely, the assumption on finite presentation entails an isomorphism between additive functors 푅-Mod → 푅픪-Mod

Hom푅(푀, −) ⊗ 푅픪 ≃ Hom푅 (푀픪, (−)픪). 푅 픪

Hence 푀픪 is projective for all 픪 implies 푀 is projective, by Remark 3.3.6 and the exactness of localizations. ⋅ 36 ⋅ Integral dependence, Nullstellensatz and flatness

Note that for finitely presented 푀, the equivalence between projectivity and flatness holds for any ring 푅. The arguments are verbatim, and this can also be deduced from the local case. We close this section by a stronger result, whose proof is referred to [8, Theorem A6.6].

Theorem 3.4.8 (Govorov–Lazard). An 푅-module is flat if and only if it is a filtered lim of −−→ free 푅-modules.

3.5 Faithful flatness and surjectivity

We begin by a contemplation on the definition of faithful flatness. Recall that 푅-Mod is the template of abelian categories, in which one can talk about the zero object 0, direct sums, kernels, cokernels, images and exactness. We say a functor between categories is faithful if it induces injections on Hom-sets. A functor between additive categories is called additive if it induces group homomorphisms between Hom-sets. An additive functor between abelian categories is exact if it preserves all exact sequences. Exact functors preserve kernels, cokernels and images.

Lemma 3.5.1. Let 퐹 ∶ 풞 → 풞′ be an additive functor between abelian categories. The following are equivalent.

(i) 퐹 is exact and faithful;

(ii) 퐹 is exact and (퐹푀 = 0 ⟺ 푀 = 0) for every object 푀 of 풞;

(iii) a sequence 푀′ → 푀 → 푀″ is exact in 풞 if and only if 퐹푀′ → 퐹푀 → 퐹푀″ is exact in 풞′.

Proof. (i) ⟹ (ii): If 퐹푀 = 0 then 퐹(id푀) = id퐹푀 = 0, and the faithfulness implies id푀 = 0 in End풞 (푀); this is possible only when 푀 = 0. (ii) ⟹ (i): Suppose 푢 ∶ 푁 → 푀 is mapped to 0 under 퐹. Then we have 퐹(im(푢)) = 0, thereby im(푢) = 0 and 푢 = 0. 푢 푣 퐹푢 퐹푣 (i) ⟹ (iii): Suppose 푀′ 푀 푀″ induces an exact sequence 퐹푀′ 퐹푀 퐹푀″. From 퐹(푣푢) = 퐹(푣)퐹(푢) = 0 we get 푣푢 = 0. Thus it makes sense to define 퐶 ∶= ker(푣)/ im(푢). One has an exact sequence

im(푢) → ker(푣) → 퐶 → 0.

Since 퐹 is exact, we deduce 퐹퐶 = 0, which implies 퐶 = 0 by (i) ⟹ (ii). (iii) ⟹ (i): Suppose 푢 ∶ 푁 → 푀 is mapped to 0 under 퐹. Consider 푣 ∶ 푀 → coker(푢). Since 퐹 preserves exact sequences and 퐹푣 is an isomorphism, we see 푣 is also an isomorphism, therefore 푢 = 0.

Proposition 3.5.2. The following are equivalent for an 푅-module 푁. (i) 푁 is faithfully flat. (ii) The functor − ⊗ 푁 is exact and faithful. 푅 §3.5 Faithful flatness and surjectivity ⋅ 37 ⋅

(iii) 푁 is flat and for every maximal ideal 픪 of 푅, we have 푁/픪푁 ≃ 푁 ⊗ 푅/픪 ≠ 0. 푅 Proof. The equivalence (i) ⟺ (ii) has just been established. An 푅-module 푀 is nonzero if and only if there exist exact sequences

0 → 푅/픞 → 푀, 푅/픞 → 푅/픪 → 0

where 픞 is a proper ideal and 픪 is a maximal over-ideal of 픞. Next, let’s show (iii) ⟹ (i) or (ii): there are exact sequences

0 → 푇 → 푀 ⊗ 푁, 푇 → 푀 ⊗ (푅/픪) → 0 푅 푅 where 푇 ∶= (푅/픞) ⊗ 푁, therefore 푀 ⊗ 푁 ≠ 0 and we apply Lemma 3.5.1. Finally, (i) or 푅 푅 (ii) ⟹ (iii) is clear. Corollary 3.5.3. Let 푅 → 푅′ be a local homomorphism1 between local rings and let 푀 be a finitely generated 푅′-module, 푀 ≠ 0. Then as an 푅-module, 푀 is faithfully flat if and only if it is flat. Proof. Let 픪, 픪′ be the maximal ideals of 푅, 푅′, respectively. Since 픪 is mapped into 픪′ = rad(푅′), the assertion follows from Proposition 3.5.2 together with Nakayama’s Lemma. This is often applied to the cases 푅 = 푅′ or 푀 = 푅′. Exercise 3.5.4 (Faithfully flat descent for flatness). Given a faithfully flat homomorphism 휑 ∶ 푅 → 푅′. If 푀 is an 푅-module such that 푀 ⊗ 푅′ is a flat (resp. faithfully flat) 푅 푅′-module, then so is 푀 over 푅. Proposition 3.5.5. Suppose 휑 ∶ 퐴 → 퐵 is a faithfully flat ring homomorphism and regard 퐵 as an 퐴-algebra. ⋄ For any 퐴-module 푀, the natural map 푀 → 푀 ⊗ 퐵 is injective; in particular, 휑 is seen 퐴 to be injective by taking 푀 = 퐴.

⋄ For any ideal 픞 ⊊ 퐴 we have 휑−1(픞퐵) = 픞.

⋄ The map 휑♯ ∶ Spec(퐵) → Spec(퐴) is surjective. Proof. Let 푁 be the kernel of 푀 → 푀⊗퐵. The 퐵-module homomorphism 푁⊗퐵 → 푀⊗퐵 퐴 퐴 퐴 is injective by flatness; it is zero on 푁 ⊗ 1, hence identically zero and we obtain 푁 = {0} by faithful flatness (Lemma 3.5.1). Let 픞 ⊂ 퐴 be a proper ideal. The previous step with 푀 ∶= 퐴/픞 implies that 퐴/픞 → (퐴/픞) ⊗ 퐵 ≃ 퐵/픞퐵 is injective, hence 휑−1(픞퐵) = 픞. 퐴 To show the surjectivity of 휑♯, consider a given 픭 ∈ Spec(퐴) and form the faithfully flat ring homomorphism 휑픭 ∶ 퐴픭 → 퐵 ⊗ 퐴픭 = 퐵픭 퐴

1 That is, the preimage of the maximal ideal 픪푅′ equals 픪푅. ⋅ 38 ⋅ Integral dependence, Nullstellensatz and flatness

−1 by base change. The previous step implies 휑픭 (픭퐵픭) = 픭퐴픭, hence 픭퐵픭 is a proper ideal −1 of 퐵픭. Any maximal over-ideal 픪0 of 픭퐵픭 satisfies 휑픭 (픪0) = 픭퐴픭. Take 픪 ∈ Spec(퐵) mapping to 픪0. From the commutative diagrams

퐵 퐵픭 Spec(퐵) Spec(퐵픭)

휑 휑 ♯ ♯ 픭 휑 휑픭

퐴 퐴픭 Spec(퐴) Spec(퐴픭)

one infers 휑♯(픪) = 픭.

Theorem 3.5.6. The following are equivalent for a ring homomorphism 휑 ∶ 퐴 → 퐵.

(i) 휑 is faithfully flat;

(ii) 휑 is flat and 휑♯ ∶ Spec(퐵) → Spec(퐴) is surjective;

(iii) 휑 is flat and for any maximal ideal 픭 ⊂ 퐴 there exists a maximal ideal 픪 ⊂ 퐵 such that 휑−1(픪) = 픭.

Proof. (i) ⟹ (ii) is contained in Proposition 3.5.5. As for (ii) ⟹ (iii), take any 픮 ∈ Spec(퐵) that pulls back to 픭 ∈ Spec(퐴), then any maximal over-ideal 픪 of 픮 also pulls back to 픭. To show (iii) ⟹ (i), apply the criterion of Proposition 3.5.2: for any 픭 ∈ MaxSpec(퐴), the existence of 픪 ↦ 픭 implies 픪 ⊃ 휑(픭) ⋅ 퐵 = 픭퐵, hence 픭퐵 ≠ 퐵. The notion of flatness was first introduced by J.-P. Serre in[15]. The surjections 휑♯ ∶ Spec(퐵) → Spec(퐴) (or rather their global avatars) for faithfully flat 휑 ∶ 퐴 → 퐵 are often employed as candidates of “coverings” in algebraic geometry, leading up to the well-known fpqc (faithfully flat + quasi-compact) and fppf (faithfully flat + finitely presented) topologies in the sense of Grothendieck. They have been indispensable tools for contemporary geometers; [17] serves as a readable introduction to this circle of ideas. Lecture 4

Going-up, going-down, gradings and filtrations

This lecture will be less self-contained than the other ones.

4.1 Going-up and going-down

In geometry, it is often crucial to know properties of a morphism between algebraically defined geometric objects. Rephrased in terms of commutative algebra, our goalisto understand the image of 휑♯ ∶ Spec(퐵) → Spec(퐴) where 휑 ∶ 퐴 → 퐵 is a given ring homomorphism.

⋄ We say the going-up property holds for 휑 if for every 픭 ⊂ 픭′ in Spec(퐴) and 픮 ∈ Spec(퐵) with 휑♯(픮) = 픭 (and we say 픮 lies over 픭...) there exists 픮′ ⊃ 픮 lying over 픭′.

⋄ We say the going-down property holds for 휑 if for every 픭 ⊂ 픭′ in Spec(퐴) and 픮′ ∈ Spec(퐵) lying over 픭′, there exists 픮 ⊂ 픮′ lying over 픭.

Pictorially:

픮′ 퐵

픮

픭′ 퐴

픭

Lemma 4.1.1 (Existence of minimal over-primes). Let 픞 be a proper ideal in a ring 푅. There exists a prime ideal 픭 which is minimal among all primes containing 픞. If 픓 is an ideal containing 픞, one can choose 픭 ⊂ 픓. ⋅ 40 ⋅ Going-up, going-down, gradings and filtrations

Proof. One easily reduces to the case 픞 = {0}. We want to use Zorn’s Lemma to find a minimal prime. It boils down to show that any chain (픭푖)푖∈퐼 of prime ideals (퐼: totally ordered set with 푗 > 푖 ⟹ 픭푖 ⊃ 픭푗) has a lower bound; we assume 픭푖 ⊂ 픓 when 픓 is prescribed. It suffices to show 픭 ∶= ⋂푖∈퐼 픭푖 is prime: if 푥푦 ∈ 픭 but there exists 푖 with 푥 ∉ 픭푖, then 푥 ∉ 픭푗 whenever 푗 ≥ 푖; in this case 푗 ≥ 푖 ⟹ 푦 ∈ 픭푗. This entails 푦 ∈ 픭. Lemma 4.1.2. The going-down property for 휑 is equivalent to the following: for every 픭 ∈ Spec(퐴) with 휑(픭)퐵 ≠ 퐵 and any minimal over-prime 픮 of 휑(픭)퐵, we have 휑♯(픮) = 픭.

Proof. Assuming going-down for 휑, let 픭, 픮 be as above. Evidently 휑♯(픮) ⊃ 휑−1(휑(픭)퐵) ⊃ 픭. If we have ⊋, then going-down guarantees the existence of 픮♭ ⊊ 픮 lying over 픭. Thus 휑−1(픮♭) = 픭 implies 픮♭ ⊃ 휑(픭)퐵, contradicting the minimality of 픮. To show the converse, consider 픭 ⊂ 픭′ with 픮′ lying over 픭′. We have 휑(픭)퐵 ⊂ 휑(픭′)퐵 ⊂ 픮′ ≠ 퐵. Take 픮 to be a minimal over-prime of 휑(픭)퐵 (which exists by Lemma 4.1.1) to verify the going-down property.

Theorem 4.1.3. Going-down holds for flat 휑 ∶ 퐴 → 퐵.

′ ′ ′ Proof. Consider 픭 ⊂ 픭 and 픮 lying over 픭 in the setting of going-down. First, 퐵픮′ is flat over 퐴픭′ by Proposition 3.3.9. Secondly, 퐴픭′ → 퐵픮′ is faithfully flat since it is local by Theorem 3.5.6 (iii), therefore induces a surjection on spectra. Take any prime of 퐵픮′ mapping to 픭퐴픭 ∈ Spec(퐴픭′ ) and to 픮 ∈ Spec(퐵). In view of the commutative diagrams

퐵 퐵픮′ Spec(퐵) Spec(퐵픮′ )

휑 휑 ′ ♯ 휑♯ 픭 휑 픭′

퐴 퐴픭′ Spec(퐴) Spec(퐴픭′ )

we see 픮 is the required prime in going-down.

Theorem 4.1.4 (Krull–Cohen–Seidenberg). Suppose the ring 퐵 is integral over its subring 퐴. The following holds.

(i) The map Spec(퐵) → Spec(퐴) given by 픮 ↦ 픮 ∩ 퐴 is surjective.

(ii) There are no inclusion relations in any fiber of Spec(퐵) → Spec(퐴).

(iii) Going-up holds for 퐴 ↪ 퐵.

(iv) If 퐴 is a local ring and 픭 ∈ MaxSpec(퐴), then the fiber {픮 ∶ 픮 ∩ 퐴 = 픭} equals MaxSpec(퐵).

(v) Assume 퐴, 퐵 are domains and 퐴 is normal. Then going-down holds for 퐴 ↪ 퐵.

(vi) Assume furthermore that 퐵 is the integral closure of 퐴 in a normal field extension 퐿 ⊃ 퐾 ∶= Frac(퐴), then 훤 ∶= Aut(퐿/퐾) acts transitively on each fiber of Spec(퐵) → Spec(퐴).

The last assertion should be familiar to readers with a background in algebraic number theory. §4.1 Going-up and going-down ⋅ 41 ⋅

Proof. (iv): Let 픮 ∈ MaxSpec(퐵) and 픭0 ∶= 픮 ∩ 퐴. We know 퐵/픮 is a field, integral over its subring 퐴/픭0. We claim that 퐴/픭0 is also a field, therefore 픭0 = 픭. Let 푥 ∈ 퐴/픭0 ∖{0}. 1 푛 1 푛−1 Its inverse in 퐵/픮 satisfies an integral relation ( 푥 ) +푎푛−1( 푥 ) +⋯+푎0 = 0 over 퐴/픭0. 푛−1 1 Multiplying both sides by 푥 yields 푥 ∈ (퐴/픭0)[푥]. Conversely, we have to show any 픮 ∈ Spec(퐵) with 픮 ∩ 퐴 = 픭 is maximal. Again, there is an integral extension of domains 퐴/픭 ↪ 퐵/픮. Consider 푦 ∈ 퐵/픮 satisfying 푛 푛−1 푦 + 푎푛−1푦 + ⋯ + 푎0 = 0, with the smallest possible 푛. If 푦 ≠ 0 then 푎0 ≠ 0. Since 퐴/픭 is a field, the recipe to produce 푦−1 ∈ (퐴/픭)[푦] is well-known. (i), (ii): Fix 픭 and consider the inclusion 퐴픭 ↪ 퐵픭 = 퐵 ⊗ 퐴픭 which is still integral 퐴 (note that 퐴 ∖ 픭 is a multiplicative subset of 퐵, and 퐵픭 is nonzero). We are reduced to the case 퐴 is local with maximal ideal 픭. By (iv) the fiber of 픭 in Spec(퐵) is nothing but MaxSpec(퐵). This establishes (i) and (ii) since there are no inclusions among maximal ideals. (iii): Consider 픭 ⊂ 픭′ and 픮 over 픭 in the setting of going-up. Then (i) is applicable to 퐴/픭 ↪ 퐵/픮 and yields the required 픮′ ∈ Spec(퐵/픮) ↪ Spec(퐵). (vi): Observe that every 휎 ∈ 훤 induces an 퐴-automorphism of 퐵. Let 퐾′ ∶= 퐿훤 . By (infinite) Galois theory we know 퐿/퐾′ is Galois and 퐾′/퐾 is purely inseparable. Let 퐴′ be the integral closure of 퐴 in 퐾′. First observe that

Spec(퐴′) ⟶ Spec(퐴), 픭′ ↦ 픭 = 픭′ ∩ 퐴

is a bijection. Indeed, 퐾′ ≠ 퐾 only when 푝 ∶= char(퐾) > 0, in which case the inverse 푚 is given by 픭′ = {푡 ∈ 퐴′ ∶ 푡푝 ∈ 픭, 푚 ≫ 0}. Thus we assume henceforth that 퐿/퐾 is Galois. Deal with the case [퐿 ∶ 퐾] < ∞ first. Consider 픮, 픮′ ∈ Spec(퐵) in the fiber over 픭. Suppose on the contrary that 훤픮 does not meet 픮′, then by (ii) we have 픮′ ⊄ 휎(픮) for ′ each 휎 ∈ 훤. By the prime avoidance (Proposition 1.1.5), there exists 푥 ∈ 픮 ∖ ⋃휎 휎(픮), since 훤 is finite. Now define the norm 푦 ∶= 푁퐿/퐾(푥) ∈ 퐾, which is some positive power (namely [퐿 ∶ 퐾]푖) of ∏휎∈훤 휎(푥), hence belongs to 퐵. Notice that ⋄ 퐴 normal implies 푦 ∈ 퐴; ⋄ 푥 ∉ 휎−1(픮) for all 휎 ∈ 훤 implies 푦 ∉ 픭; ⋄ however 푦 ∈ 픮′ ∩ 퐴 = 픭 since 푥 ∈ 픮′. Contradiction. Now suppose [퐿 ∶ 퐾] is infinite and 픮, 픮′ in the fiber over 픭. We need to use the Krull topology on 훤. For every finite, Galois subextension 퐸/퐾 of 퐿/퐾, define the set

풯 (퐸) ∶= {휎 ∈ 훤 ∶ 휎(픮 ∩ 퐸) = 픮′ ∩ 퐸} .

By the finite case we know 풯 (퐸) ≠ ∅. Furthermore, ⋄ 풯 (퐸) is closed in 훤 (it is the preimage of some subset of Gal(퐸/퐾)); ⋄ 퐸′ ⊂ 퐸 ⟹ 풯 (퐸′) ⊃ 풯 (퐸); ⋄ the intersection of all 풯 (퐸) is nonempty (by the compactness of 훤 and the previous step). ′ Taking 휎 ∈ ⋂퐸 풯 (퐸) gives 휎(픮) = 픮 . (v): Define 퐿0 ∶= Frac(퐵) and 퐾 ∶= Frac(퐴). Then 퐿0/퐾 is algebraic and we may take a normal closure 퐿 of 퐿0 over 퐾. Let 퐶 be the integral closure of 퐴 (thus of 퐵) in 퐿. Consider the setting 픭 ⊂ 픭′ and 픮 ∩ 퐴 = 픭 of going-down. Take any 픯 ∈ Spec(퐶) ′ mapping to 픭. By (iii) for 퐴 → 퐶 we obtain 픯1 ∈ Spec(퐶) such that 픯1 ↦ 픭 and ⋅ 42 ⋅ Going-up, going-down, gradings and filtrations

′ 픯1 ⊃ 픯. Next, take 픯2 ∈ Spec(퐶) mapping to 픮 ; by (vi) there exists 휎 ∈ Aut(퐿/퐾) with 휎(픯1) = 픯2. One can check that 픮 ∶= 휎(픯) ∩ 퐵 is the required prime ideal. Explained pictorially: 휎

픯 픯1 픯2 퐶

픮′ 퐵

픭 픭′ 퐴

we first construct 픯 and then 픯1 by going-up, then “tilt” it via some 휎 to match 픯1 with ′ some chosen 픯2 above 픮 , so that 휎(픯) ∩ 퐵 produces the required going-down: Exercise 4.1.5. Let 퐴 ⊂ 퐵 be an integral extension of rings. Show that Spec(퐵) → Spec(퐴) is a closed map (Cf. Proposition 4.2.1.) Hint: Let 픟 ⊂ 퐵 be a proper ideal, then 퐴/픟∩퐴 ↪ 퐵/픟 is still integral. Reduce the problem to showing that 푉({0퐵}) = Spec(퐵) has closed image in Spec(퐴).

4.2 Subsets in the spectrum

Proposition 4.2.1. Let 휑 ∶ 퐴 → 퐵 be a ring homomorphism with going-up property and suppose 퐵 is Noetherian. Then 휑♯ ∶ Spec(퐵) → Spec(퐴) is a closed map: it maps closed subsets to closed subsets.

Proof. Consider a closed subset 푉(픟) of Spec(퐵). First, every 픮 ∈ Spec(퐵) with 픮 ⊃ 픟 lies over a minimal over-prime of 픟, by Lemma 4.1.1. Secondly, 퐵 is Noetherian implies Ass(퐵/픟) is finite; in particular there are only finitely many minimal over-primes 픮1, … , 픮푛 of 픟. ♯ ♯ Set 픭푖 ∶= 휑 (픮푖) for all 푖. By going-up, 푉(픭푖) is contained in 휑 (푉(픟)). On the other ♯ ♯ hand, every 픭 = 휑 (픮) with 픮 ∈ 푉(픟) lies over some 픭푖 = 휑 (픮푖) by the foregoing ♯ 푛 discussion. This shows 휑 (푉(픟)) = ⋃푖=1 푉(픭푖) is closed. Corollary 4.2.2. Suppose 퐵 is Noetherian and integral over a subring 퐴. Then Spec(퐵) → Spec(퐴) is a closed surjection with finite fibers.

Proof. Apply Theorem 4.1.4 with Proposition 4.2.1 to show that Spec(퐵) → Spec(퐴) is closed and surjective. To show the finiteness of the fiber over 픭 ∈ Spec(퐴), note that the preimage of 푉(픭) in Spec(퐵) equals 푉(픭퐵). Since there are no inclusions in the fiber over 픭, every element in that fiber must be a minimal over-prime of 픭퐵. We have seen in the proof of Proposition 4.2.1 that there are only finitely many such minimal-over primes. Note that the “closed surjection” part applies to any integral extension of rings. See Exercise 4.1.5. In order to obtain further results of this type, we have to introduce more notions. Let 푅 be a ring. §4.2 Subsets in the spectrum ⋅ 43 ⋅

Definition 4.2.3. For 픭, 픭′ ∈ Spec(푅) satisfying 픭 ⊂ 픭′, we say 픭 is a generalization of 픭′, and 픭′ is a specialization of 픭.

To make geometric meaning from it, being “specialized” signifies that there are “more equations” in 픭′, therefore it corresponds a smaller embedded geometric object. For example, in 푅 = ℂ[푋, 푌] the prime ideal (푋, 푌) is a specialization of (푋), as the origin 푋 = 푌 = 0 belongs to the line 푋 = 0.

Lemma 4.2.4. With respect to the Zariski topology, 픭 is a generalization of 픭′ if and only if 픭′ ∈ {픭}.

Proof. The condition 픭′ ∈ {픭} means that for every ideal 픞, if 픭 ⊃ 픞 then 픭′ ⊃ 픞. Taking 픞 = 픭 yields 픭′ ⊃ 픭, and the converse is even easier. A subset is called stable under specialization (resp. generalization) if the specialization (resp. generalization) of any member still belongs to that set. The following is straightforward.

Lemma 4.2.5. Any closed subset is stable under specialization; any open subset is stable under generalization.

Definition 4.2.6. Suppose 푅 is Noetherian. A subset is called locally closed if it is the intersection of an open with a closed subset, called constructible if it is a finite union of locally closed subsets. Closed subsets of the form 푉(픭), where 픭 ∈ Spec(푅), are called irreducible; in this case we call 픭 the generic point of 푍, which is uniquely characterized as the point which generalizes every member of 푍.

⋄ The foregoing definition is standard only for 푅 Noetherian. The general definition in EGA differs.

⋄ These notions can be applied to any topological space 푋. In practice one usually suppose 푋 to be – Noetherian: the closed subsets satisfy descending chain condition, – sober: every irreducible has a generic point, in order to get interesting results. This explains our Noetherian assumption.

⋄ If 한 is algebraically closed, 푅 = 한[푋1, … , 푋푛]/픞 corresponds to an affine algebraic variety 풳 ⊂ 한푛 and we work with MaxSpec(푅) instead of Spec(푅), then a subset 퐸 ⊂ 풳 being locally closed means that it can be defined by a formula using the usual language of algebraic operations over ℂ, with coordinate variables 푋1, … , 푋푛 and the symbols =, ≠, but without using the quantifiers ∃, ∀. For example, the formula

(¬푋 = 0) ∨ (푋 = 0 ∧ 푌 = 0)

defines the constructible subset 퐸 ∶= {(푥, 푦) ∈ 한2 ∶ 푥 ≠ 0} ∪ {(0, 0)}, which is neither closed nor open. Note that 퐸 is the image of the polynomial map 한2 → 한2 given by (푥, 푦) ↦ (푥, 푥푦). ⋅ 44 ⋅ Going-up, going-down, gradings and filtrations

Exercise 4.2.7. Show that the set of constructible subsets is stable under finite ∪, finite ∩ and taking complements.

Exercise 4.2.8. Show that irreducible closed subsets 푍 admit the following topological characterization: if 푋 = 퐴 ∪ 퐵 with 퐴, 퐵 closed, then either 푋 = 퐴 or 푋 = 퐵.

Suppose 푅 is Noetherian. Given any closed subset 푍 ⊂ Spec(푅), we may write 푍 = 푍1 ∪ ⋯ ∪ 푍푛 with each 푍푖 irreducible. One way to do this is to use the primary decomposition for 픞, where we assume 푍 = 푉(픞); then 푍1, … , 푍푛 will correspond to the minimal elements in Ass(푅/픞). One can show by purely topological means that such an irreducible decomposition is unique if we require 푖 ≠ 푗 ⟹ 푍푖 ⊄ 푍푗. See [10, I.1.5].

Lemma 4.2.9. Let 퐸 be a subset of Spec(푅) where 푅 is a Noetherian ring. The following are equivalent:

(i) 퐸 is constructible;

(ii) for every irreducible subset 푍 of Spec(푅), either 푍∩퐸 is not dense in 푍 or 푍∩퐸 contains a nonempty open subset of 푍.

Proof. Omitted. See [11, (6.C)].

Theorem 4.2.10 (C. Chevalley). Let 휑 ∶ 퐴 → 퐵 be a ring homomorphism such that 퐴 is Noetherian and 퐵 is a finitely generated 퐴-algebra. Then 휑♯ maps constructible subsets to constructible subsets.

Proof. Omitted. See [11, (6.E)].

Now we can give a partial converse to Lemma 4.2.5, albeit not in the strongest form.

Proposition 4.2.11. Suppose 푅 is Noetherian and 퐸 is a constructible subset of Spec(푅). If 퐸 is stable under specialization (resp. generalization), then 퐸 is closed (resp. open) in Spec(푅).

Proof. It suffices to treat the specialization-stable case by taking complements. Write the Zariski-closure 퐸̄ as a finite union of irreducibles components 푍, without inclusion relations. For each irreducible component 푍, notice that 푍 ∩ 퐸 is dense in 푍 for all 푍, ̄ since otherwise 퐸 = ⋃푍(푍 ∩ 퐸) = ⋃푍 푍 ∩ 퐸 would lead to another irreducible decomposition. Thus 푍 ∩ 퐸 contains a nonempty open of 푍 by Lemma 4.2.9. This open subset of 퐸 must contain the generic point of 푍. As 퐸 is stable under specialization, we obtain 푍 ⊂ 퐸. This being true for all 푍, we deduce that 퐸 = 퐸̄.

Proposition 4.2.12. Consider a ring homomorphism 휑 ∶ 퐴 → 퐵 satisfying going-down. Sup- pose 퐴 is Noetherian and 퐵 is a finitely generated 퐴-algebra, then 휑♯ ∶ Spec(퐵) → Spec(퐴) is an open map.

Proof. Let 푈 = Spec(퐵) ∖ 푉(픞) be an open subset. Going-down implies that 휑♯(푈) is stable under generalization. It suffices to show 휑♯(푈) is constructible, and this is the content of Chevalley’s Theorem 4.2.10. §4.3 Graded rings and modules ⋅ 45 ⋅

4.3 Graded rings and modules

Let (훤, +) be a commutative monoid. In most cases we consider 훤 = ℤ≥0.

Definition 4.3.1. A 훤-graded ring is a ring 푅 whose underlying additive group is endowed with a decomposition 푅 = ⨁훾∈훤 푅훾, such that 푅훾푅휂 ⊂ 푅훾+휂 for all 훾, 휂 ∈ 훤. For 푅 as above, a 훤-graded 푅-module is an 푅-module 푀 whose underlying additive group decomposes as 푀 = ⨁훾∈훤 푀훾, such that 푅훾푀휂 ⊂ 푀훾+휂 for all 훾, 휂 ∈ 훤; in particular, 푅 itself is a 훤-graded 푅-module. If 푥 ∈ 푀훾 ∖ {0}, we say 푥 is homogeneous of degree 훾.

We will often omit 훤 when there is no worry of confusion. Note that if 0 is allowed to be homogeneous, as people sometimes do, it will be homogeneous of any degree.

Exercise 4.3.2. Show that in a graded ring 푅 we always have 1 ∈ 푅0, provides that (훤, +) satisfies the cancellation law: 훾 + 휂 = 휂 ⟺ 훾 = 0. Hint: let 푒0 be the component of 1푅 in degree 0, argue that 푥푒0 = 푥 = 푒0푥 for all homogeneous 푥 ∈ 푅. The condition 1 ∈ 푅0 is sometimes built into the definition of graded rings.

Definition 4.3.3. A graded submodule of a graded 푅-module 푀 is a submodule 푁 with 푁 = ⨁훾(푁 ∩ 푀훾), which gives rise to a natural grading 푁훾 ∶= 푁 ∩ 푀훾 on 푁.

For graded 푁 ⊂ 푀, the quotient 푅-module 푀/푁 = ⨁훾 푀훾/푁훾 is again graded. As a special case, we have the notion of graded ideals of 푅 (also known as homogeneous ideals), and the quotient ring 푅/픞 with respect to graded 픞 inherits the evident grading. Let 푁 be an 푅-submodule of 푀 and suppose 푀 is graded. The following are easily seen to be equivalent:

(i) 푁 ⊂ 푀 is graded;

(ii) 푁 is generated by homogeneous elements;

(iii) if 푥 = ∑훾 푥훾 ∈ 푁 with each 푥훾 homogeneous of degree 훾, then ∀ 푥훾 ∈ 푁.

Example 4.3.4. Let 퐴 be a ring and 푅 ∶= 퐴[푋1, … , 푋푛]. Then 푅 is naturally ℤ≥0-graded by degrees: for each 푑 ∈ ℤ≥0, let 푅푑 be the set of homogeneous polynomials of total degree 푑. Ideals generated by homogeneous polynomials are precisely the graded ideals. The importance of this grading comes from projective algebraic geometry. On the other hand, 푅 can also be graded by monomials by taking 훤 = ℤ≥0 ×⋯×ℤ≥0 푛 푅 = 퐴 ⋅ 푋푑1 ⋯ 푋푑푛 ( copies), and we set (푑1,…,푑푛) 1 푛 Many constructions in commutative algebra can be generalized to the graded case. Let us illustrate what one can do in an important case, the primary decomposition (cf. [8, §3.5 and Exercise 3.5]). In the ℤ-graded context, it says that for a finitely generated graded module 푀 over a Noetherian graded ring, the associated primes of 푀 are all graded ideals, and one can write {0} = 푁1 ∩ ⋯ ∩ 푁푚 where 푁푖 ⊂ 푀 are graded submodules with Ass(푀/푁푖) = {픭푖}, 픭푖 ∈ Ass(푀), etc. Most of the arguments in the ungraded case carry over verbatim, and the only new technique is the following ⋅ 46 ⋅ Going-up, going-down, gradings and filtrations

Lemma 4.3.5. Let 푀 be a ℤ-graded module over a ℤ-graded ring 푅. If 푥 ∈ 푀 and 픭 ∶= ann(푥) is a prime ideal, then

(i) 픭 is a homogeneous ideal, and

(ii) 픭 = ann(푦) for some homogeneous element 푦 ∈ 푀.

Proof. We begin with (i). Let 푡 ∈ 픭 and 푥 ∈ 푀 be such that 픭 = ann(푀). Write

푡 = ∑ 푡훾, 푥 = ∑ 푥휂, 훾∈풜 휂∈ℬ

where 푡훾 ∈ 푅훾 ∖ {0} and 푥휂 ∈ 푀휂 ∖ {0} for all 훾, 휂. Denote by 훾0 and 휂0 the minimal elements in 풜 and ℬ, respectively. Homogenity amounts to 푡훾 ∈ 픭 for each 훾 ∈ 풜, and this will be done by induction on |ℬ|. By a further induction on |풜|, for fixed 푥 and 픭 푡 ∈ 픭 , this can be reduced to showing 훾0 . 푡푥 = 0 푡 푥 = 0 푥 = 푥 First of all, considerations of degrees and lead to 훾0 휂0 . If 휂0 (i.e. |ℬ| = 1 푡 ∈ (푥) = 픭 ), we obtain 훾0 ann as required. In general: ⋄ 픭 = (푡 푥) Suppose that ann 훾0 ; since the decomposition

푡 푥 = ∑ 푡 푥 훾0 훾0 휂 휂∈ℬ 휂≠휂0

involves fewer homogeneous terms, 픭 is then homogeneous by our induction hypothesis on |ℬ|.

⋄ 푠 ∈ 푅 ∖ 픭 푠(푡 푥) = 0 푠푡 ∈ 픭 푡 ∈ 픭 Suppose there exists such that 훾0 , then 훾0 , hence 훾0 . This concludes the homogeneity (i).

From the homogeneity 픭 we infer that 픭 ⊂ ann(푥휂) for each 휂 ∈ ℬ. Now that

픭 = ann(푥) ⊃ ∏ ann(푥휂), 휂∈ℬ

we have 픭 ⊃ ann(푥휂) for some 휂, hence 픭 = ann(푥휂). Take 푦 = 푥휂 to obtain (ii).

4.4 Filtrations

Now turn to filtrations. We only deal with decreasing filtrations indexed by ℤ≥0. Definition 4.4.1. A filtration on a ring 푅 is a descending sequence

푅 = 퐹0푅 ⊃ 퐹1푅 ⊃ 퐹2푅 ⊃ ⋯

푖 푗 푖+푗 of ideals such that 퐹 푅 ⋅ 퐹 푅 ⊂ 퐹 푅. Define the associated ℤ≥0-graded ring

푛 푛+1 gr퐹(푅) ∶= ⨁ ⎵⎵⎵⎵⎵퐹 푅/퐹 푅 푛≥0 푛 =∶gr퐹 푅 §4.4 Filtrations ⋅ 47 ⋅

whose multiplication is defined as follows: if 푥 ∈ 퐹푛푅/퐹푛+1푅 and 푦 ∈ 퐹푚푅/퐹푚+1푅, choose liftings ̃푥∈ 퐹푛푅 and ̃푦∈ 퐹푚푅 and define 푥푦 ∶= the image of ̃푥̃푦 in 퐹푛+푚푅/퐹푛+푚+1푅; this is readily seen to be well-defined. The multiplication of arbitrarily many homogeneous elements can be obtained in the same recipe. The datum (푅, 퐹•푅) is called a filtered ring, Given a filtered ring 푅, a filtered 푅-module 푀 is an 푅-module 푀 equipped with a descending sequence1 of submodules ⋯ ⊃ 퐹푖푀 ⊃ 퐹푖+1푀 ⊃ ⋯ , 푖 ∈ ℤ such that 퐹푖푅 ⋅ 퐹푗푀 ⊂ 퐹푖+푗푀. Define the associated graded module as the ℤ-graded

gr퐹 푅-module 푛 푛+1 gr퐹(푀) ∶= ⨁ ⎵⎵⎵⎵⎵⎵퐹 푀/퐹 푀 푛∈ℤ 푖 =∶gr퐹 푀 whose scalar multiplication is defined using liftings as above.

The subscript 퐹 in gr will often be omitted. To guarantee that gr퐹(푅) ≠ {0}, we usually impose the harmless condition 푅 = 퐹0푅 ⊋ 퐹1푅. Example 4.4.2. Let 픞 be a proper ideal of 푅, then 퐹푖푅 ∶= 픞푖 defines a filtration on 푅, called the 픞-adic filtration. Definition 4.4.3. Equip 푅 with the 픞-adic filtration. A filtered 푅-module 푀 is called 픞-stable if 픞 ⋅ 퐹푖푀 = 퐹푖+1푀 for 푖 ≫ 0. Recall that 픞 ⋅ 퐹푖푀 ⊂ 퐹푖+1푀 holds for all 푀, which is a part of our assumption. As an easy example, set 퐹푖푀 ∶= 픞푖푀 for 푖 ≥ 1, and 퐹≤0푀 ∶= 푀; this is the 픞-adic filtration on 푀, which is obviously 픞-stable. Proposition 4.4.4. Suppose 픞 is a proper ideal of 푅. If 푅 is Noetherian, so is gr(푅) with respect to the 픞-adic filtration. In fact gr(푅) is finitely generated over gr0(푅). 2 Proof. Let 푥1, … , 푥푛 be generators of 픞, with images 푖̄푥 ∈ 픞/픞 . Then gr(푅) is generated 0 by 1̄푥 , …푛 ̄푥 over 푅/픞 = gr 푅, which is also Noetherian. Now apply Hilbert’s Basissatz.

Proposition 4.4.5. Let 픞 be a proper ideal of 푅 and 푀 a finitely generated 푅-module. Suppose 푀 is endowed with an 픞-stable filtration such that 퐹푖푀 is finitely generated for each 푖, and 퐹≤0푀 = 푀. Then gr(푀) is a finitely generated gr(푅)-module. 푖 푖+1 푖 Proof. Take 푛 such that 픞 ⋅ 퐹 푀 = 퐹 푀 for all 푖 ≥ 푛. Then in gr(푀) = ⨁푖 gr 푀 we have (픞/픞2) ⋅ gr푖 푀 = gr푖+1 푀, 푖 ≥ 푛. Therefore it suffices to take generators from gr0 푀, … , gr푛 푀, each of whom is finitely generated over 푅/픞 = gr0 푅. 1 In view of later applications, the filtration on a module is indexed by ℤ instead of ℤ≥0. ⋅ 48 ⋅ Going-up, going-down, gradings and filtrations

4.5 Theorems of Artin–Rees and Krull

Conserve the conventions in the previous section on filtrations, etc.

Definition 4.5.1 (Morphisms between filtered objects). Let (퐴, 퐹•퐴) and (퐵, 퐹•퐵) be filtered rings. A morphism between them means a ring homomorphism 휑 ∶ 퐴 → 퐵 satisfying 휑(퐹푖퐴) ⊂ 퐹푖퐵 for all 푖. Similarly, suppose 퐴 is filtered and let 푀, 푁 be filtered 퐴-modules. A morphism 푀 → 푁 means a homomorphism 휓 ∶ 푀 → 푁 of 푅-modules satisfying 휓(퐹푖푀) ⊂ 퐹푖푁 for all 푖.

This makes the filtered rings and the filtered modules over a filtered ring intocat- egories. Obviously, morphisms 휑 between filtered objects induce graded morphisms gr 휑 between the associated graded objects. Therefore we obtain a functor from the category of filtered rings or modules into their graded avatars. Remark 4.5.2. Suppose 휑 ∶ 푀 → 푁 is a morphism between filtered 퐴-modules. The quotient 푀/ ker(휑) inherits a filtration from 푀, whereas the submodule im(휑) inherits one from 푁. When the natural isomorphism 푀/ ker(휑) → im(휑) is an isomorphism between filtered modules, or equivalently

∀푖 ∈ ℤ, 휑(퐹푖푀) = 휑(푀) ∩ 퐹푖푁, we say 휑 is a strict morphism. It is often useful to relate properties of a filtered module or morphism to its graded counterpart. Propositions 4.4.4 and 4.4.5 are such examples. Here is an example for 푖 the other direction. We say that a filtration on 푀 is exhaustive if ⋃푖 퐹 푀 = 푀, separating 푖 (or Hausdorff ) if ⋂푖 퐹 푀 = {0} Proposition 4.5.3. Suppose that 휑 ∶ 푀 → 푁 is a morphism between filtered 푅-modules. If 푀 is exhaustive and separating, and gr 휑 is injective, then 휑 is also injective.

Proof. Let 푥 ∈ ker(휑). There exists 푛 such that 푥 ∈ 퐹푛푀. Regard 푥 + 퐹푛+1푀 as an element of gr푛 푀. Then gr(휑)(푥 + 퐹푛+1푀) = 휑(푥) + 퐹푛+1푁 = 0, so 푥 ∈ 퐹푛+1푀. 푘 Iterating this argument, we have 푥 ∈ ⋂푘≥푛 퐹 푀 = {0}. See Lemma 5.2.3 for the case of surjections. In what follows, 픞 always denotes a proper ideal of a ring 푅.

Definition 4.5.4 (Blow-up algebras and modules). Introduce an indeterminate 푋 and define the ℤ≥0-graded 푅-algebra

푛 푛 Bl픞푅 ∶= ⨁ 픞 푋 ⊂ 푅[푋]. 푛≥0

푖 If an 푅-module 푀 is endowed with a filtration (퐹 푀)푖≥0 compatible with 픞, we define

Bl(푀) ∶= ⨁(퐹푛푀) ⊗ 푋푛 ⊂ 푀 ⊗ 푅[푋]. 푛≥0 푅

Clearly, Bl(푀) is a graded Bl픞푅-module. §4.5 Theorems of Artin–Rees and Krull ⋅ 49 ⋅

Remark 4.5.5. The indeterminate 푋 is somehow a placeholder. Enlarge Bl픞푅 to 퐵̃ by setting the negative graded pieces to be 푅 ⋅ 푋<0, and set 푇 = 푋−1. We see that 퐵̃ is actually an 푅[푇]-algebra, sometimes called the Rees algebra of 픞. Under the specialization 푇 = 0 we get 퐵̃ 픞푛 ≃ ⨁ ⋅ 푇−푛 ≃ gr(푅). 푛+1 푇퐵̃ 푛≥0 픞 On the other hand, inverting 푇 yields

퐵[푇̃ −1] = ⨁ 푅 ⋅ 푇−푛 = 푅[푇±1]. 푛∈ℤ

This reflects a well-known deformation construction in geometry; gr(푅) is actually the graded 푅-algebra corresponding to the normal cone defined by 픞 ⊂ 푅. We refer to [9, §5.1] for details.

Lemma 4.5.6. Consider a ring 푅 with proper ideal 픞, together with a filtered 푅-module 푀, assume furthermore that each 퐹푖푀 is finitely generated over 푅. The following are equivalent:

(i) Bl(푀) is finitely generated over Bl픞푅; (ii) the filtration on 푀 is 픞-stable. (Definition 4.4.3)

Proof. (i) ⟹ (ii): Choose homogeneous generators 푥1, … , 푥푛 ∈ Bl(푀) with degrees 푑1, … , 푑푛 respectively. It is then routine to see that

푖+1 푖+1 푖 푖 푖 ≥ max{푑1, … , 푑푛} ⟹ (퐹 푀)푋 = 픞푋 ⋅ (퐹 푀)푋 , that is, 픞 ⋅ 퐹푖푀 = 퐹푖+1푀 for these 푖. (ii) ⟹ (i): Suppose 픞 ⋅ 퐹푖푀 = 퐹푖+1푀 for 푖 ≥ 푑, then Bl(푀) is generated by 푗 푗 푗 ⨁푗≤푑(퐹 푀)푋 , and each 퐹 푀 is finitely generated over 푅 = (Bl픞푅)0.

Theorem 4.5.7 (Artin–Rees). Let 푅 be a Noetherian ring endowed with 픞-adic filtration. Let 푀 be a finitely generated 푅-module and 푁 ⊂ 푀 an 푅-submodule. Then the filtration on 푁 induced by the 픞-adic filtration of 푀, namely 퐹푖푁 ∶= 픞푖푀 ∩ 푁, is 픞-stable.

Proof. Define Bl(푁) using the induced filtration 퐹푖푁 ∶= 픞푖푀 ∩ 푁, which is a submodule of the finitely generated Bl픞푅-module Bl(푀) (Lemma 4.5.6). If 픞 = (푎1, … , 푎푚) than Bl픞푅 = 푅[푎1푋, … , 푎푚푋] ⊂ 푅[푋], hence Noetherian by Hilbert’s Basissatz. We deduce • that Bl(푁) is finitely generated over Bl픞푅. In turn, this implies 퐹 푁 is an 픞-stable filtration on 푁 by Lemma 4.5.6.

푛 Theorem 4.5.8. For 푅, 픞, 푀 as in the previous theorem, we set 푁 ∶= ⋂푛≥0 픞 푀. Then 픞푁 = 푁.

Proof. Since the induced filtration on 푁 is 픞-stable by Theorem 4.5.7, for 푛 ≫ 0 we have

푁 = 픞푛푀 ∩ 푁 = 픞 ⋅ (픞푛−1푀 ∩ 푁) = 픞푁.

The assertion follows. ⋅ 50 ⋅ Going-up, going-down, gradings and filtrations

푛 Corollary 4.5.9 (Krull). If 픞 ⊂ rad(푅), then ⋂푛≥0 픞 푀 = {0} for any finitely generated 푛 푅-module 푀. In particular ⋂푛≥0 픞 = {0} whenever 픞 ⊂ rad(푅). Proof. Theorem 4.5.8 together with Nakayama’s lemma imply 푁 = {0}.

Corollary 4.5.10 (Krull’s Intersection Theorem). Let 푅 be a Noetherian domain and 픞 a 푛 proper ideal. Then ⋂푛≥0 픞 = {0}. 푛 Proof. Define 푁 ∶= ⋂푛≥0 픞 ⊂ 푅. By Theorem 4.5.8 we have 픞푁 = 푁, thus there exists 푟 ∈ 픞 with 1 + 푟 ∈ ann(푁) by Nakayama’s Lemma (Theorem 1.3.5). As 픞 is proper, 1 + 푟 cannot be zero. Since 푅 is a domain containing 푁, the only possibility is 푁 = {0} as asserted. Lecture 5

From completions to dimensions

The main references are [11, 8].

5.1 Completions

Consider a ring 푅 together with a family of ideals ℐ ≠ ∅, such that for any 퐼, 퐽 ∈ ℐ there exists 퐾 ∈ ℐ with 퐾 ⊂ 퐼 ∩ 퐽. This turns 푅 into a topological ring, characterized by the property that ℐ forms a local base of open neighborhoods of 0. Recall that being a topological ring means that addition, multiplication and 푥 ↦ −푥 are all continuous. By standard arguments, 푅 is Hausdorff if and only if {0} is closed, if and only if ⋂퐼∈ℐ 퐼 = {0}. To simplify matters, we assume that ⋄ the family ℐ is countable, so that the topological properties (accumulation points, etc.) are detected by convergence of sequences as in the case of metric spaces; ⋄ furthermore, we may arrange that ℐ = {퐼 ⊃ 퐽 ⊃ 퐾 ⊃ ⋯}, in other words our topology comes from filtrations. Without the countability assumption, the sequences will have to be replaced by filters. It makes sense to talk about topological 푅-modules for a topological ring 푅. By replacing filtration by ideals by filtration by 푅-submodules subject to the usual compatibility relation 퐹푖푅 ⋅ 퐹푗푀 ⊂ 퐹푖+푗푀, the recipe above applies to 푅-modules as well. Given 푁 ⊂ 푀, the topology so obtained on 푀 passes to 푀/푁 by taking the quotient topology, or equivalently the quotient filtration (퐹•푀 + 푁)/푁. If the filtration in question is 퐼-adic, where 퐼 ⊊ 푅 is an ideal, we obtain the 퐼-adic topology on rings and modules. An 푅-module 푀 equipped with a topology as above is complete if every Cauchy sequence (푥푛)푛≥1 has a limit; a Cauchy sequence (푥푛)푛≥1 is a sequence satisfying

∀퐼 ∈ ℐ, ∃푁 푖, 푗 ≥ 푁 ⟹ 푥푖 − 푥푗 ∈ 퐼.

As in the familiar case of metric spaces, one has the completion of 푀. It is actually a morphism 푀 → 푀̂ with 푀̂ complete Hausdorff, characterized by the following universal ⋅ 52 ⋅ From completions to dimensions

property:

휑∶ cont. homo. 푀 푀̂ 푀 퐿 ⇝ ∃! ̂휑 complete Hausdorff 휑 퐿 The uniqueness results immediately, and the formation of 푀 ↦ 푀̂ is seen to be functorial in 푀. If 푀 is already complete Hausdorff, one may take 푀̂ = 푀. Certainly, the same applies to the ring 푅. Exercise 5.1.1. Suppose that 푅 is complete Hausdorff with respect to the 퐼-adic topology, where 퐼 is a proper ideal. Show that every element of the form 푢 + 푥, 푢 ∈ 푅× and 푥 ∈ 퐼, is invertible. From the algebraic perspective, the completion of a filtered 푅-module 푀 = 퐹0푀 ⊃ 퐹1푀 ⊃ ⋯ can be constructed as the projective limit 푀̂ ∶= lim 푀/퐹푖푀 ←−− 푖≥1 푖 푖 = {(푥푖)푖≥1 ∶ 푖 ≤ 푗 ⟹ 푥푖 ≡ 푥푗 (mod 퐹 푀)} ⊂ ∏ 푀/퐹 푀. 푖≥1 The morphism 푀 → 푀̂ is the diagonal map. The topology on 푀̂ arises from the filtration 푖 푖 퐹 푀̂ ∶= ker [푝푖 ∶ 푀̂ → 푀/퐹 푀] = {(푥푛)푛 ∈ 푀̂ ∶ 푖 ≤ 푘 ⟹ 푥푖 = 0} , so that the preimage of 퐹푖푀̂ in 푀 is precisely 퐹푖푀. In the case where 푀 = 푅 and 퐹푖푅 ̂ 푖 are ideals, we obtain the complete Hausdorff ring 푅, which is a subring of ∏푖≥1 푅/퐹 푅. Since the filtrations on 푅 and 푀 are assumed compatible, 푀̂ is an 푅̂-module. Example 5.1.2. Fix a prime number 푝. The completion of ℤ with respect to the ideal 푝ℤ is nothing but the ring ℤ푝 of 푝-adic integers. Similarly, the completion of 한[푋] with respect to (푋) is isomorphic to the 한-algebra 한J푋K. Exercise 5.1.3. Describe the kernel of 푀 → 푀̂ and show 푀 ↪ 푀̂ if and only if 푀 is Hausdorff. Exercise 5.1.4. Show that the topology of 푀̂ is the restriction of the product topology of 푖 푖 ∏푖 푀/퐹 푀, provided that each 푀/퐹 푀 is endowed with the discrete topology. Show ̂ 푖 that 푀 is a closed subspace of ∏푖 푀/퐹 푀 Lemma 5.1.5. Let 푀 be a complete 푅-module with respect to some filtration 퐹•푀. For any submodule 푁, the quotient 푀/푁 is also complete with respect to the quotient topology, or equivalently with respect to the quotient filtration (퐹•푀 + 푁)/푁.

Proof. Let 푛̄푥 be a Cauchy sequence in 푀/푁. Choose preimages 푀 ∋ 푥푛 ↦푛 ̄푥 for all 푖(푛) 푛. We have 푛+1̄푥 −푛 ̄푥 ∈ 퐹 푀 + 푁 where lim푛→∞ 푖(푛) = ∞, therefore we can write 푖(푛) ′ 푥푛+1 −푥푛 = 푦푛 +훿푛 where 푦푛 ∈ 퐹 푀 and 훿푛 ∈ 푁. We contend that 푥푛 ∶= 푥1 +∑푖<푛 푦푖 ′ ′ is a Cauchy sequence in 푀. Indeed, for any 푖 > 푗 we have 푥푖 − 푥푗 = ∑푗≤푘<푖 푦푘, which inf푘 푖(푘) ′ lies in 퐹 푀. This implies (푥푛)푛 is a Cauchy sequence, hence has a limit 푥 ∈ 푀. It ′ ′ is also clear that 푥푛 ↦푛 ̄푥 . Hence 푛̄푥 has a limit ̄푥= 푥 mod 푁 in 푀/푁. §5.1 Completions ⋅ 53 ⋅

Let us turn to the exactness of completion. This should be understood in the broader framework of lim of arbitrary projective systems. For simplicity, we only consider 퐼-adic ←−− topologies on finitely generated modules over a Noetherian ring. Observe that any homomorphism 휑 ∶ 푀 → 푁 between 푅-modules is automatically 픞-adically continuous, for that 휑(픞푛푀) ⊂ 픞푛푁.

Proposition 5.1.6. Let 푅 be a Noetherian ring and 퐼 ⊊ 푅 an ideal. Suppose 0 → 푀′ → 푀 → 푀″ → 0 is an exact sequence of finitely generated 푅-modules, each term equipped with the 퐼-adic topology. The completed sequence 0 → 푀̂ ′ → 푀̂ → 푀″̂ → 0 is also exact.

Consequently, completion preserves the exactness of sequences formed by finitely generated 푅-modules. This is what makes completion so useful. Proof. We shall show (i) the topology on 푀′ induced from 푀 is the same as the 퐼-adic topology; (ii) the quotient topology on 푀″ = 푀/푀′ is 퐼-adic; (iii) the completion 푀̂ ′ is naturally identified with the closure of the image of 푀′ in 푀̂ ; (iv) 푀̂ ″ is naturally identified with the quotient of 푀̂ by 푀̂ ′. (i) is a direct consequence of Artin–Rees Theorem 4.5.7: for 푛 ≫ 0 we have

퐼푛푀′ ⊂ 푀′ ∩ 퐼푛푀 = 퐼(푀′ ∩ 퐼푛−1푀) ⊂ 퐼푛−1푀′, (5–1)

and this suffices to identify the resulting topologies. (ii) is immediate. As for (iii), we may embed 푀′ into 푀 and work with the induced topology. Realize 푀̂ as lim 푀/퐼푛푀. As a topological module 푀̂ ′ equals ←−−푛 푀′ ̂ ′ lim ′ 푛 = { ̂푥= (푥푛)푛 ∈ 푀 ∶ ∀푘, 푥푘 comes from 푀 } ←−−푛 푀 ∩ 퐼 푀 = { ̂푥∈ 푀̂ ∶ ∀푘 ∃푦 ∈ 푀′ s.t. ̂푥∈ 푦 + 퐹푘푀}̂ which is readily seen to be the closure of the image of 푀′. For (iv), the quotient 푀/̂ 푀̂ ′ is Hausdorff since 푀̂ ′ is a closed submodule. It is also complete by Lemma 5.1.5. There is a natural homomorphism 푀/푀′ → 푀/̂ 푀̂ ′. Given any continuous homomorphism 휑 ∶ 푀/푀′ → 푁 to a complete Hausdorff 푅-module 푁, we may pull it back to 푀 → 푁, which corresponds to a unique continuous 푀̂ → 푁 that is trivial on the image of 푀′; but such a homomorphism must also vanish on the closure 푀̂ ′. This yields the required ̂휑∶ 푀/̂ 푀̂ ′ → 푁 in the universal property. For any 푅-module 푀 endowed with 퐼-adic topology, there is a canonical homomorphism 푀 ⊗ 푅̂ → 푀̂ . Indeed, 푀̂ = lim 푀/퐼푛푀 is a 푅̂ = lim 푅/퐼푛-module by 푅 ←−−푛 ←−−푛

(푟푛)푛≥1 ⋅ (푥푛)푛≥1 = (푟푛푥푛)푛≥1, (푟푛)푛 ∈ 푅,̂ (푥푛)푛 ∈ 푀,̂ hence the 푅-homomorphism 푀 → 푀̂ gives rise to 푀⊗푅̂ → 푀̂ by the universal property 푅 of base change. This homomorphism is continuous if 푀 ⊗ 푅̂ is filtered by the images 푅 of 푀 ⊗ (퐹•푅)̂ , which makes it into a topological 푅̂-module. 푅 ⋅ 54 ⋅ From completions to dimensions

Theorem 5.1.7. Suppose 푅 is Noetherian. Let 푀 be a finitely generated 푅-module endowed with the 퐼-adic topology. The homomorphism 푀 ⊗ 푅̂ → 푀̂ is then an isomorphism. 푅

Proof. Write down a finite presentation 푅⊕푎 → 푅⊕푏 → 푀 → 0. By the naturality of the homomorphism above, the right-exactness of ⊗ and Proposition 5.1.6, we have a commutative diagram

푅⊕푎 ⊗ 푅̂ 푅⊕푎 ⊗ 푅̂ 푀 ⊗ 푅̂ 0 푅 푅 푅

푅̂⊕푎 푅̂⊕푏 푀̂ 0

∧ with exact rows. Completion commutes with direct sums: (푁1 ⊕ 푁2) = 푁̂1 ⊕ 푁̂2 canonically (easy, and the categorical reason is that completion is left adjoint to oblivion 푅-CompHausMod → 푅-TopMod). Therefore the first two vertical arrows may be identified with the canonical arrow 푅⊕⋆ ⊗ 푅̂ → 푅̂ ⊕⋆, ⋆ ∈ {푎, 푏}, which is an isomorphism of 푅 topological 푅̂-modules. We infer that 푀 ⊗ 푅̂ →∼ 푀̂ topologically. 푅

Remark 5.1.8. In fact 푀 ⊗ 푅̂ →∼ 푀̂ is also a homeomorphism. It suffices to observe that 푅 in the rows of the commutative diagram above, 푀 ⊗ 푅̂ and 푀̂ are both realized as 푅 quotient topological 푅̂-modules, and that 푅⊕⋆ ⊗ 푅̂ →∼ 푅̂ ⊕⋆ is a homeomorphism. The 푅 second point has been observed in the proof of Proposition 5.1.6, and the first follows from the fact completed modules carry the 퐼-adiĉ topology. See Proposition 5.2.2. We do not need this result. In the following statements, 푅 is Noetherian and an ideal 퐼 ⊊ 푅 is chosen.

Corollary 5.1.9. The canonical homomorphism 푅 → 푅̂ is flat.

Proof. Flatness can be tested on short exact sequences of the form 0 → 픞 → 푅 → 푅/픞 → 0 where 픞 is a finitely generated ideal of 푅. Its base-change to 푅̂ is the same as completion, and completion is an exact functor by Proposition 5.1.6.

Corollary 5.1.10. Assume 푅 is 퐼-adically complete Hausdorff. Then every finitely generated 푅-module 푀 is 퐼-adically complete Hausdorff, and any submodule 푁 ⊂ 푀 is closed.

Proof. For the first assertion: the completion of 푀 can be identified with the composite 푀 = 푀 ⊗ 푅 → 푀 ⊗ 푅̂ → 푀̂ , which is bijective. Therefore every Cauchy sequence in 푀 푅 푅 has a limit in 푀. As to the second assertion, recall that the 퐼-adic topology on 푁 is the same as the one restricted from 푀 by (5–1). It remains to notice that complete subspaces must be closed. §5.2 Further properties of completion ⋅ 55 ⋅

5.2 Further properties of completion

Let 푅 be a Noetherian ring and 푀 a finitely generated 푅-module. Fix an ideal 퐼 ⊊ 푅. Unless otherwise specified, the topologies and completions are always 퐼-adic.

Proposition 5.2.1. Let 픞 be an ideal, then 픞푀̂ = 픞푀̂ =푀 ̂픞 ̂ as submodules of 푀̂ . Consequently 푀/픞̂ 푀̂ ≃ (푀/픞푀)∧ canonically.

Proof. By the exactness of completion (Proposition 5.1.6), we may realize ̂픞 as an ideal of 푅̂; in fact it is the image 픞푅̂ of 픞 ⊗ 푅̂ → 푅 ⊗ 푅̂ = 푅̂. Hence 픞푅̂ =. ̂픞 Now consider the 푅 푅 commutative diagram 픞 ⊗ 푀 ⊗ 푅̂ 픞 ⊗ 푀̂ 푅 푅 푅

픞푀 ⊗ 푅̂ 푀̂ 푅

The upper horizontal arrow is an isomorphism by Theorem 5.1.7, therefore the diagonal arrow has image equal to 픞푀̂ . The lower horizontal arrow is just the completion of 픞푀 ↪ 푀, thus injective with image (픞푀)∧ by Proposition 5.1.6. A comparison yields (픞푀)∧ = 픞푀̂ . Also note that 푀̂픞 ̂ = 픞푅̂푀̂ = 픞푀̂ . The final assertion results from the exactness of completion.

Since the 퐼-adic topology on 푀/퐼푛푀 is discrete, as a special case (픞 = 퐼푛) we deduce the natural identifications

푀/퐼푛푀 = 푀/퐼̂ 푛푀̂ = 푀/̂ 퐼푛̂ 푀,̂ ∀푛 ≥ 0, ̂ ̂ gr퐼(푅) = gr퐼(푅) = gr퐼(̂ 푅), ̂ ̂ gr퐼(푀) = gr퐼(푀) = gr퐼(̂ 푀).

Proposition 5.2.2. For any finitely generated 푅-module 푀, the topology on 푀̂ coincides with the 퐼-adiĉ one.

푛 Proof. Consider the closure of the image of 퐼 푀 in 푀̂ . It is readily seen to be {(푥푘)푘 ∶ 푖 ≤ 푛 푛 푛 ⟹ 푥푖 = 0} = 퐹 푀̂ . On the other hand, we have seen that this closure is 퐼̂푀 ⊂ 푀̂ . By virtue of Proposition 5.2.1, we have 퐼̂푛푀 = 퐼̂푛푀̂ and 퐼̂푛 = 퐼푛푅̂ = (퐼푅)̂ 푛 = 퐼푛̂ .

Lemma 5.2.3. Consider a homomorphism 휑 ∶ 퐿 → 푁 between filtered modules over some ring, such that 퐿 is complete, 푁 is Hausdorff and exhaustive (see §4.5) with respect to their filtrations, and gr(휑) ∶ gr(퐿) → gr(푁) is surjective. Then 휑 is also surjective.

Proof. Let 푦 ∈ 퐹푑푁, we may take 푥 ∈ 퐹푑퐿 such that 푦′ ∶= 푦 − 휑(푥) ∈ 퐹푑+1푁. Next, take 푥′ ∈ 퐹푑+1퐿 with 푦″ ∶= 푦′ − 휑(푥′) ∈ 퐹푑+2푁, and so forth. Use the completeness of 퐿 to ′ ″ define 푥∞ ∶= 푥 + 푥 + 푥 + ⋯, which maps to 푦 since 푁 is Hausdorff.

Proposition 5.2.4. The ring 푅̂ is also Noetherian, and 퐼̂ ⊂ rad(푅)̂ . ⋅ 56 ⋅ From completions to dimensions

Proof. Let 프 be any ideal of 푅̂, equipped with the filtration 퐹푛프 ∶= 퐼푛̂ ∩ 프. We have ̂ to show 프 is finitely generated. Since gr퐼(푅) = gr퐼(̂ 푅) is Noetherian, so is gr퐹(프). 푑 푑푖 ̄ 푖 Take 푡1, … , 푡푛 ∈ 프, 푡푖 ∈ 퐹 프, whose images 푡푖 in gr퐹 (프) generates gr퐹(프). Using an appropriately shifted filtration on 퐿 ∶= 푅̂ ⊕푛, we obtain a filtered homomorphism 휑 ∶ 퐿 → 프 with image (푡1, … , 푡푛), such that gr(휑) is surjective. Now apply the previous Lemma to obtain the first assertion. One of the characterizations of Jacobson radical says that 퐼̂ ⊂ rad(푅)̂ if and only if 1 − 퐼̂ ⊂ 푅̂ ×. This is verified by noting that (1 − 푡)−1 = 1 + 푡 + 푡2 + ⋯ converges 퐼-adically. This proves the second assertion.

Proposition 5.2.5. The map 픭 ↦ 픭̂ furnishes an injection from 푉(퐼) to Spec(푅)̂ satisfying 1∶1 푅/픭 ≃ 푅/̂ 픭̂ (as rings). It restricts to a bijection MaxSpec(푅) ∩ 푉(퐼) MaxSpec(푅)̂ . Consequently, if 푅 is local (resp. semi-local), so is 푅̂.

Proof. Since 픭 ⊃ 퐼, the 퐼-adic topology on 푅/픭 is discrete. By Proposition 5.2.1, 푅/̂ ̂푝≃ (푅/픭)∧ = 푅/픭, and here the isomorphism even respects ring structures. Therefore 픭̂ is a prime ideal. Moreover, it is maximal if and only if 픭 is. Claim: 픭 is the preimage of 픭̂ = lim 픭/(퐼푛 ∩ 픭) under 푅 → 푅̂ = lim 푅/퐼푛. Indeed, lying in that preimage amounts ←−−푛 ←−−푛 to 푥 ∈ 퐼푛 + 픭 = 픭, for all 푛 ≥ 1. Injectivity follows. Lemma 5.2.4 implies that every 프 ∈ MaxSpec(푅)̂ contains 퐼,̂ therefore is open by Proposition 5.2.2. Since 푅̂ is Noetherian, 프 is also closed by Corollary 5.1.10. As 푅 → 푅̂ has dense image, we conclude that 프 equals the completion of its preimage 픪 ∈ 푉(퐼) ⊂ Spec(푅). By the previous paragraph, 픪 is a maximal ideal.

5.3 Hilbert–Samuel polynomials

For a graded ring 푅 = ⨁훾 푅훾 (we always assume 1 ∈ 푅0) and a given 휂, we may define its twist 푅(휂) as the graded 푅-module

푅(휂)훾 ∶= 푅훾+휂.

To generate a graded 푅-module 푀 by finitely many homogeneous elements 푥1, … , 푥푛, of degrees 휂1, … , 휂푛 respectively, is equivalent to giving a surjection of graded 푅-modules

푛 ⨁푖=1 푅(−휂푖) 푀 (5–2) (… , 0, 1⎵ , 0, …) 푥푖. 푖−th slot

Hereafter, we assume that

푁 ⋄ everything is graded by 훤 = (ℤ≥0, +) for some fixed 푁,

⋄ 푅0/푅0 ∩ ann(푀) is an Artinian ring,

⋄ 푅 is finitely generated over 푅0. §5.3 Hilbert–Samuel polynomials ⋅ 57 ⋅

The appearance of ann(푀) is harmless since 푅 can be safely replaced by 푅/픞픫픫(푀), which is legitimate since ann(푀) is a graded ideal of 푅. To see this, write ann(푀) as the intersection of ann(푥) where 푥 ranges over the homogeneous elements of 푀, and observe that ann(푥) must be graded.

Lemma 5.3.1. For 푅 as above and 푀 a finitely generated graded 푅-module, each graded piece 푀훾 is an 푅0-module of finite length.

Proof. Using (5–2) this is readily reduced to the case 푀 = 푅(−휂), and then to 푀 = 푅. Write 푅 = 푅0[푥1, … , 푥푛] where each 푥푖 is homogeneous of degree 푑푖. Given 훾, the 푅0- 푎1 푎푛 module 푀훾 is generated by monomials 푥1 ⋯ 푥푛 with ∑푖 푎푖푑푖 = 훾 and 푎1, … , 푎푛 ∈ ℤ≥0; this admits only finitely many solutions (푎1, … , 푎푛). We conclude that 푀훾 has finite length since 푅0/푅0 ∩ ann(푀) is an Artinian ring.

Recall that saying a module 푁 over a ring 퐴 has finite length means that there exists a composition series

푁 = 푁0 ⊃ ⋯ ⊃ 푁푛 = {0}, ∀푁푖/푁푖+1 is simple.

The unique number (Jordan–Hölder Theorem) is called the length of 푁, denoted by ℓ퐴(푁). The length function is additive in short exact sequences. When 퐴 is a field we have ℓ퐴 = dim퐴.

Definition 5.3.2. For 푅 and 푀 as in Lemma 5.3.1, we define the functions

휒(푀, 훾) ∶= ℓ (푀 ), 훾 ∈ 훤 ∶= ℤ푁 푅0 훾 ≥0 with values in ℤ≥0.

One sees immediately that for a short exact sequence 0 → 푀′ → 푀 → 푀″ → 0 of finitely generated graded 푅-modules, we have 휒(푀, 훾) = 휒(푀′, 훾) + 휒(푀″, 훾) for all 훾 ∈ 훤. This extends to alternating sums of 휒 in finite exact sequences. One can control the behavior of 휒(푀, ⋅) by forming the Poincaré series 푃푀(X) = 훾 ∑훾∈훤 휒(푀, 훾)X ; see [7, 6.D]. Here we shall restrict to the case 푁 = 1, i.e. 훤 = ℤ≥0, in order to gain more control of 휒(푀, 훾). As a preparation, we say a function 퐻 ∶ ℤ → ℂ is a quasi-polynomial of period 휛 if its restriction to each congruence class modulo 휛 coincides with a polynomial function (necessarily unique); the degree of 휒 is defined by taken the maximum among congruence classes. In particular, a quasi-polynomial of period 1 is just a polynomial.

Theorem 5.3.3. Assume 훤 = ℤ≥0. Suppose 푅 is generated by homogeneous elements 푥1, … , 푥푛 over 푅0. There exists a unique quasi-polynomial 퐻푀 of degree ≤ 푛 − 1, with coefficients in ℚ and period 푒 ∶= lcm(deg 푥1,…, deg 푥푛), such that

휒(푀, 훾) = 퐻푀(훾), |훾| ≫ 0.

Proof. Uniqueness is clear. We construct 퐻푀 by induction on the minimal number of generators 푛. If 푛 = 0 then 푅 = 푅0 and 푀훾 = 0 for |훾| ≫ 0, in which case 퐻푀 = 0. ⋅ 58 ⋅ From completions to dimensions

For 푛 ≥ 1, write 푅 = 푅0[푥1, … , 푥푛] as usual. We may assume that 푥푖 ≠ 0 has degree 휂푖, for 푖 = 1, … , 푛. Fix 푖 and define the graded modules

⋅푥푖 ⋅푥푖 푍 ∶= ker (푀 푀(휂푖)) , 푌 ∶= coker (푀(−휂푖) 푀)

which are again finitely generated, so that we have the exact sequence

0 → 푍 → 푀 → 푀 → 푌 → 0, 훾 ∈ 훤. 훾 훾 훾+휂푖 훾+휂푖

Since 푥푖 annihilates 푍 and 푌, induction hypothesis entails

휒(푀, 훾 + 휂푖) − 휒(푀, 훾) = 휒(푌, 훾 + 휂푖) − 휒(푍, 훾),

the right-hand side being quasi-polynomials of period lcm(… , ̂휂푖, …) for large |훾| and of degrees ≤ 푛−2, since 푥푖 acts trivially on 푍 and 푌. Doing this for all 푖 yields difference equations that witness the polynomiality of 휒(푀, 훾) for |훾| ≫ 0 in every congruence class modulo 푒.

In particular, if 푅 is generated by 푅1 over 푅0, the period 푒 = 1 and we have the notion of Hilbert–Samuel polynomials.

Example 5.3.4. For 푅 = 푀 = 한[푋1, … , 푋푛] graded by total degree, where 한 is a field, our assumptions are readily verified. We see 휒(푀, 훾) = dim한 한[푋1, … , 푋푛]deg=훾 for 훾+푛−1 all 훾 ∈ ℤ≥0, which equals ( 푛−1 ) by high school combinatorics. Hence the Hilbert– 푋+푛−1 Samuel polynomial is 퐻푀(푋) = ( 푛−1 ) ∈ ℚ[푋].

5.4 Definition of Krull dimension

Let 푅 be a ring.

Definition 5.4.1 (Height and dimension). For any prime ideal 픭 of 푅, define its height ht(픭) as the supremum of the lengths of prime chains

픭 = 픭0 ⊋ 픭1 ⊋ ⋯ ⊋ 픭푛, length ∶= 푛.

For any ideal 픞 of 푅, we define ht(픞) ∶= inf{ht(픭) ∶ 픭 ⊃ 픞}.

Define the Krull dimension of 푅 to be dim 푅 ∶= sup픭∈Spec(푅) ht(픭). The following results are immediate.

⋄ The zero prime in a domain has height 0.

⋄ Fields have dimension zero. In fact, a ring has dimension zero if and only if every prime ideal is maximal.

⋄ We have ht(픭) = dim 푅픭 for every prime ideal 픭 ⊂ 푅. ⋄ For any ideal 픞 we have dim 푅 ≥ dim(푅/픞) + ht(픞). §5.4 Definition of Krull dimension ⋅ 59 ⋅

Exercise 5.4.2. Verify the last property above.

Exercise 5.4.3. Show that every principal ideal domain which is not a field has dimension one.

More generally, we define the dimension of an 푅-module 푀 as

dim 푀 ∶= dim(푅/ann(푀)), dim{0} ∶= −∞.

For a short exact sequence 0 → 푀′ → 푀 → 푀″ → 0 we have dim 푀′, dim 푀″ ≤ dim 푀.

Lemma 5.4.4. Suppose 푅 is Noetherian. The following are equivalent for a finitely generated 푅-module 푀 ≠ {0}.

(i) dim 푀 = 0.

(ii) 푅/ann(푀) is Artinian.

(iii) 푀 has finite length.

Proof. (i) ⟺ (ii) is already known: recall that a Noetherian ring is Artinian if and only if its prime ideals are all maximal (Corollary 1.4.3). Let us show (i) or (ii) ⟹ (iii). By writing 푀 = 푀1 + ⋯ + 푀푛 where each 푀푖 is generated by one element, we may assume 푀 ≃ 푅/픞 for some ideal 픞 = ann(푀). It has been shown that 푅/픞 has finite length as a module since it is an Artinian ring. (iii) ⟹ (i). Upon modulo ann(푀) we may assume ann(푀) = {0}. Take any minimal prime 픭 in 푅. As ann(푀) = {0} we have 푀픭 ≠ {0}. Therefore 픭 is a minimal element of Supp(푀), hence belongs to Ass(푀). We may embed 푅/픭 into 푀. The 푅- module 푅/픭 has finite length since 푀 does, therefore 푅/픭 is an Artinian ring. This implies 픭 is a maximal ideal, therefore dim 푅 = 0 since every prime in 푅 lies over a minimal prime.

Our strategy is to study the Krull dimension via completions and Hilbert polynomials. As a preparation, we begin with the local, or more generally the semi-local rings.

Definition 5.4.5. Let 푅 be a Noetherian semi-local ring (i.e. there are finitely many maximal ideals 픪1, … , 픪푛). Let 푀 ≠ {0} be a finitely generated 푅-module. We say an ideal 퐼 is a parameter ideal for 푀 if 퐼 ⊂ rad(푅) and 푀/퐼푀 has finite length.

Parameter ideals are often called ideals of definition. Here we follow the terminolo- gies of [8].

Exercise 5.4.6. Show that 퐼 is a parameter ideal for 푅 if and only if there exists 푘 with

rad(푅)푘 ⊂ 퐼 ⊂ rad(푅).

Show that such an ideal is a parameter ideal for every 푀. Hint: If 퐼 ⊃ rad(푅)푘, every 푘 prime ideal 픭 ⊃ 퐼 must contain (픪1 ⋯ 픪푛) , hence 픭 = 픪푖 for some 푖. Conversely, show that in an Artinian ring we have rad(푅)푘 = 0 for 푘 ≫ 0, using Corollary 1.4.3. Hint: for Artinian rings, rad(푅) equals the nilpotent radical, and is finitely generated. ⋅ 60 ⋅ From completions to dimensions

Dimension theory for modules can be built solely on the parameter ideals for 푅, but we opt to introduce the general notion here. Hereafter we fix a Noetherian semi-local ring 푅 and a finitely generated 푅-module 푀 ≠ {0}. Lemma 5.4.7. An ideal 퐼 ⊂ rad(푅) is a parameter ideal for 푀 if and only if there exists 푘 with rad(푅)푘 ⊂ 퐼 + ann(푀). In this case 푅/(퐼 + ann(푀)) is an Artinian ring. In particular, rad(푅) is a parameter ideal for any 푀. Proof. First we claim that 푉(ann(푀/퐼푀)) = Supp(푀/퐼푀) equals 푉(퐼 + ann(푀)). By the exactness of localizations together with Nakayama’s Lemma, we have

Supp(푀/퐼푀) = Supp(푀) ∩ {픭 ∶ 퐼푅픭 ⊊ 푅픭}; the last term equals Supp(푀) ∩ 푉(퐼) = 푉(ann(푀) + 퐼), thereby proving our claim. By applying to 푀/퐼푀 the Lemma 5.4.4, 퐼 ⊂ rad(푅) being a parameter ideal for 푀 is equivalent to any one of the following 푅 is Artinian ⟺ 푉(ann(푀/퐼푀)) ⊂ MaxSpec(푅) ann(푀/퐼푀) ⟺ 푉(퐼 + ann(푀)) ⊂ MaxSpec(푅) 푅 ⟺ 푅̄ ∶= is Artinian. 퐼 + ann(푀) If 푅̄ is Artinian, then the image of rad(푅) in 푅̄ is contained in rad(푅)̄ , and we know rad(푅)̄ 푘 = 0 for large 푘. Conversely, suppose rad(푅)푘 ⊂ 퐼 + ann(푀). We claim that 푅/rad(푅)푘 is Artinian: rad(푅) contains the product 픪1 ⋯ 픪푘 of all maximal ideals, so every over-prime of 푘 rad(푅) is some 픪푖, thus maximal. This shows that 푀/퐼푀 has finite length by the previous equivalences and Lemma 5.4.4. Given an parameter ideal 퐼 for 푀. The 퐼-adic grading gives rise to the graded objects 퐼푛 퐼푛푀 gr (푅) = ⨁ , gr (푀) = ⨁ . 퐼 푛+1 퐼 푛+1 푛≥0 퐼 푛≥0 퐼 푀 Recall that 0 ⋄ gr퐼(푅) is finitely generated over gr퐼 (푅) = 푅/퐼 as an algebra and is Noetherian (Proposition 4.4.4); 1 ⋄ more precisely, gr퐼(푅) is generated by gr퐼 (푅) over 푅/퐼. ⋄ gr퐼(푀) is a finitely generated gr퐼(푅)-module (Proposition 4.4.5); 0 0 ⋄ the ring gr퐼 (푅) = 푅/퐼 becomes Artinian after modulo gr퐼 (푅) ∩ ann(gr퐼(푀)), which contains (ann(푀) + 퐼)/퐼 (use Lemma 5.4.7). Upon recalling Lemma 5.3.1, it are justified to define

푛−1 푛 푗 푗+1 휒(푀, 퐼; 푛) ∶= ℓ푅/퐼푛 (푀/퐼 푀) = ∑ ℓ푅/퐼(퐼 푀/퐼 푀), 푛 ∈ ℤ≥0. 푗=0

By the theory of Hilbert–Samuel polynomials, 푛 ↦ 휒(푀, 퐼; 푛) is a polynomial 퐻퐼(푀, ⋅) whenever 푛 ≫ 0, with deg 퐻퐼(푀, ⋅) bounded by the minimal number of generators of 0 gr퐼(푅) over gr퐼 (푅) = 푅/퐼. §5.4 Definition of Krull dimension ⋅ 61 ⋅

Lemma 5.4.8. Let 푀 ≠ {0} be a finitely generated 푅-module with parameter ideal 퐼.

(i) The degree 푑(푀) of 퐻퐼(푀, ⋅) is independent of the choice of the parameter ideal 퐼. (ii) In a short exact sequence 0 → 푀′ → 푀 → 푀″ → 0 of finitely generated 푀-modules, we have ′ ″ deg 퐻퐼(푀 , ⋅), deg 퐻퐼(푀 , ⋅) ≤ deg 퐻퐼(푀, ⋅), ′ ″ and 퐻퐼(푀, ⋅) − 퐻퐼(푀 , ⋅) − 퐻퐼(푀 , ⋅) has degree < 푑(푀). Proof. (i): To compare the graded objects associated to two parameter ideals 퐼, 퐽, we apply the characterization in Lemma 5.4.7: it suffices to take 퐽 = rad(푅), so that

퐽푚 + ann(푀) ⊂ 퐼 + ann(푀) ⊂ 퐽 + ann(푀) for some 푚 ≥ 1. This implies 휒(푀, 퐽; 푛) ≤ 휒(푀, 퐼; 푛) and 휒(푀, 퐼; 푛) ≤ 휒(푀, 퐽; 푚푛) for all 푛 ≥ 0. Whence (i). (ii): For the first part, note that 푀/퐼푛푀 → 푀″/퐼푛푀″ is surjective, so 휒(푀″, 퐼; 푛) ≤ ′ 푛 ′ 푛 푀′∩퐼푛푀 휒(푀, 퐼; 푛). On the other hand, 푀 /퐼 푀 → 푀/퐼 푀 has kernel 퐼푛푀′ . For 푛 ≥ 푛0 ≫ 0, Artin–Rees (Theorem 4.5.7) gives

푀′ ∩ 퐼푛푀 퐼푛−푛0 (푀′ ∩ 퐼푛0 푀) 퐼푛−푛0 푀′ ℓ ( ) = ℓ ( ) ≤ ℓ ( ) 퐼푛푀′ 퐼푛푀′ 퐼푛푀′ ′ ′ 푀 푀 ′ ′ = ℓ ( ) − ℓ ( ) = 휒(푀 , 퐼; 푛) − 휒(푀 , 퐼; 푛 − 푛0) 퐼푛푀′ 퐼푛−푛0 푀′

′ ′ which has degree inferior to 퐻퐼(푀 , ⋅). Hence deg 퐻퐼(푀 , ⋅) ≤ deg 퐻퐼(푀, ⋅). To establish the second part of (ii), we consider 휒(푀, 퐼; 푛)−휒(푀″, 퐼; 푛) which equals 푀 푀 푀′ + 퐼푛푀 푀′ ℓ ( ) − ℓ ( ) = ℓ ( ) = ℓ ( ). 퐼푛푀 푀′ + 퐼푛푀 퐼푛푀 푀′ ∩ 퐼푛푀 By Artin–Rees, the rightmost term is squeezed between ℓ(푀′/퐼푛푀′) = 휒(푀′, 퐼; 푛) and ℓ(푀′/퐼푛−푘푀′) = 휒(푀′, 퐼; 푛 − 푘) for some 푘 independent of 푛 ≫ 0. Hence 휒(푀, 퐼; 푛) − 휒(푀″, 퐼; 푛) is a polynomial with the same leading term as 휒(푀′, 퐼; 푛), for 푛 ≫ 0.

Define 푠(푀) to be the smallest integer 푠 such that there exist 푡1, … , 푡푠 ∈ rad(푅) with 푠 푀/ ∑푖=1 푡푖푀 of finite length. In other words, 푠(푀) is the minimal number of generators 푠 for parameter ideals for 푀. Observe that 푀/ ∑푖=1 푡푖푀 ≠ {0}, otherwise Nakayama’s Lemma will lead to 푀 = {0}. Theorem 5.4.9. For any finitely generated nonzero 푅-module 푀, we have dim 푀 = 푠(푀) = 푑(푀). In particular dim 푀 is finite. Proof. We argue inductively on 푑(푀) to show dim 푀 ≤ 푑(푀). If 푑(푀) = 0 then 퐼푛푀 = 퐼푛+1푀 = ⋯ for 푛 ≫ 0. Corollary 4.5.9 implies 퐼푛푀 = {0}, hence 푀 = 푀/퐼푛푀 has finite length and dim 푀 = 0 by Lemma 5.4.4. Now assume 푑(푀) ≥ 1. Take a minimal 픭 ∈ Ass(푀) verifying dim(푅/픭) = dim 푀, so that 푅/픭 ↪ 푀. As 푑(푅/픭) ≤ 푑(푀), we are reduced to the case 푀 = 푅/픭. Consider a chain of prime ideals 픭 = 픭0 ⊊ ⋯ ⊊ 픭푚 ⋅ 62 ⋅ From completions to dimensions

in 푅. We claim that 푚 ≤ 푑(푅/픭). We may surely suppose 푚 ≥ 1. Take 푡 ∈ 픭1 ∖ 픭0. Reduction modulo 푅푡 + 픭 yields a prime chain of length 푚 − 1, namely 픭 픭 1 ⊊ ⋯ 푚 푅푡 + 픭 푅푡 + 픭 in 푅/(푅푡 + 픭). Hence dim(푅/푅푡 + 픭) ≥ 푚 − 1. In view of the exactness of

푡 푅 0 → 푅/픭 푅/픭 → → 0, 푅푡 + 픭 Lemma 5.4.8 (ii) implies that 푑(푅/푅푡 + 픭) < 푑(푅/픭). By induction we deduce 푑(푅/픭) > 푑(푅/푅푡 + 픭) ≥ dim(푅/푅푡 + 픭) ≥ 푚 − 1, hence 푑(푅/픭) ≥ 푚 as required. Next, let us show 푠(푀) ≤ dim 푀. Set 푟 ∶= dim 푀. We contend that there exist 푡1, … , 푡푟 ∈ rad(푅) such that 푀/(푡1, … , 푡푟)푀 has finite length; therefore 푠 ≤ 푟. When 푟 = 0 this follows from Lemma 5.4.4. For 푟 > 0, we have rad(푅) ⊄ 픭 for any minimal 픭 ∈ Ass(푀) verifying dim 푅/픭 = dim 푀, for otherwise 픭 will contain, thus equal to a maximal ideal as 푅 is semi-local, and we would get dim 푀 = 0. Using prime avoidance (Proposition 1.1.5 applied to 퐼 ∶= rad(푅) and the primes 픭 above), we may pick 푡1 ∈ rad(푅) that does not belong to any 픭 above. From ann(푀/푡1푀) ⊃ 푅푡1 + ann(푀) and our choice of 푡1,, we see dim 푀/푡1푀 < dim 푀. Our claim results from induction on 푟. 푠 We finish the proof by showing 푑(푀) ≤ 푠(푀). Suppose 푀/ ∑푖=1 푡푖푀 ≠ {0} has finite length. We contend that for any 푡 ∈ rad(푅) we have 푑(푀) ≥ 푑(푀/푡푀) ≥ 푑(푀) − 1. (5–3)

If this holds, we can look at the sequence 푀, 푀/푡1푀, 푀/(푡1푀 + 푡2푀), …: at each step 푠 푑(⋯) drops at most by one; at the end 퐿 ∶= 푀/ ∑푗=1 푡푗푀 we have 푑(퐿) = 0: indeed, as 퐿 has finite length, ℓ(퐿/퐼푛퐿) is uniformly bounded by ℓ(퐿) so that 푑(퐿) = 0. Thus 푑(푀) ≤ 푠 as expected. To prove (5–3), first note that 푑(푀) ≥ 푑(푀/푡푀) is known. Take any parameter ideal 퐼 ∋ 푡. We bound 휒(푀/푡푀, 퐼; 푛) as follows 푀 푀 푡푀 + 퐼푛푀 ℓ ( ) = ℓ ( ) − ℓ ( ). 푡푀 + 퐼푛푀 퐼푛푀 퐼푛푀 Note that 푀 푀 푡푀 푡푀 + 퐼푛푀 ↠ →∼ ≃ 퐼푛−1푀 {푥 ∈ 푀 ∶ 푡푥 ∈ 퐼푛푀} 푡푀 ∩ 퐼푛푀 퐼푛푀 푦 ↦ 푡푦. Hence 휒(푀/푡푀, 퐼; 푛) ≥ 휒(푀, 퐼; 푛) − 휒(푀, 퐼; 푛 − 1) for 푛 ≫ 0, proving the second in- equality in (5–3). ̂ Corollary 5.4.10. Under the same assumptions, we have dim푅 푀 = dim푅̂ 푀, where we take 퐼-adic completions. ̂ ̂ Proof. Proposition 5.2.1 gives identifications gr퐼(푅) = gr퐼(̂ 푅) and gr퐼(푀) = gr퐼(̂ 푀); moreover 푅̂ is still semi-local and 퐼 ̂is still a parameter ideal for 푀̂ (see Proposition 5.2.4, 5.2.5). Since 푑(푀) and 푑(푀)̂ are read from these graded modules, they are equal. These results will be applied to the case 푀 = 푅 in the next section. §5.5 Krull’s theorems and regularity ⋅ 63 ⋅

5.5 Krull’s theorems and regularity

We still assume 푅 is a Noetherian ring.

Theorem 5.5.1 (Krull). Suppose 픞 = (푡1, … , 푡푟) is a proper ideal of 푅, then for every minimal over-prime ideal 픭 of 픞, we have ht(픭) ≤ 푟.

From Definition 5.4.1 we infer that ht(픞) ≤ 푟. The special case 푟 = 1 says that every principal ideal (푡) ≠ 푅 has height at most one (exactly one if 푡 is not a zero divisor — use Theorem 2.2.5 (ii)); this is called the Hauptidealsatz.

Proof. We work in 푅픭 and 퐼 ∶= 픞푅픭 = (푡1, … , 푡푟). Since 퐼 ⊂ rad(푅픭) and 푅픭/퐼 has dimension zero, thus Artinian, 퐼 is a parameter ideal. We conclude by Theorem 5.4.9 and ht(픭) = dim 푅픭.

Now assume 푅 is Noetherian and local with maximal ideal 픪. The parameter ideals of 푅 are precisely those squeezed between 픪 and 픪푘 for some 푘 ≥ 1, by Lemma 5.4.7. Set 푑 ∶= dim 푅, which is finite by Theorem 5.4.9. The same theorem tells us that we can generate some parameter ideal 퐼 ⊂ 픪 (namely 푅/퐼 Artinian) by elements 푡1, … , 푡푑 ∈ 픪. These elements form a system of parameters of 푅.

Proposition 5.5.2. Suppose 푅 is a Noetherian local ring with a system of parameters 푡1, … , 푡푑, which generate a parameter ideal 퐼. For any 0 ≤ 푖 ≤ 푑 we have dim(푅/(푡1, … , 푡푖)) = 푑 − 푖, and 푡푖+1, … , 푡푑 form a system of parameters for 푅/(푡1, … , 푡푖).

Proof. Consider the sequence 푅, 푅/(푡1), 푅/(푡1, 푡2), … , 푅/(푡1, … , 푡푑). Recall the formal- ism in Theorem 5.4.9: at each stage 푑(푅/ ⋯) drops at most by one, by (5–3). After 푑 steps we arrive at 푅/퐼 with dim(⋅) = 푑(⋅) = 푠(⋅) = 0, since it has finite length. Hence 푑 drops exactly by one at each stage. The remaining assertions are immediate.

A natural question arises: when can we assure 퐼 = 픪?

Definition 5.5.3. We say a Noetherian local ring 푅 is a regular local ring if 픪 can be generated by 푑 = dim 푅 elements 푡1, … , 푡푑. In this case we say 푡1, … , 푡푟 form a regular system of parameters.

In particular, 픪 = {0} if 푅 is regular local with dim 푅 = 0. J K Exercise 5.5.4. Let 한 be a field. The 한-algebra of formal power series 한 푋1, … , 푋푑 is a regular local ring. Indeed, it is Noetherian with maximal ideal 픪 = (푋1, … , 푋푑), and 2 푋1, … , 푋푑 form a regular system of parameters. On the other hand, 픪/픪 has a 한-basis formed by the images of 푋1, … , 푋푑. One way to determine its dimension and prove its 푛 푛+1 regularity is to calculate the functions 푛 ↦ dim한(픪 /픪 ) explicitly, i.e. count the monomials in 푑 variables with total degree 푛. You should get a polynomial in 푛 with degree 푑 − 1, cf. Exercise 5.3.4.

Theorem 5.5.5. For any Noetherian local ring 푅 with maximal ideal 픪 and residue field 한, we 2 have dim 푅 ≤ dim한 픪/픪 . Equality holds if and only if 푅 is a regular local ring. ⋅ 64 ⋅ From completions to dimensions

The 한-vector space 픪/픪2 is called the Zariski cotangent space of Spec(푅), in honor of Oscar Zariski (1899–1986). Picture taken from Wikimedia Commons.

Proof. By Nakayama’s Lemma (more precisely, Corollary 1.3.6), 픪/픪2 can be generated over 한 by 푠 elements if and only if 픪 can be generated over 푅 by 푠 elements, for any 푠 ∈ ℤ≥0. Hence Theorem 5.4.9 imposes the bound 푠 ≥ dim 푅, and equality holds if and only if 푅 admits a regular system of parameters. Due to time constraints, we cannot say too much about regular local rings. Below is one of their wonderful properties.

Theorem 5.5.6. Regular local rings are integral domains.

Proof. Induction on 푑 ∶= dim 푅. If dim 푅 = 0 then 픪 = {0}, hence 푅 is a field. Assume hereafter that 푑 > 0. We know there are only finitely many minimal prime ideals and 2 dim한 픪/픪 ≥ 1. By prime avoidance (Proposition 1.1.5) applied to 퐼 ∶= 픪, the minimal prime ideals and 픪2, there exists 푡 ∈ 픪 ∖ 픪2 that does not lie in any minimal prime ideal. Put 푅′ ∶= 푅/(푡) with maximal ideal 픪′ = 픪/(푡); our choice of 푡 together with Proposition 5.5.2 imply

dim 푅′ = dim 푅 − 1, ′ ′ 2 2 dim한 픪 /(픪 ) = dim한 픪/픪 − 1 = dim 푅 − 1.

Hence 푅′ is still regular local, and by induction it is a domain. This implies (푡) is prime. Take any minimal prime 픭 below (푡); note that 푡 ∉ 픭 by construction. To show 푅 is a domain it suffices to prove 픭 = {0}. Indeed, every 푠 ∈ 픭 can be written as 푠 = 푎푡, 푎 ∈ 푅. Since 푡 ∉ 픭, we must have 푎 ∈ 픭. Hence 픭 = 푡픭 ⊂ 픪픭. Nakayama’s Lemma (Theorem 1.3.5) implies 픭 = {0}. Lecture 6

Dimension of finitely generated algebras

The main reference is [11, §13].

6.1 Dimensions in fibers

Consider a homomorphism 휑 ∶ 퐴 → 퐵, which induces 휑♯ ∶ Spec(퐵) → Spec(퐴) on prime spectra. Given 픭 ∈ Spec(퐴), we are interested in the fiber (휑♯)−1(픭); the prime ideals 픮 therein are described by 휑−1(픮) = 픭, or equivalently: 휑−1(픮) ∩ (퐴 ∖ 픭) = ∅, 픮 ⊃ 휑(픭). Adopt the convention that a zero ring has Spec = ∅. The first condition says that 픮 comes from Spec(퐵픭), where 퐵픭 is the localization with respect to 휑(퐴 ∖ 픭) (possibly zero). The second condition then says that 픮퐵픭 lies over the image of 픭퐴픭. Set

휅(픭) ∶= 퐴픭/픭퐴픭. The fiber (휑♯)−1(픭) is then identified with the spectrum of

퐵 ⊗ 휅(픭) = (퐵 ⊗ 퐴픭) ⊗ 휅(픭) = 퐵픭 ⊗ (퐴픭/픭퐴픭), 퐴 퐴 퐴픭 퐴픭

which is empty if and only if 퐵 ⊗ 휅(픭) is zero. This also equips (휑♯)−1(픭) with an extra 퐴 structure: it is the spectrum of an explicit quotient ring of 퐵픭. ♯ −1 Observe that for all 픮 ∈ (휑 ) (픭), the localization of 퐵픭 ⊗ 휅(픭) at the image of 픮 is 퐴픭 canonically isomorphic to 퐵픮 ⊗ 휅(픭). 퐴픭 Proposition 6.1.1. Assume 퐴, 퐵 to be Noetherian. Let 픮 ∈ Spec(퐵) and 픭 ∶= 휑♯(픮). We have

(i) dim(퐵픮) ≤ dim(퐴픭) + dim 퐵픮 ⊗ 휅(픭) (note that the existence of 픮 ensures that 퐵픭 ⊗ 퐴픭 퐴픭 휅(픭) ≠ {0}, hence 퐵픮 ⊗ 휅(픭) ≠ {0}); 퐴픭 ⋅ 66 ⋅ Dimension of finitely generated algebras

(ii) equality holds if going-down holds for 휑;

(iii) if going-down holds and 휑♯ is surjective, then dim(퐵) ≥ dim(퐴), and for all ideal 픞 ⊊ 퐴 we have 휑(픞)퐵 ≠ 퐵 and ht(픞) = ht(휑(픞)퐵).

It is crucial to notice that dim 퐵픮 ⊗ 휅(픭) = ht(픮퐵픮/휑(픭)퐵픮) = ht(픮/휑(픭)퐵). 퐴픭 This bounds the source dimension of a morphism by the target dimension plus the fiber dimension, localized both at 픭 and 픮 ∈ (휑♯)−1(픭). In order to have an equality, a certain submersion-like condition on 휑♯ is evidently required; this explains the going- down condition. Cf. [11, (6.H)].

Proof. We have an induced local homomorphism 퐴픭 → 퐵픮 since 픮 ↦ 픭. Since (i) and (ii) depend only on this induced homomorphism, we may assume from the outset that 퐴, 퐵 are local with maximal ideals 픭, 픮, and 휑 is a local homomorphism. Let 푑 ∶= dim 퐴 푘 and take a parameter ideal 퐼 = (푡1, … , 푡푑) of 퐴, so that 픭 ⊂ 퐼 ⊂ 픭 for some 푘. It follows that √휑(픭)퐵 = √휑(퐼)퐵, therefore dim(퐵 ⊗ 휅(픭)) = dim(퐵/휑(픭)퐵) = dim(퐵/휑(퐼)퐵); 퐴 denote this number as 푒. Take a 푠1, … , 푠푒 ∈ 픮 whose images generate a parameter ideal for 퐵/휑(퐼)퐵, and put 퐽 ∶= (휑(푡1), … , 휑(푡푑), 푠1, … , 푠푒). Then 퐵/퐽 is Artinian, therefore dim 퐵 ≤ 푑 + 푒 establishes (i). As for (ii), we conserve the same hypotheses and take a prime chain 픮 = 픮0 ⊋ ⋯ ⊋ 픮푒 with 픮푒 ⊃ 휑(픭)퐵 in 퐵, as well as a prime chain 픭 = 픭0 ⊋ ⋯ ⊋ 픭푑 in 퐴. Note that −1 휑 (픮푒) = 픭 since 푒 = dim 퐵/휑(픭)퐵. By applying going-down repeatedly to the chain 픭푖, we obtain a prime chain in 퐵

−1 픮푒 ⊋ ⋯ ⊋ 픮푒+푑, 휑 (픮푒+푖) = 픭푖.

Concatenation with 픮0 ⊋ ⋯ gives dim(퐵) = 푑 + 푒. This shows (ii). The first assertion of (iii) results from (ii). To show the remaining one, let usshow 휑(픞)퐵 ≠ 퐵: if 휑♯(픮) = 픭 ⊃ 픞, then 픮 ⊃ 휑(픞)퐵. Next, take a minimal over-prime −1 픮 ⊃ 휑(픞)퐵 with ht(픮) = ht(휑(픞)퐵). With 픭 ∶= 휑 (픮) ⊃ 픞, we must have dim 퐵픮 ⊗ 휅(픭) = ht(픮/휑(픭)퐵) = 0 by the minimality of 픮. An application of (ii) yields

ht(휑(픞)퐵) = ht(픮) = ht(픭) ≥ ht(픞).

To obtain ≤, choose 픭 ⊃ 픞 with ht(픭) = ht(픞) and take 픮 ∈ Spec(퐵) with 픭 = 휑−1(픮); this implies 픮 ⊃ 휑(픭)퐵 ⊃ 휑(픞)퐵. Upon shrinking 픮, we may even assume 픮 is minimal over 휑(픭)퐵, i.e. ht(픮/휑(픭)퐵) = 0. Using (ii), this entails ht(픞) = ht(픭) = ht(픮) ≥ ht(휑(픞)퐵).

The minimal primes in a ring 퐵 may have different heights when the scheme Spec(퐵) is not equi-dimensional. §6.2 Calculation for polynomial algebras ⋅ 67 ⋅

Going-down holds for flat 휑 by Theorem 4.1.3, therefore the dimension equality

dim(퐵픮) = dim(퐴픭) + dim 퐵픮 ⊗ 휅(픭) 퐴픭

holds for flat ring homomorphisms. Remark 6.1.2. In general, if 퐵 is a finitely generated algebra over Noetherian 퐴 such that Spec(퐵) → Spec(퐴) is a closed map, the fiber dimension 픭 ↦ dim 퐵/픭퐵 is an upper semi-continuous function on the target space Spec(퐴). More concretely, the fiber dimension is non-decreasing under specialization of 픭. Cf. [8, §14.3] or [11, (13.E)]. Try to understand this phenomenon intuitively. Proposition 6.1.3. Suppose 퐵 is integral over a subring 퐴. (i) We have dim 퐴 = dim 퐵. (ii) If we assume moreover that 퐴, 퐵 are both Noetherian, then ht(픮) ≤ ht(픮 ∩ 퐴) for every 픮 ∈ Spec(퐵). (iii) Furthermore, if going-down also holds for 퐴 ↪ 퐵, we have ht(퐽) = ht(퐽 ∩ 퐴) for every ideal 퐽 ⊊ 퐵. Proof. Going-up holds and Spec(퐵) ↠ Spec(퐴) in the situation of (i) by Theorem 4.1.4, hence dim 퐵 ≥ dim 퐴 by lifting prime chains. To prove ≤, observe that 픮 ⊊ 픮′ implies 픮∩퐴 ⊊ 픮′ ∩퐴 since there are no inclusion relations in the fibers of Spec(퐵) → Spec(퐴). As for (ii), note that ht(픮) ≤ ht(픭) + ht(픮/픭퐵) where 픭 ∶= 픮 ∩ 퐴; as the are no inclusion relations in fibers, the last term must be 0. Now assume going-down and consider (iii). Take 픮 ∈ Spec(퐵) with ht(픮) = ht(퐽). Put 픭 ∶= 픮 ∩ 퐴 ⊃ 퐽 ∩ 퐴. Again, since there are no inclusions in the fiber over 픭 of Spec(퐵) ↠ Spec(퐴), we have dim(퐵픮/픭퐵픮) = 0. Proposition 6.1.1 (ii) implies ht(픮) = ht(픭), therefore ht(퐽) ≥ ht(퐽 ∩ 퐴). On the other hand, for any 픭 ⊃ 퐽 ∩ 퐴 with ht(픭) = ht(퐽 ∩ 퐴), since 퐴/퐽 ∩ 퐴 ↪ 퐵/퐽 is integral, there exists 픮 ⊃ 퐽 with 픮 ∩ 퐴 = 픭. Together with Proposition 6.1.1 (i) and ht(픮/픭퐵) = 0, this implies ht(퐽) ≤ ht(픮) ≤ ht(픭) = ht(퐽 ∩ 퐴) by (ii).

6.2 Calculation for polynomial algebras

Let us apply the results from the previous section to elucidate the Krull dimension of polynomial algebras.

Theorem 6.2.1. Let 퐴 be a Noetherian ring, we have dim 퐴[푋1, … , 푋푛] = dim 퐴 + 푛 for any 푛 ≥ 0. In particular, dim 퐴[푋1, … , 푋푛] = 푛 if 퐴 is Artinian (eg. a field). Proof. Evidently we may assume 푛 = 1. We shall apply Proposition 6.1.1 to 퐴 ↪ 퐵 = 퐴[푋]. Take any 픭 ∈ Spec(퐴) and let 픮 be a maximal element in {픮′ ∈ Spec(퐵) ∶ 픮 ∩ 퐴 = 픭}. Put 휅 ∶= 휅(픭). It suffices to show that 퐵픮 ⊗ 휅 has dimension one, since 퐵 is free 퐴픭 hence flat over 퐴, and Proposition 6.1.1 will imply

dim 퐵픮 = dim 퐴픭 + 1 ⋅ 68 ⋅ Dimension of finitely generated algebras

and taking supremum over 픭 ∈ Spec(퐴) gives the result. −1 ′ −1 Indeed, put 퐵픭 ∶= 퐵[(퐴 ∖ 픭) ] = 퐴픭[푋] and 픮 ∶= 픮[(퐴 ∖ 픭) ] ∈ Spec(퐵픭). As ′ ′ ′ ′ 픮 ⊃ 픭퐴픭, we have 픮 ∶= 픮 ⊃ 픭퐴픭 ∈ Spec(휅[푋]), and 픮 is maximal in the fiber over {0} of Spec(휅[푋]) → Spec(휅), i.e. in MaxSpec(휅[푋]). Localization in stages yields

퐵픮 퐵픭 퐵픮 ⊗ 휅 ≃ ≃ ( ) ≃ 휅[푋] ′ . 퐴 픭퐵 픭퐴 퐵 픮 픭 픮 픭 픭 픮′ As 휅[푋] is a principal ideal domain which is not a field, every maximal ideal thereof ′ has height one. Hence dim 휅[푋]픮′ = ht(픮 ) = 1.

Corollary 6.2.2. Let 한 be a field, then for every 0 ≤ 푖 ≤ 푛 we have ht(푋1, … , 푋푖) = 푖 in 한[푋1, … , 푋푛].

Proof. The prime chain {0} ⊂ (푋1) ⊊ ⋯ ⊊ (푋1, … , 푋푛) has length 푛 = dim 한[푋1, … , 푋푛]. Thus for each 0 ≤ 푖 ≤ 푛, the chain {0} ⊂ (푋1) ⊊ ⋯ ⊊ (푋1, … , 푋푖) has maximal length among all prime chains starting with (푋1, … , 푋푖).

Combined with Theorem 6.2.1, we see that for 한 a field, 푅 ∶= 한[푋1, … , 푋푛] and 픭 ∶= (푋1, … , 푋푖), the equality ht(픭) + dim 푅/픭 = dim 푅.

holds. This will be generalized to finitely generated domains over 한. We record another simple consequence for later use. Corollary 6.2.3. Let 한 be a field. Any 한-algebra 퐴 with 푛 generators has finite dimension ≤ 푛.

Proof. Writing 퐴 = 한[푋1, … , 푋푛]/퐼 for some ideal 퐼, we have dim 퐴 ≤ 한[푋1, … , 푋푛] since every prime chain in 퐴 lifts to 한[푋1, … , 푋푛]. Now apply Theorem 6.2.1.

6.3 Noether normalization and its consequences

Fix a field 한. A few preparatory results are in order.

Lemma 6.3.1. Suppose 한 is a field and 푡 ∈ 한[푋1, … 푋푒] ∖ 한. There exist 푡1, … , 푡푒−1 ∈ 한[푋1, … , 푋푒] such that 한[푋1, … , 푋푒] is finitely generated as a module over the 한-subalgebra 푆 ∶= 한[푡1, … , 푡푒−1, 푡].

푘푖 Proof. We seek 푡푖 of the form 푋푖 − 푋푒 where 푘 is a large integer. Then 푡 can be uniquely × expressed as a polynomial of 푡1, … , 푡푒−1, 푋푒. We claim that upon modifying 푡 by 한 , which is clearly harmless, one can choose 푘 such that 푡 is monic as an element of 한[푡1, … , 푡푒−1][푋푒], say of some degree 훿. If this is the case,

훿 푗 푡 = 푋푒 + ∑ (polynomial in 푡1, … , 푡푒−1) 푋푒 0≤푗<훿 says that 푋푒 is integral over 푆, and then 한[푋1, … , 푋푒] is generated as an 푆-module by 훿−1 1, 푋푒, … , 푋푒 . §6.3 Noether normalization and its consequences ⋅ 69 ⋅

To choose 푘, one stares at the expansion

1 푎1 푒−1 푎푒−1 1 푒−1 푎1 푎푒 푘 푘 푎푒 푎푒+푎1푘 +⋯+푎푒−1푘 푋1 ⋯ 푋푒 = (푡1 + 푋푒 ) ⋯ (푡푒−1 + 푋푒 ) ⋅푋푒 = 푋푒 +mixed terms.

If 푘 > max{푎1, … , 푎푒}, the exponent of 푋푒 is simply the base 푘 expression with digits 푎1 푎푒 푎푒, 푎1, … , 푎푒−1. Now write 푡 as a linear combination of monomials 푋1 ⋯ 푋푒 and ex- pand them in terms of 푡1, … , 푡푒−1, 푋푒. From the observation above, different (푎1, … , 푎푒) × contributes a different exponent of 푋푒 whenever 푘 ≫ 0. Adjusting 푡 by 한 , we get the asserted property. Lemma 6.3.2. Let 픞 be a nonzero ideal of a domain 푅, then dim(푅/픞) + 1 ≤ dim 푅.

Proof. Any prime chain 픭0 ⊋ ⋯ ⊋ 픭푛 in 푅 with 픭푛 ⊃ 픞 can be extended to a prime chain length 푛 + 1, namely by adjoining 픭푛+1 ∶= {0}. Theorem 6.3.3 (E. Noether, M. Nagata). Let 퐵 be a finitely generated 한-algebra of dimension 푛. Consider a chain of proper ideals 퐼1 ⊊ ⋯ ⊊ 퐼푚, with dim(퐵/퐼푗) = 푑푗 and 푑1 > ⋯ > 푑푚 ≥ 0. Then there exist a 한-subalgebra 퐴 ⊂ 퐵 together with an isomorphism 퐴 ≃ 한[푋1, … , 푋푛], satisfying ⋄ 퐵 is a finitely generated 퐴-module, in particular 퐵 is integral over 퐴; ⋄ 퐼 ∩ 퐴 ≃ (푋 , … , 푋 ) under the isomorphism above, for each 1 ≤ 푗 ≤ 푚. 푗 푑푗+1 푛

Note that the assumption 퐼푗 ⊊ 퐼푗+1 is merely for convenience. Allowing 퐼푗 = 퐼푗+1 and 푑푗 = 푑푗+1 for some 푗 is surely possible..

Proof. Write 퐵 = 한[푌1, … , 푌푟]/퐽, where 푟 ≥ 푛 (Corollary 6.2.3), and denote the preimage of 퐼푗 in 한[푌1, … , 푌푟] by 퐼푗̃ . The first step is to reduce to the case 퐽 = {0}. To see this, we adjoin 퐼0̃ = {0} (with 푑0 = 푛) into the ideal chain; it may happen that 퐼0̃ = 퐼1̃ , but that’s harmless. Suppose we can find 퐴̃ ⊂ 한[푌1, … , 푌푟], 퐴̃ ≃ 한[푋1, … , 푋푟] with the required properties relative to 퐼•̃ . Taking quotient by 퐽, we obtain the corresponding properties for 퐼•. Indeed, the passage from 퐴̃ to 퐴 ∶= 퐴/(̃ 퐴̃ ∩ 퐽) truncates the variables 푋푛+1, … , 푋푟, whereas

퐼푗̃ ∩ (퐴̃ + 퐽) (퐼푗̃ ∩ 퐴)̃ + 퐽 퐼푗̃ ∩ 퐴̃ 퐼푗 ∩ 퐴 = = ≃ ; 퐽 퐽 퐽 ∩ 퐴̃ try to convince yourself of the middle equality. Secondly, having reduced to the case 퐵 = 한[푌1, … , 푌푟] (thus 푟 = 푛), it suffices to pick 푥1, … , 푥푛 ∈ 퐵 such that

(a) 퐵 is finitely generated as a module over 퐴 ∶= 한[푥1, … , 푥푛], and 퐼 ∩ 퐴 ⊃ (푥 , … , 푥 ) 푗 (b) 푗 푑푗+1 푛 for all . Indeed, (a) implies Frac(퐴) and Frac(퐵) have the same transcendence degree 푛 over 한, therefore 푥1, … , 푥푛 must be algebraically independent over 한. On the other hand, (퐵/퐼 ) = (퐴/퐼 ∩ 퐴) 퐼 ∩ 퐴 ⊋ (푥 , … , 푥 ) dim 푗 dim 푗 by (a) and Proposition 6.1.3; but if 푗 푑푗+1 푛 퐴/퐼 ∩ 퐴 한[푥 , … , 푥 ] then 푗 is a proper quotient of the domain 1 푑푗 , therefore would have dimension < 푑푗 by Lemma 6.3.2. We shall construct 푥1, … , 푥푛 step by step. Suppose 0 ≤ 푒 ≤ 푛 and that we have ′ ′ produced elements 푥1, … , 푥푒 and 푥푒+1, … , 푥푛 in 퐵 satisfying ⋅ 70 ⋅ Dimension of finitely generated algebras

′ ′ (i) 퐵 is finitely generated as a module over 한[푥1, … , 푥푒, 푥푒+1, … , 푥푛] =∶ 푆푒;

(ii) 퐼푗 ∩ 푆푒 ⊃ (푥푒+1, … , 푥푛) when 푑푗 ≤ 푒; 퐼 ∩ 푆 ⊃ (푥 , … , 푥 ) 푑 ≥ 푒 푗 = 1, … , 푚 (iii) 푗 푒 푑푗+1 푛 when 푗 (with ). ′ For the initial case 푒 = 푛, simply take 푥푖 ∶= 푌푖. Our aim is 푒 = 0. Let us explain the induction step from 푒 ≥ 1 to 푒−1. The prior argument based on transcendence degrees implies the algebraic independence among

′ ′ 푥1, … , 푥푒, 푥푒+1, … , 푥푛. ′ If 푒 ≤ 푑푗 for all 푗, we are done. Otherwise set 푗 ∶= min{푗 ∶ 푒 > 푑푗′ }, we contend that

′ ′ 퐼푗 ∩ 한[푥1, … , 푥푒] ≠ {0}.

If not, we would have 퐼푗 ∩ 푆푒 = (푥푒+1, … , 푥푛) since 퐼푗 ∩ 푆푒 ⊃ (푥푒+1, … , 푥푛) by (ii). We have dim(푆푒/퐼푗 ∩ 푆푒) = dim(퐵/퐼푗) = 푑푗 by integrality, whilst dim(푆푒/(푥푒+1, … , 푥푛)) = 푒. Contradiction. ′ ′ Now take 푥푒 ∈ 퐼푗 ∩ 한[푥1, … , 푥푒] ∖ {0}. Note that 푥푒 ∉ 한 as 퐼푗 ≠ 퐵. By Lemma 6.3.1, ″ ″ ′ ′ ′ ′ we may choose 푥1, … , 푥푒−1 ∈ 한[푥1, … , 푥푒] such that 한[푥1, … , 푥푒] is finitely generated ″ ″ over 한[푥1, … , 푥푒−1, 푥푒] as a module. It remains to verify that the new sequence ″ ″ 푥1, … , 푥푒−1, 푥푒, … , 푥푛 satisfies (i)—(iii) above with 푒 − 1 replacing 푒. ″ First, 푆푒 is a finitely generated module over its subalgebra 한[푥1, … , 푥푛] by construc- ′ tion, hence so is 퐵 and (i) follows. Next, let 1 ≤ 푗 ≤ 푚. If 푑푗′ > 푒 − 1, the procedure ″ 푥 , … , 푥 퐼 ′ ∩ 한[푥 , … , 푥 ] 푑 ′ < 푒 above does not affect 푑푗+1 푛, so they belong to 푗 1 푛 . If 푗 , setting ″ ′ 푗 ∶= min{푗 ∶ 푒 > 푑푗″ } we have 푗 ≥ 푗 and

″ ″ 퐼푗′ ∩ 한[푥1, … , 푥푛] ⊃ 퐼푗 ∩ 한[푥1, … , 푥푛] ⊃ (푥푒+1, … , 푥푛) + (푥푒). All in all, we obtain (ii) and (iii). Corollary 6.3.4 (Dimension formula). Let 퐵 be a finitely generated algebra over a field 한. Suppose 퐵 is a domain, then for all 픮 ∈ Spec(퐵) we have

dim(퐵/픮) + ht(픮) = dim 퐵.

Proof. Choose a subalgebra 퐴 of 퐵 as in Theorem 6.3.3 and put 픭 = 픮 ∩ 퐴. Since 퐵 is a domain and 퐴 is normal, the Cohen–Seidenberg Theorem 4.1.4 asserts the going-down property for 퐴 ↪ 퐵. Proposition 6.1.3 implies that dim 퐴 = dim 퐵, dim 퐴/픭 = dim 퐵/픮 and ht(픭) = ht(픮), so we are again reduced to the case 퐵 = 한[푋1, … , 푋푛] and 픮 = (푋푑+1, … , 푋푛). This is known by Corollary 6.2.2. The dimension formula allows us to compute dim 퐵 by choosing any prime 픮, a prime chain of longest length below 픮 (i.e. in 퐵픮) and another one above 픮 (i.e. in 퐵/픮); their concatenation will then be a prime chain in 퐵 with maximal length dim 퐵. By applying the dimension formula to 픮, 픮′ ⊂ 퐵 and to 픮′/픮 ⊂ 퐵/픮, it follows that ht(픮′) = ht(픮) + ht(픮′/픮) for all 픮′ ⊋ 픮 in 퐵. §6.3 Noether normalization and its consequences ⋅ 71 ⋅

For a general Noetherian domain 퐵, we call a prime chain maximal if it is not properly contained in any prime chain. A priori, a maximal prime chain does not necessarily have length equal to dim 퐵. If

∀픮′ ⊃ 픮 ∶ primes, ht(픮′) = ht(픮) + ht(픮′/픮),

′ we say 퐵 is a catenary domain. This means that ht(픮 /픮) = dim 퐵픮′ /픮퐵픮′ is the common length of all maximal prime chains between 픮′ and 픮. In particular, maximal prime chains in 퐵픮′ are automatically longest. As shown above, finitely generated domains over a field are catenary. For a finer analysis of catenary and universally catenary rings, we refer to [11, §14].

Corollary 6.3.5. Suppose 퐵 is a domain finitely generated over a field 한. Set 퐿 ∶= Frac(퐵).

Then dim 퐵 = tr.deg한(퐿) and it is the common length of maximal prime chains. Proof. Choose a subalgebra 퐴 of 퐵 as in Theorem 6.3.3. Since the field Frac(퐵) is a finite extension of Frac(퐴) and dim 퐴 = dim 퐵 by Proposition 6.1.3, the first assertion reduces immediately to the case 퐵 = 한[푋1, … , 푋푛], which is obvious. As to the second assertion, consider a maximal prime chain 픮0 ⊋ ⋯ ⊋ 픮푛 = {0} in 퐵. Maximality implies ht(픮푖/픮푖+1) = 1 for all 푖 and dim 퐵/픮0 = 0. Applying Corollary 6.3.4 repeatedly, we see

dim 퐵 = dim 퐵/픮푛 = dim 퐵/픮푛−1 + ht(픮푛−1/픮푛) = 푛−1 dim 퐵/픮푛−2 + ht(픮푛−1/픮푛−2) + ht(픮푛−1/픮푛) = ⋯ = dim 퐵/픮0 + ∑ ht(픮푖/픮푖+1) 푖=0

which equals 푛. Remark 6.3.6. Recall that finitely generated domains over an algebraically closed field 한 are objects “opposite” to the irreducible affine 한-varieties. It is instructive to make a comparison with the analytic theory when 한 = ℂ. Let 풳 be a compact connected complex manifold of (complex) dimension 푛. Denote by ℳ(풳) the field of meromorphic functions on 풳. Siegel proved that tr.degℂ(ℳ(풳)) ≤ 푛. When 풳 is a projective algebraic ℂ-variety, equality holds and we have ℳ(풳) = Frac(퐴) if Spec(퐴) is any open dense affine subscheme in 풳. In general, the abundance of meromorphic functions on 풳 is a subtle issue, cf. the case of Riemann surfaces (푛 = 1). Compact connected

complex manifolds with tr.degℂ(ℳ(풳)) = dimℂ 풳 are called Moishezon manifolds. Non-algebraic Moishezon manifolds do exist, and Moishezon proved that a Moishe- zon manifold is projective, hence algebraic, if and only if it is Kähler. Following M. Artin and D. Knutson, one can enlarge the category of ℂ-schemes into that of algebraic spaces over ℂ, and there is an analytification functor 풳 ↦ 풳an that sends algebraic spaces of finite type over ℂ to complex analytic varieties. M. Artin [2, §7] showed that the analytification establishes an equivalence between the category of smooth proper algebraic spaces of finite type over ℂ and that of Moishezon manifolds. Therefore, such manifolds still retain an algebraic flavor: they are quotients of certain ℂ-schemes by étale equivalence relations. ⋅ 72 ⋅ Dimension of finitely generated algebras Lecture 7

Serre's criterion for normality and depth

References: [8, §11] and [3, X.1]. Except in the last section, we will try to avoid the use of depth as in [8].

7.1 Review of discrete valuation rings

Let 푅 be an integral domain. Definition 7.1.1. A discrete valuation on a field 퐾 is a surjective map 푣 ∶ 퐾× → ℤ such that ⋄ 푣(푥푦) = 푣(푥) + 푣(푦), and ⋄ 푣(푥 + 푦) ≥ min{푣(푥), 푣(푦)} for all 푥, 푦 ∈ 퐾×, where we set 푣(0) = +∞ for convenience. We say a domain 푅 with 퐾 ∶= Frac(푅) is a discrete valuation ring, abbreviated as DVR, if there exists a discrete valuation 푣 such that 푅 = 푣−1([0, +∞]). We say 푡 ∈ 푅 is a uniformizer if 푣(푡) = 1. It follows immediately that 푅× = 푣−1(0). Uniformizers are unique up to 푅×. Note that 푣(퐾×) = ℤ implies that 푅 cannot be a field.

Example 7.1.2. The ring ℤ푝 of 푝-adic integers (푝: prime number) together with the usual 푝-adic valuations are standard examples of DVR. The algebra of formal power J K 푛 series 한 푋 are also DVR: the valuation of ∑푛 푎푛푋 is the smallest 푛 such that 푎푛 ≠ 0. More generally, in the geometric context, discrete valuations can be defined by looking at the vanishing order of rational/meromorphic functions along subvarieties of codimension one with suitable regularities. Lemma 7.1.3. Let 푅 be a discrete valuation ring with valuation 푣 and uniformizer 푡. Every ideal 픞 ≠ {0} of 푅 has the form (푡푟) for a unique 푟 ≥ 0. In particular, 푅 is a local principal ideal domain which is not a field, hence is of dimension 1. ⋅ 74 ⋅ Serre’s criterion for normality and depth

Proof. Take 푟 ∶= min{푣(푥) ∶ 푥 ∈ 픞}. In the exercises below, we assume 푅 is a discrete valuation ring with valuation 푣.

Exercise 7.1.4. Show that 푡 is a uniformizer if and only if it generates the maximal ideal of 푅.

Exercise 7.1.5. Reconstruct 푣 from the ring-theoretic structure of 푅.

Recall that a regular local ring 푅 with dim 푅 = 1 is a Noetherian local ring whose maximal ideal 픪 is principal and nonzero; elements generating 픪 are called the regular parameters for 푅.

Proposition 7.1.6. Suppose 푡 is a regular parameter in a regular local ring 푅 of dimension one, then 푅 is a domain, and every element 푥 ∈ 푅 ∖ {0} can be uniquely written as 푥 = 푡푟푢 with 푟 ≥ 0 and 푢 ∈ 푅×. This makes 푅 into a discrete valuation ring by setting 푣(푥) = 푟, for which 푡 is a uniformizer. Therefore 푅 is a discrete valuation ring. Conversely, every discrete valuation ring is regular local of dimension 1.

Proof. From Theorem 5.5.6 we know regular local rings are Noetherian domains. By applying Krull’s Intersection Theorem (Corollary 4.5.10) to the powers of (푡), we see that 푟 ∶= sup{푘 ≥ 0 ∶ 푥 ∈ (푡)푘} is finite. Write 푥 = 푡푟푢. Since 푅× = 푅 ∖ 픪, we see 푢 ∈ 푅×. As to uniqueness, suppose 푡푟푢 = 푡푠푤 with 푟 ≥ 푠, then 푡푟−푠 = 푢−1푤 ∈ 푅× implies 푟 = 푠, hence 푢 = 푤 as 푅 is a domain. As every element of Frac(푅)× is uniquely expressed as 푡푟푢 with 푟 ∈ ℤ, one readily checks that 푣(푡푟푢) = 푟 satisfies all the requirements of discrete valuation. The converse direction has been addressed in Lemma 7.1.3. To recap, in dimension one we have

regular local ring ⟺ discrete valuation ring.

This will be related to normality later on.

Exercise 7.1.7. Explain that the regular local rings of dimension 0 are just fields.

7.2 Auxiliary results on the total fraction ring

Let 푅 be a ring, henceforth assumed Noetherian. If there exist a non zero-divisor 푡 ∈ 푅 and 픭 ∈ Ass(푅/(푡)), we say 픭 is associated to a non zero-divisor.

Lemma 7.2.1. Let 푀 be a finitely generated 푅-module. An element 푥 ∈ 푀 is zero if and only if its image in 푀픭 is zero for every maximal element 픭 in Ass(푀). Proof. Suppose 푥 ≠ 0. Since 푀 is Noetherian, among ideals of the form ann(푦) there is a maximal one containing ann(푥), and we have seen in Lemma 2.2.3 that such an ideal 픭 belongs to Ass(푀). Since ann(푥) ⊂ 픭, we have 푥/1 ∈ 푀픭 ∖ {0}. Call a ring reduced if it has no nilpotent element except zero. §7.2 Auxiliary results on the total fraction ring ⋅ 75 ⋅

Lemma 7.2.2. Suppose 푅 is reduced, then Ass(푅) consists of minimal primes.

Proof. As 푅 is reduced, {0} = √0푅 is the intersection of minimal prime ideals 픭1, 픭2,… (all lying in Ass(푅), hence finite in number). By the theory of primary decompositions, one infers that Ass(푅) = {픭1, …}.

For the next result, we denote by 푇 the set of non zero-divisors of 푅. Recall that the total fraction ring 퐾(푅) is 푅[푇−1]; this is the largest localization such that 푅 → 퐾(푅) is injective, and 퐾(푅) = Frac(푅) when 푅 is a domain. The map 픭 ↦ 픭퐾(푅) sets up an order-preserving bijection Ass(푅) →∼ Ass(퐾(푅)): indeed, if 픭 ∋ 푡 for some 푡 ∈ 푇, then 픭 cannot belong to Ass(푅) because the union of Ass(푅) equals 푅 ∖ 푇.

∼ Lemma 7.2.3. Let 푅 be reduced. Then 퐾(푅) → ∏픭 퐾(푅/픭) as 푅-algebras, where 픭 ranges over the minimal prime ideals of 푅. For any multiplicative subset 푆 ⊂ 푅 there is a canonical isomorphism of 푅[푆−1]-algebras

퐾(푅[푆−1]) →∼ 퐾(푅)[푆−1].

In other words, the formation of total fraction ring commutes with localizations.

Proof. Each element in 퐾(푅) = 푅[푇−1] is either a zero-divisor or invertible. The set of zero-divisors of 퐾(푅) is the union of minimal prime ideals 픭푖퐾(푅) of 퐾(푅) (where Ass(푅) = {픭1, … , 픭푚} by an earlier discussion), therefore each prime ideal of 퐾(푅) must equal some 픭푖퐾(푅), by prime avoidance (Proposition 1.1.5). Hence 픭1퐾(푅), … , 픭푚퐾(푅) are also the maximal ideals in 퐾(푅), with zero intersection. Chinese Remainder The- 푚 orem entails that 퐾(푅) ≃ ∏푖=1 퐾(푅)/픭푖퐾(푅). To conclude the first part, notice that −1 퐾(푅)/픭푖퐾(푅) = (푅/픭푖)[푇 ]; this is a field in generated by an isomorphic copyof 푅/픭푖 since 픭푖 ∩ 푇 = ∅, hence equals Frac(푅/픭푖). As for the second part, one decomposes 퐾(푅[푆−1]) and 퐾(푅)[푆−1] by the previous step, noting that ⋄ 푅[푆−1] is reduced; −1 −1 ⋄ Ass(푅[푆 ]) = {픭푖푅[푆 ] ∶ 1 ≤ 푖 ≤ 푚, 픭푖 ∩ 푆 = ∅} consists of minimal primes; −1 −1 −1 ⋄ 퐾 (푅[푆 ]/픭푖푅[푆 ]) ≃ 퐾(푅/픭푖) = 퐾(푅/픭푖)[푆 ] when 픭푖 ∩ 푆 = ∅, by the arguments above; −1 ⋄ 퐾(푅/픭푖)[푆 ] = {0} when 픭푖 ∩ 푆 ≠ ∅. A term-by-term comparison finishes the proof.

We use Lemma 7.2.3 to interchange 퐾(⋅) and localizations in what follows.

Lemma 7.2.4. Suppose 푅 is reduced. Then 푥 ∈ 퐾(푅) belongs to 푅 if and only if its image in 퐾(푅)픭 = 퐾(푅픭) belongs to 푅픭 for every prime 픭 associated to a non zero-divisor.

Proof. Only the “if” direction requires a proof. Write 푥 = 푎/푡 with 푡 not a zero-divisor. Suppose that 푎 ∉ (푡), i.e. 푎 does not map to zero in 푅/(푡). Lemma 7.2.1 asserts there exists 픭 ∈ Ass(푅/(푡)) such that 푎 does not map to 0 ∈ (푅/(푡))픭 = 푅픭/푡푅픭. It follows that the image of 푎/푡 in 퐾(푅픭) does not lie in 푅픭. ⋅ 76 ⋅ Serre’s criterion for normality and depth

7.3 On normality

Fix a Noetherian ring 푅.

Exercise 7.3.1. Suppose 푅 is a domain, 퐾 ∶= Frac(푅). If 푅 = ⋂푖∈퐼 푅푖 where {푅푖 ⊂ 퐾}푖∈퐼 are subrings such that Frac(푅푖) = 퐾 and 푅푖 is normal, for each 푖, then 푅 is normal. Proposition 7.3.2. Let 푅 be a Noetherian domain. Then 푅 is normal if and only if for every principal ideal (푡) ⊂ 푅 and every 픭 ∈ Ass(푅/(푡)), the ideal 픭푅픭 is principal. Proof. Assume the conditions above. To prove the normality of 푅, it suffices to use 푅 = ⋂ 푅픭 where 픭 ranges over the primes associated to nonzero principal ideals (consequence of Lemma 7.2.4). Indeed, each 푅픭 is regular, hence normal by Proposition 7.1.6, therefore so is their intersection by the previous exercise. Conversely, assume 푅 is normal and let 픭 ∈ Ass(푅/(푡)) with 푡 ≠ 0, we have to show 픭푅픭 is principal. Upon replacing 푅 by 푅픭 and recalling how associated primes behave under localization, we may even assume 푅 is local with maximal ideal 픭. Express 픭 as the annihilator of some ̄푥∈ 푅/(푡) with 푥 ∈ 푅. Define the fractional ideal

픭−1 ∶= {푦 ∈ Frac(푅) ∶ 푦픭 ⊂ 푅}.

It is an 푅-submodule of Frac(푅) containing 푅. Define the 푅-submodule 픭−1픭 of Frac(푅) in the obvious way. Clearly 픭 ⊂ 픭−1픭 ⊂ 푅. By maximality of 픭, exactly one of the ⊂ is equality. If 픭픭−1 = 픭, every element of 픭−1 is integral over 푅, hence 픭−1 ⊂ 푅 by normality (integrality is “witnessed” by the module 픭). From 픭푥 ⊂ (푡) we see 푥/푡 ∈ 픭−1 = 푅; this would imply ̄푥= 0 and 픭 = 푅, which is absurd. Therefore we must have 픭픭−1 = 푅. This implies that 픭푦 ⊄ 픭 for some 푦 ∈ 픭−1, hence 픭푦 = 푅 since 푅 is local. Hence 픭 = 푦−1푅 ≃ 푅 is principal.

Corollary 7.3.3. The following are equivalent for a local domain 푅:

(i) 푅 is normal of dimension 1;

(ii) 푅 is a regular local ring of dimension 1;

(iii) 푅 is a discrete valuation ring.

Proof. (iii) ⟹ (i). We have seen that discrete valuation rings are principal ideal rings of dimension 1, therefore also normal by unique factorization property. (i) ⟹ (ii). Under the normality assumption, choose any 푡 ∈ 푅 ∖ {0}. Since dim 푅 = 1 and {0} is a prime ideal, the associated prime of (푡) can only be the maximal ideal 픪, which is principal by Proposition 7.3.2. This shows that 푅 is regular local. (ii) ⟹ (iii) is included in Proposition 7.1.6.

Corollary 7.3.4. Let 푅 be a Noetherian normal domain. Then

푅 = ⋂ 푅픭 ht(픭)=1

inside Frac(푅). §7.4 Serre’s criterion ⋅ 77 ⋅

Proof. Evidently ⊂ holds. By Lemma 7.2.4 together with Proposition 7.3.2, 푅 can be written as an intersection of 푅픭 where 픭 is associated to some non zero-divisor, such that 픭푅픭 is principal; it suffices to show ht(픭) = 1. From 픭 ≠ {0} we see ht(픭) ≥ 1; on the other hand, by Hauptidealsatz or by the discussion on regular local rings, we see ht(픭) = ht(픭푅픭) ≤ 1.

7.4 Serre’s criterion

Lemma 7.4.1. Let 푅 be a Noetherian ring. Suppose that ⋄ the primes in Ass(푅) are all minimal, and ⋄ 푅픭 is a field for every minimal prime ideal 픭, then 푅 is reduced.

Proof. Take a minimal primary decomposition {0} = 퐼1 ∩ ⋯ ∩ 퐼푚 with Ass(푅/퐼푗) =

{픭푗 = √퐼푗} and Ass(푅) = {픭1, … , 픭푚}. By the properties of primary ideals, 픭푗 = √퐼푗 ⊃ 퐼푗 푗 픭 푅 for all . By assumption each 푗 is minimal, and 픭푗 is a field. From the uniqueness of 퐼 = [푅 → 푅 ] non-embedded components in primary decompositions, 푗 ker 픭푗 is a prime 푚 contained in 픭푗, hence 퐼푗 = 픭푗. We deduce that {0} = ⋂푗=1 픭푗, thereby showing √0푅 = {0}.

Theorem 7.4.2 (J.-P. Serre). A Noetherian ring 푅 is a finite direct product of normal domains if and only if the following two conditions hold.

⊳ R1 The localization of 푅 at every prime ideal of height 1 (resp. 0) is a discrete valuation ring (resp. a field).

⊳ S2 For every non zero-divisor 푡 of 푅, the primes in Ass(푅/(푡)) are all of height 1; the primes in Ass(푅) are all of height 0.

The condition R1 means regularity in codimension ≤ 1. The condition S2 is often rephrased in terms of depth, which will be discussed in Proposition 7.5.11.

Proof. We begin with the ⟹ direction. Suppose 푅 = 푅1 × ⋯ × 푅푛 where each 푅푖 is 푛 a normal domain. As is well-known, Spec(푅) = ⨆푖=1 Spec(푅푖) as topological spaces: to be precise, the elements of Spec(푅) take the form 픭 = 푅1 × ⋯ × 픭푖 × ⋯ × 푅푛, where 픭푖 ∈ Spec(푅푖). We have (픭) = (픭 ), 푅 ≃ (푅 ) . ht ht 푖 픭 푖 픭푖 Furthermore, one easily checks that

푛 Ass(푅/(푡)) = ⨆ Ass(푅푖/(푡푖)), 푡 = (푡1, … , 푡푛) ∈ 푅 푖=1 compatibly with the description above. This reduces the verification of S2 to the case of normal domains, which is addressed in Proposition 7.3.2. The condition R1 is implied by Corollary 7.3.3 since normality is preserved under localizations. ⋅ 78 ⋅ Serre’s criterion for normality and depth

Assume conversely R1 and S2. They imply the conditions of Lemma 7.4.1, hence 푅 is reduced. Now for every prime 픭 associated to a non zero-divisor, we have ht(픭) = 1 and 푅픭 is a normal domain by R1 ∧ S2. By Lemma 7.2.4 (as 푅 is reduced), 푅 is integrally closed in 퐾(푅): indeed, if 푥 ∈ 퐾(푅) is integral over 푅, so is its image in 퐾(푅픭) = 푚 퐾(푅)픭 for every 픭 as above, therefore lies in 푅픭. Decompose 퐾(푅) = ∏푖=1 퐾(푅/픭푖) as in Lemma 7.2.3. The idempotents 푒푖 ∈ 퐾(푅) associated to this decomposition are trivially 2 integral over 푅: 푒푖 − 푒푖 = 0, hence 푒푖 ∈ 푅 for all 푖. It follows that 푅 = 푅푒1 + ⋯ + 푚 푅푒푚 ∏푖=1 푅푒푖 ⊂ 퐾(푅) and one easily checks that 푅푒푖 = 푅/픭푖. Finally, since 푅 is integrally closed in 퐾(푅), the decomposition above implies 푅/픭푖 is integrally closed in 퐾(푅/픭푖). All in all, we have written 푅 as a direct product of normal domains.

Exercise 7.4.3. Recall that for an 푅-module 푀, a prime ideal 픭 ∈ Ass(푀) is called embedded if 픭 is not a minimal element in Ass(푀). Show that for 푀 = 푅, embedded primes are primes in Ass(푅) with height > 0. For 푀 = 푅/(푡) where 푡 is not a zero-divisor, embedded primes are primes in Ass(푅/(푡)) with height > 1. Use this to rephrase S2 as follows: there are no embedded primes in Ass(푅/(푡)) (푡 not a zero-divisor) or in Ass(푅).

Exercise 7.4.4. Suppose a ring 푅 is isomorphic to a direct product ∏푖∈퐼 푅푖. Show that 푅 |퐼| = 1 퐼 = {푖 } 푅 is a domain if and only if (say 0 ), and 푖0 is a domain. Corollary 7.4.5. A Noetherian domain 푅 is normal if and only if R1 and S2 hold for 푅.

Proof. Immediate from the previous exercise and Theorem 7.4.2.

7.5 Introduction to depth

Based on [3], we give a brief account on the notion of depth. Let 푅 be a ring and 푀 be an 푅-module, 푀 ≠ {0}. Recall the Ext-functors

푛 푛 Ext푅(푋, 푌) ∶= 퐻 (푅ℋom(푋, 푌)) = Hom퐷+(푅-Mod)(푋, 푌[푛]).

Definition 7.5.1 (Depth of a module). Let 퐼 be a proper ideal of 푅. We define the depth of 푀 relative to 퐼 as

푛 depth퐼(푀) ∶= inf {푛 ≥ 0 ∶ Ext푅(푅/퐼, 푀) ≠ 0}

with values in ℤ≥0 ⊔ {+∞}. Proposition 7.5.2. For 퐼, 푀 as above, the following are equivalent:

(i) depth퐼(푀) = 0;

푥 (ii) for all 푥 ∈ 퐼, the homomorphism 푀 푀 is not injective;

(iii) Ass(푀) ∩ 푉(퐼) ≠ ∅. §7.5 Introduction to depth ⋅ 79 ⋅

푥 Proof. In each case we have 푀 ≠ {0}. If (i) holds, then 푀 푀 vanishes on the image of some nonzero 푅/퐼 → 푀, hence (ii). If (ii) holds, the union of Ass(푀) will cover 퐼, and (iii) follows by prime avoidance. Finally, suppose 픭 ∈ Ass(푀) ∩ 푉(퐼), there is an embedding 푅/픭 ↪ 푀, which yields a non-zero 푅/퐼 → 푀.

Definition 7.5.3. A sequence 푥1, … , 푥푛 ∈ 푅 is called an 푀-regular sequence of length 푛 if (푥1, … , 푥푛)푀 ⊊ 푀 and

푥푘 0 → 푀/(푥1, … , 푥푘−1) 푀/(푥1, … , 푥푘−1)

is exact for all 1 ≤ 푘 ≤ 푛.

Lemma 7.5.4. Let 푀 be an 푅-module, 푥1, … , 푥푟 be an 푀-regular sequence lying in an ideal 퐼 ⊊ 푅. We have depth퐼(푀) = 푟 + depth퐼(푀/(푥1, … , 푥푟)푀). Proof. The case 푟 = 1 follows by staring at the long exact sequence attached to 0 → 푥1 푀 푀 → 푀/푥1푀 → 0. The general case follows by induction on 푟. Theorem 7.5.5. Assume 푅 Noetherian, 푀 finitely generated and 퐼 ⊊ 푅.

(i) depth퐼(푀) is the supremum of the lengths of 푀-regular sequences with elements in 퐼.

(ii) Suppose depth퐼(푀) < +∞. Every 푀-regular sequence with elements in 퐼 can be extended to one of length depth퐼(푀). (iii) The depth of 푀 relative to 퐼 is finite if and only if 푉(퐼) ∩ Supp(푀) ≠ ∅, or equivalently 퐼푀 ≠ 푀.

Proof. To prove (i) and (ii), by the previous Lemma we are reduced to show that

depth퐼(푀) > 0 implies the existence of 푥 ∈ 퐼 which is not a zero-divisor of 푀; this follows from Proposition 7.5.2. Now pass to the word “equivalently” in (iii). We have 푀 ≠ 퐼푀 if and only if (푀/퐼푀)픭 = 푀픭/퐼픭푀픭 ≠ 0 for some prime ideal 픭. That quotient always vanishes when 픭 ⊅ 퐼, in which case 퐼픭 = 푅픭. On the other hand, when 픭 ∈ 푉(퐼) we have 퐼픭 ⊂ rad(푅픭), thus the non-vanishing is equivalent to 푀픭 ≠ {0} by Nakayama’s Lemma 1.3.5. The “if” direction of (iii) is based on the following fact

퐼푀 ≠ 푀 ⟹ depth퐼(푀) < +∞ which will be proved in the next lecture (Theorem 8.3.2) using Koszul complexes. As for the “only if” direction, 푉(퐼) ∩ Supp(푀) = ∅ implies 퐼 + ann(푀) = 푅, but the elements 푛 in 퐼 + ann(푀) annihilate each Ext (푅/퐼, 푀), hence depth퐼(푀) = +∞.

Corollary 7.5.6. With the same assumptions, let (푥1, … , 푥푟) be an 푀-regular sequence with 푥푖 ∈ 퐼. It is of length depth퐼(푀) if and only if Ass(푀/(푥1, … , 푥푟)푀) ∩ 푉(퐼) ≠ ∅.

Proof. The sequence has length depth퐼(푀) if and only if 푅-module 푀/(푥1, … , 푥푟)푀 has depth zero, so it remains to apply Proposition 7.5.2. ⋅ 80 ⋅ Serre’s criterion for normality and depth

Jean-Louis Koszul (1921—2018) created the Koszul complexes in order to define a cohomology theory for Lie algebras; this device turns out to be a general, convenient construction in homological algebra, which will be discussed in the next lecture. The study of “Koszulness” in a broader (eg. operadic) context is now an active area of research. J.-L. Koszul is also a second-generation member of the Bourbaki group. Source: by Konrad Jacobs - Oberwolfach Photo Collection, ID 2273.

Corollary 7.5.7. Let 푅 be a Noetherian local ring with maximal ideal 픪, and 푀 ≠ {0} a finitely

generated 푅-module, then depth픪(푀) ≤ dim 푀. Proof. Consider the following situation: 푥 ∈ 픪 is not a zero-divisor for 푀 ≠ {0}. In the discussion of dimensions, we have seen that 푑(푀) ≥ 푑(푀/푥푀) ≥ 푑(푀/푥푀) − 1, where 푑(⋅) is the degree of Hilbert–Samuel polynomial; on the other hand, since the 푥 alternating sum of Hilbert–Samuel polynomials in 0 → 푀 푀 → 푀/푥푀 → 0 has degree < 푑(푀), we infer that 푑(푀/푥푀) = 푑(푀)−1. Hence dim(푀/푥푀) = dim(푀)−1.

By relating depth to 푀-regular sequences, we deduce depth픪(푀) ≤ dim 푀. Definition 7.5.8 (Cohen–Macaulay modules). Let 푅 be a Noetherian ring. A finitely generated 푅-module 푀 is called Cohen–Macaulay if

depth (푀 ) = dim 푀 픪퐴픪 픪 픪 for every 픪 ∈ MaxSpec(푅) ∩ Supp(푀). We say 푅 is a Cohen–Macaulay ring if it is Cohen–Macaulay as a module.

Example 7.5.9. Regular local rings are Cohen–Macaulay, although we do not prove this here. Another important class of Cohen–Macaulay rings is the algebra of invariants 퐴퐺 where 퐴 is the algebra of regular functions on an affine 한-variety 풳 with rational singularities (eg. 퐴 = 한[푋1, … , 푋푛]) with an action by a reductive 한-group 퐺 (finite §7.5 Introduction to depth ⋅ 81 ⋅

groups allowed), and we assume char(한) = 0. Here is the reason: Boutot [6] proved the GIT quotient 풳//퐺 has rational singularities as well, hence is Cohen–Macaulay; in characteristic zero this strengthens an earlier theorem of Hochster–Roberts. These algebras are interesting objects from the geometric, algebraic or even combinatorial perspectives.

Exercise 7.5.10. Show by using Proposition 7.5.2 that depth픪(푅) = 0 if 푅 is local with maximal ideal 픪 and has dimension zero. Proposition 7.5.11. The condition S2 in Theorem 7.4.2 is equivalent to

ht(픭) ≥ 푖 ⟹ depth (푅 ) ≥ 푖 픭푅픭 픭 for all 픭 ∈ Spec(푅) and 푖 ∈ {1, 2}. Proof. Assume the displayed conditions. We first show that every 픭 ∈ Ass(푅) has height zero. Upon localization we may assume 푅 local with maximal ideal 픭, thus 픭 has depth zero by Proposition 7.5.2; our conditions force ht(픭) = 0. Next, consider 픭 ∈ Ass(푅/(푡)) with 푡 non zero-divisor. Note that 푡 remains a non zero-divisor in 푅픭, and 픭푅픭 ∈ Ass(푅픭/(푡)). Hence depth(푅픭) = depth(푅픭/(푡)) + 1 = 1 since depth픭(푅픭/(푡)) = 0 by Proposition 7.5.2. Our conditions force ht(픭) ≤ 1 and 푡 ∈ 픭 implies ht(픭) > 0. Conversely, assume S2. Suppose ht(픭) ≥ 1. If 푅픭 has depth zero then Ass(푅픭) ∩ 푉(픭푅픭) ≠ ∅, which implies 픭푅픭 ∈ Ass(푅픭) thus 픭 ∈ Ass(푅), contradicting S2. Next, suppose ht(픭) ≥ 2. If 푅픭 has depth ≤ 1, the standard property

∀푖 ≥ 0, Ext푖 (푋 , 푌 ) ≃ Ext푖 (푋, 푌) 푅픭 픭 픭 푅 픭 valid for Noetherian 푅 and finitely generated 푅-modules 푋, 푌, yields

0 ≤ depth (푅) ≤ depth (푅 ) ≤ 1. 픭 픭푅픭 픭

′ ′ If depth픭(푅) = 0, there exists of 픭 ⊃ 픭 and 픭 ∈ Ass(푅) by Proposition 7.5.2, but S2 ′ implies ht(픭) ≤ ht(픭 ) = 0 which is absurd. If depth픭(푅) = 1, there exists a non zero- ′ divisor 푥 in 푅 such that depth픭(푅/(푥)) = 0. The same argument furnishes 픭 ⊃ 픭 such that 픭′ ∈ Ass(푅/(푥)). Now S2 implies ht(픭) ≤ ht(픭′) = 1, again a contradiction.

Now the conditions R1 and S2 can be generalized to arbitrary 푘 ∈ ℤ≥0:

⊳ Rk ht(픭) ≤ 푘 implies 푅픭 is a regular local ring, for every 픭 ∈ Spec(푅);

⊳ Sk depth(푅픭) ≥ min{푘, ht(픭)} for every 픭 ∈ Spec(푅). One readily checks their compatibility with Proposition 7.5.11. Note that S0 is trivial, whilst 푅 is Cohen–Macaulay if and only if it satisfies Sk for all 푘. Example 7.5.12. From Theorem 7.4.2, we see that any finite direct product of normal domains of dimension ≤ 2 is Cohen–Macaulay. Exercise 7.5.13. Show that S1 holds if and only if there are no embedded primes in Ass(푅). ⋅ 82 ⋅ Serre’s criterion for normality and depth Lecture 8

Some aspects of Koszul complexes

This lecture is a faithful replay of the relevant sections of [4] and [3]; the main aim here is to complete the proof of Theorem 7.5.5. In what follows, we work with a chosen ring 푅. We impose no Noetherian or finiteness conditions here.

8.1 Preparations in homological algebra

For any 푅-module 퐿, denote its exterior algebra over 푅 by 푛 ⋀ 퐿 ∶= ⨁ ⋀ 퐿. 푛≥0 푛 ⊗푛 It is the quotient of the tensor algebra 푇(퐿) = ⨁푛≥0(푇 (퐿) ∶= 퐿 ) by the graded ideal generated by the pure tensors ⋯ ⊗ 푥 ⊗ 푥 ⊗ ⋯ , 푥 ∈ 퐿. 0 The multiplication operation in ⋀ 퐿 is written as ∧. Note that ⋀ 퐿 = 푇0(퐿) = 푅 by convention. The traditional notion of exterior algebras encountered in differential geometry is recovered when ℚ ⊂ 푅. 푛+1 Given 푢 ∈ Hom푅(퐿, 푅), one can define the corresponding contractions 푖푢 ∶ ⋀ 퐿 → 푛 ⋀ 퐿, given concretely as 푛 푖 푖푢(푥0 ∧ ⋯ ∧ 푥푛) = ∑(−1) 푢(푥푖) ⋅ 푥0 ∧ ⋯ ̂푥푖 ⋯ ∧ 푥푛 푖=0 where ̂푥푖 means 푥푖 is omitted. It is routine to check that 푖푢 satisfies 푖푢 ∘ 푖푢 = 0, thereby giving rise to a chain complex. Definition 8.1.1. Let 퐿 and 푢 be as above. Define the corresponding Koszul complex as • 퐾•(푢) ∶= (⋀ 퐿, 푖푢). For any 푅-module 푀, put

퐾•(푢; 푀) ∶= 푀 ⊗ 퐾•(푢), 푅 • 퐾 (푢; 푀) ∶= Hom퐴(퐾•(푢), 푀) ⋅ 84 ⋅ Some aspects of Koszul complexes

which is naturally a chain (resp. cochain) complex in positive degrees; here one re- gards 푀 as a complex in degree zero. These definitions generalize to the case of any complex 푀, and are functorial in 푀. The reader might have encountered the following result in differential geometry. 푛 Proposition 8.1.2 (Homotopy formula). For any 푥 ∈ 퐿 and 휔 ∈ ⋀ 퐿, we have

푖푢(푥 ∧ 휔) + 푥 ∧ (푖푢(휔)) = 푢(푥)휔.

Proof. Consider 휔 = 푥1 ∧ ⋯ ∧ 푥푛. Put 푥0 ∶= 푥. The left-hand side equals

푛 푖 ∑(−1) ⋅ 푢(푥푖) ⋯ ∧ ̂푥푖 ∧ ⋯ 푖=0 whereas the right-hand side equals

푛 푖+1 ∑(−1) ⋅ 푢(푥푖)푥0 ∧ ⋯ ∧ ̂푥푖 ∧ ⋯ . 푖=1

The terms with a ̂푥푖 (non-existant — hopefully this won’t generate metaphysical issues) where 푖 > 0 cancel out. We are left with 푢(푥0)푥1 ∧ ⋯ ∧ 푥푛 = 푢(푥)휔. Obviously, the same formula extends to all 휔 ∈ ⋀ 퐿 by linearity. Proposition 8.1.3. Set 픮 ∶= 푢(퐿), which is an ideal of 푅. Then 픮 annihilates each homology • (resp. cohomology) of 퐾•(푢; 푀) (resp. 퐾 (푢; 푀)). Again, this generalizes to general complexes 푀. Proof. Given 푡 ∈ 픮, the homotopy formula implies that the endomorphism 휔 ↦ 푡휔 of 퐾•(푢) is homotopic to zero, hence so are the induced endomorphisms of 퐾•(푢; 푀) and 퐾•(푢; 푀) by standard homological algebra. Proposition 8.1.4. Suppose 퐿 is projective over 푅 and 0 → 푀′ → 푀 → 푀″ → 0 is exact. Then there is a natural short exact sequence of complexes

0 → 퐾•(푢; 푀′) → 퐾•(푢; 푀) → 퐾•(푢; 푀″) → 0

which gives rise to a long exact sequence of cohomologies of the Koszul complexes in question. 푛 Proof. Standard. It suffices to note that 퐿 is projective implies each graded piece ⋀ 퐿 of ⋀ 퐿 is projective as well. 푛 Exercise 8.1.5. Justify the assertion above concerning the projectivity of ⋀ 퐿. Hint: 푛 suppose 퐿 is a direct summand of a free module 퐹, show that ⋀ 퐿 is a direct summand 푛 푛 of ⋀ 퐹 and ⋀ 퐹 is free. Similar properties hold for the homological version when 퐿 is flat over 푅. One needs 푛 the property that ⋀ 퐿 is flat if 퐿 is. 푛 Exercise 8.1.6. Prove the assertion above concerning flatness of ⋀ 퐿. Consult the proof in [4, p.15] if necessary. §8.2 Auxiliary results on depth ⋅ 85 ⋅

8.2 Auxiliary results on depth

We fix an ideal 퐼 ⊂ 푅.

Proposition 8.2.1. For every family {푀훽}훽∈ℬ of 푅-modules, we have

⎜⎛ ⎟⎞ depth ⎜∏ 푀훽⎟ = inf depth (푀훽). 퐼 훽 퐼 ⎝ 훽 ⎠

푛 푛 Proof. This follows from Ext푅(푅/퐼, ∏푖 푀푖) = ∏푖∈퐼 Ext푅(푅/퐼, 푀푖), as is easily seen by taking a projective resolution of 푅/퐼 and using the fact the Hom푅 preserves direct products in the second variable.

Remark 8.2.2. By stipulation, the empty product is 0, the zero object of the category 푅-Mod. In parallel we define inf ∅ ∶= ∞, so that Proposition 8.2.1 remains true in this case, since the zero module has infinite depth.

Proposition 8.2.3. Suppose 푁 is an 푅-module annihilated by some 퐼푚, where 푚 ≥ 1. Then 푖 Ext푅(푁, 푀) = 0 whenever 푖 < depth퐼(푀). 푖 Proof. To show Ext푅(푁, 푀) = 0 for 푖 < depth퐼(푀), we begin with the case 푚 = 1. This case follows by a dimension-shifting argument based on the short exact sequence

0 → 퐾 → (푅/퐼)⊕퐼 → 푀 → 0,

<0 together with induction on 푖 (note that Ext푅 (푁, 푀) = 0 for trivial reasons). The case of general 푚 follows by a standard dévissage using

0 → 퐽푁 → 푁 → 푁/퐽푁 → 0

and the associated long exact sequence.

푚 Lemma 8.2.4. Suppose 퐽 is an ideal satisfying 퐽 ⊃ 퐼 for some 푚 ≥ 1. Then depth퐼(푀) ≤ depth퐽(푀).

푖 Proof. From the previous Proposition, we have Ext푅(푅/퐽, 푀) = 0 for 푖 < depth퐼(푀) since 퐼푚 annihilates 푅/퐽. The assertion follows upon recalling the definition of depth.

The following technical results will be invoked in the proof of Theorem 8.3.2.

Proposition 8.2.5. Let 퐶• be a cochain complex of 푅-modules such that 푛 ≪ 0 ⟹ 퐶푛 = 푛 0. Let ℎ ∈ ℤ such that for all integers 푛 ≤ 푘 ≤ ℎ, the depth of 퐶 with respect to 퐽푘 ∶= ann(퐻푘(퐶•)) is > 푘 − 푛, then 퐻≤ℎ(퐶•) = 0.

In what follows, we write 퐻푛 = 퐻푛(퐶•) = 푍푛/퐵푛 in the usual notation for homological algebra. ⋅ 86 ⋅ Some aspects of Koszul complexes

Proof. Assume on the contrary that there exists 푘 ≤ ℎ with 퐻<푘 = 0 whereas 퐻푘 ≠ 0. 푘 Write 퐽 = 퐽푘. As 퐽 annihilates the nonzero 푅-module 퐻 , the criterion of depth-zero 푘 modules (Proposition 7.5.2) implies that depth퐽(퐻 ) = 0. By assumption depth퐽(퐶푘) > 푘−푘 = 0, it follows that depth퐽(푍푘) > 0 since 푍푘 ⊂ 퐶푘, by applying the aforementioned criterion of depth-zero. From the short exact sequence

0 → 퐵푘 → 푍푘 → 퐻푘 → 0

we deduce distinguished triangles in 퐷(푅-Mod)

+1 ℋom(푅/퐽, 퐵푘) → ℋom(푅/퐽, 푍푘) → ℋom(푅/퐽, 퐻푘) , +1 ℋom(푅/퐽, 퐻푘)[−1] → ℋom(푅/퐽, 퐵푘) → ℋom(푅/퐽, 푍푘) .

As the leftmost and rightmost terms of the last line are in 퐷≥1(푅-Mod), we see

푘 depth퐽(퐵 ) ≥ 1;

0 푘 0 푘 1 푘 moreover, the piece Ext푅(푅/퐽, 푍 ) → Ext푅(푅/퐽, 퐻 ) → Ext푅(푅/퐽, 퐵 ) from the long ex- 푘 act sequence shows that depth퐽(퐵 ) = 1. Now for 푛 < 푘 we have short exact sequences

0 → 퐵⎵푛 → 퐶푛 → 퐵푛+1 → 0. =푍푛 Again, one infers from the distinguished triangle

+1 ℋom(푅/퐽, 퐵푛+1)[−1] → ℋom(푅/퐽, 퐵푛) → ℋom(푅/퐽, 퐶푛) ,

푛 the assumption depth퐽(퐶 ) > 푘 − 푛 and descending induction on 푛 that 푛 < 푘 ⟹ depth퐽(퐵푛) = 푘 − 푛 + 1. This is impossible since 퐵≪0 = 0 has infinite depth. • Corollary 8.2.6. Let 퐼 ⊂ 푅 be an ideal, 퐶 be a cochain complex with 푛 ≪ 0 ⟹ 퐶푛 = 0, 푛 • 푛 and ℎ ∈ ℤ. Suppose that 푛 ≤ ℎ implies 퐼 ⋅ 퐻 (퐶 ) = 0 and depth퐼(퐶 ) > ℎ − 푛. Then 퐻≤ℎ(퐶•) = 0.

푘 • Proof. For 푘 ≤ ℎ we have 퐽푘 ∶= ann(퐻 (퐶 )) ⊃ 퐼. Hence Lemma 8.2.4 entails that 푛 ≤ 푘 ≤ ℎ ⟹ depth (퐶푛) ≥ depth (퐶푛) > ℎ − 푛 ≥ 푘 − 푛. 퐽푘 퐼 Now apply Proposition 8.2.5.

8.3 Koszul complexes and depth

The simplest Koszul complexes are defined as follows. Given 푥 ∈ 푅, we form the cochain complex in degrees {0, 1}

푥 퐾(푥) ∶= [푅 푅] . §8.3 Koszul complexes and depth ⋅ 87 ⋅

More generally, for any 푅-module 푀, viewed as a complex concentrated in degree zero, and a family x = (푥훼)훼∈풜 of element of 푅, we define the associated Koszul complex as

퐾•(x; 푀) ∶= 퐾•(푢; 푀), where we take 푢 ∶ 푅⊕풜 → 푅 corresponding to x in Definition 8.1.1. Unfolding definitions, we have ⎧ ℎ ⊕풜 ℎ {Hom푅 (⋀ (푅 ), 푀) , ℎ ≥ 0 퐾 (x; 푀) = ⎨ ⎩{0, ℎ < 0. ℎ Therefore 퐾 (x; 푀) consists of families 푚(훼1, … , 훼ℎ) ∈ 푀 that are alternating in the variables 훼1, … , 훼ℎ ∈ 풜. The differential is

휕ℎ ∶ 퐾ℎ(x; 푀) ⟶ 퐾ℎ+1(x; 푀) ⎡ ℎ ⎤ 푚 ⟼ (훼 , … , 훼 ) ↦ ∑(−1)푗푥 ⋅ 푚(… , ̂훼 , …) . ⎢ 0 ℎ 훼푗 푗 ⎥ ⎣ 푗=0 ⎦

We shall abbreviate the cohomologies of 퐾•(x; 푀) as the Koszul cohomologies. When 풜 = {1, … , 푛} we revert to

• 퐾 (푥1, … , 푥푛; 푀) ∶= 푀 ⊗ 퐾(푥1) ⊗ ⋯ ⊗ 퐾(푥푛)

with the well-known sign convention. Also note that 푅⊕풜 is projective, hence the Proposition 8.1.4 can always be applied to Koszul cohomologies. Let 퐼 denote the ideal generated by {푥훼 ∶ 훼 ∈ 풜}. The reader is invited to verify that

• • ⋄ 퐾 (x, ∏훽∈ℬ 푀훽) = ∏훽∈ℬ 퐾 (x; 푀훽) for any family {푀훽}훽∈ℬ of 푅-modules, and same for their cohomologies;

• ⋄ the 0-th cohomology of 퐾 (x; 푀) is Hom푅(푅/퐼, 푀) = {푥 ∈ 푀 ∶ 퐼푥 = 0}, the 퐼-torsion part of 푀;

⋄ if |퐴| = 푛, the 푛-th cohomology of 퐾•(x; 푀) is 푀/퐼푀.

These facts will cast in our later arguments.

Lemma 8.3.1. Each cohomology of 퐾•(x; 푀) is annihilated by 퐼.

Proof. Apply Proposition 8.1.3 by observing that 픮 = 퐼 in our setting.

Theorem 8.3.2. Let 푀 be an 푅-module and x = {푥훼}훼∈풜 be a family of elements of 푅, which generate an ideal 퐼 of 퐴. Then depth퐼(푀) equals

푛 • inf {푛 ≥ 0 ∶ 퐻 (퐾 (x; 푀)) ≠ 0} ∈ ℤ≥0 ⊔ {∞}.

Consequently, if 퐼 ⊂ 푅 is an ideal generated by elements 푥1, … , 푥푛 and 퐼푀 ≠ 푀, then we have

depth퐼(푀) ≤ 푛. ⋅ 88 ⋅ Some aspects of Koszul complexes

ℎ Proof. Let 푑 ∶= depth퐼(푀). Since 퐾 (x; 푀) is a direct product of copies of 푀 (possibly the empty product = 0), by Proposition 8.2.1 its depth equals either 푑 or ∞. Combining Lemma 8.3.1 and Corollary 8.2.6 (with ℎ = 푑 − 1), we see that 퐻<푑(퐾•(x; 푀)) = 0. It remains to show 퐻푑(퐾•(x; 푀)) ≠ 0 provided that 푑 < ∞, which we assume from now onwards. The case 푑 = 0 is clear since there exists 픭 ∈ Ass(푀) ∩ 푉(퐼), therefore 푅/픭 ↪ 푀 0 • and ∃푥 ∈ 푀 with 픭푥 ⊃ 퐼푥 = {0}, whence 퐻 (퐾 (x; 푀)) ≠ 0. Now suppose 푑 ∈ ℤ≥1 푑 • • and assume 퐻 (퐾 (x; 푀)) = 0. Take a free resolution 퐹• → 푅/퐼 → 0 and put 퐶 ∶= Hom푅(퐹•, 푀), so that 푖 • 푖 퐻 (퐶 ) ≃ Ext푅(푅/퐼, 푀), 푖 ≥ 0. Hence for 푖 < 푑 we have short exact sequences

0 → 퐵푖 → 퐶푖 → 퐵푖+1 → 0

as usual; recall 퐵푖, 푍푖 ⊂ 퐶푖. Note that each 퐶푖 is a direct product of copies of 푀, hence 퐻≤푑(퐾•(x; 퐶푖)) = 0. Indeed, 퐻<푑(퐾•(x; 푀)) = 0 has been settled in the first step, whilst 퐻푑(퐾•(x; 푀)) is the hypothesis to be refuted. It then follows from the long exact sequence for Koszul cohomologies (Proposition 8.1.4) that

(푠 ≤ 푑) ∧ (푖 < 푑) ⟹ 퐻푠(퐾•(x; 퐵푖+1)) ↪ 퐻푠+1(퐾•(x; 퐵푖)).

Suppose 푖 < 푑. As 퐵0 = 0, an iteration yields

퐻푑−푖(퐾•(x; 퐵푖+1)) ↪ ⋯ ↪ 퐻푑+1(퐾•(x; 퐵0)) = 0.

In particular, 푖 = 푑 − 1 gives rise to 퐻1(퐾•(x; 퐵푑)) = 0. Hence the short exact sequence 0 → 퐵푑 → 푍푑 → 퐻푑 → 0 (for 퐶•) together with Proposition 8.1.4 give rise to

퐻0(퐾•(x; 푍푑)) ↠ 퐻0(퐾•(x; 퐻푑)).

푑 푑 • 푑 As 퐻 ∶= 퐻 (퐶 ) ≃ Ext푅(푅/퐼, 푀) is nonzero and annihilated by 퐼, the right-hand side is nonzero. Hence 퐻0(퐾•(x; 푍푑)) ≠ 0 and then 퐻0(퐾•(x; 퐶푑)) ≠ 0, as the zeroth Koszul cohomology is nothing but the 퐼-torsion part. Finally, recall that 퐶푑 ≠ {0} is a direct product of copies of 푀, consequently

퐻0(퐾•(x; 푀)) ≠ 0.

However, this contradicts the earlier result that 퐻<푑(퐾•(x; 푀)) = 0 since 푑 ≥ 1. This yields an alternative characterization of depth. It also completes the proof of Theorem 7.5.5 as promised in the previous lecture. §8.3 Koszul complexes and depth ⋅ 89 ⋅

《晚⻛》, 李桦, 木刻版画, 年. ⋅ 90 ⋅ Some aspects of Koszul complexes Bibliography

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A going-up, 39 ann푅(푀), ann푅(푥), 15 gr퐹(푀), 46 Artinian, 11 graded, 45 Ass(푀), 16 associated prime, 16 H Hauptidealsatz, 63 B height, 58 blow-up algebra, 48 Hilbert’s basis theorem, 11 C Hilbert–Samuel polynomial, 58 catenary, 71 homogeneous, 45 Cayley–Hamilton theorem, 10 Chevalley’s theorem, 44 I Cohen–Macaulay module, 80 integral closure, 26 completion, 52 integrality, 25 constructible subset, 43 invariant theory, 13 coprimary module, 18 J D Jacobson radical, 10 depth, 78, 87 Jacobson ring, 28 dimension, 58 dimension formula, 70 K discrete valuation ring, 73 퐾•(푢; 푀), 84 퐾•(x; 푀), 87 E Koszul complex, 83, 87 embedded prime, 17 Krull’s intersection theorem, 50 exterios algebra, 83 F L faithful functor, 36 length, 11, 57 faithfully flat, 31, 36 local ring, 3 field of fractions, 6 localization, 5 filtration, 46 M 픞-adic, 47 푀[푆−1], 5 finitely presented, 7 MaxSpec(푅), 4 flat, 31 minimal prime ideal, 39 G Moishezon manifolds, 71 going-down, 39 multiplicative subset, 5 ⋅ 94 ⋅ INDEX

N Nakayama’s lemma, 10 nilpotent radical, 9 Noether normalization, 69 Noetherian, 11 normal, 26 Nullstellensatz, 13, 29

P parameter ideal, 59 primary decomposition, 19 primary module, 18 prime avoidance, 4

R reduced ring, 9, 74 Rees algebra, 49 regular local ring, 63 regular sequence, 79

S SageMath, 23 Serre’s criterion, 77 Spec(푅), 4 Supp(푀), 15 support, 15

T topological ring, 51 퐼-adic, 51 total fraction ring, 6

U uniformizer, 73

V 푉(픞), 4 variety, 12

Z Zariski topology, 4 zero divisor, 16 Zorn’s Lemma, 3, 20, 40