a collaborative, open source textbook on commutative algebra

7/31/2019 A Collaborative, Open Source Textbook on Commutative Algebra

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The CRing Project

A collaborative, open source textbook on commutative algebra.

http://people.fas.harvard.edu/~amathew/cr.html

The following people have contributed to this work.

Shishir AgrawalEva BelmontZev ChonolesRankeya DattaAnton GeraschenkoSherry GongFrancois GreerDarij GrinbergAise Johan de JongAdeel Ahmad KhanHolden LeeGeoffrey LeeAkhil MathewRyan ReichWilliam WrightMoor Xu
http://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.html


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Introduction

This is a massively collaborative, open source textbook on commutative algebra. Theproject is currently in its infancy, and needs contributions!

The latest version of this document, along with its LATEX source code, is available athttp://people.fas.harvard.edu/~amathew/cr.html.

License

Copyright (C) 2010 CRing Project. Permission is granted to copy, distribute and/ormodify this document under the terms of the GNU Free Documentation License, Version1.2 or any later version published by the Free Software Foundation; with no InvariantSections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is includedin the section entitled GNU Free Documentation License.

Prerequisites

This book is intended to be accessible to undergraduates. The prerequisite is a basicacquaintance with modern algebra. While even the notion of a ring is introduced fromscratch, it is done so rather rapidly, and the reader is advised to consult another source.In addition, we do not hesitate to use the language of categories, which is developed inChapter 0. The book is intended to provide preparation to study textbooks on algebraicgeometry such as [Har77].

The project

This project was started by several undergraduates in an attempt to create a collaborative,open source textbook on commutative algebra. It started with a collection of class notes

live-TEXed by Akhil Mathew taken in a course taught by Jacob Lurie at Harvard in thefall of 2010. The idea for the present project came from the Stacks project [ dJea10].

The main website for this project is http://people.fas.harvard.edu/~amathew/cr.html. In addition, there is a git repository at http://cring.adeel.ru. The git repositoryprovides slightly newer versions of the source code and contains a log of revisions.

Discussion of the CRing project can happen at the blog, http://cringproject.wordpress.com.

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http://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://cring.adeel.ru/http://cringproject.wordpress.com/http://cringproject.wordpress.com/http://cringproject.wordpress.com/http://cringproject.wordpress.com/http://cringproject.wordpress.com/http://cring.adeel.ru/http://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.htmlhttp://people.fas.harvard.edu/~amathew/cr.html


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Corrections

Please email corrections to [email protected].

Contributions

The following people have contributed to this work.

Shishir AgrawalEva BelmontZev ChonolesRankeya DattaAnton GeraschenkoSherry GongFrancois Greer

Darij GrinbergAise Johan de JongAdeel Ahmad KhanHolden LeeGeoffrey LeeAkhil MathewRyan ReichWilliam WrightMoor Xu

A list of contributions submitted via email is be maintained at http://people.fas.

harvard.edu/~amathew/contrib.html. They will also be accessible in the git logs.

How to contribute

To contribute, email submissions to [email protected]. Contributions do nothave to be polished; they can be rough sketches written for any purpose at allhalf-finished homework writeups, term papers, blog posts, and others are all welcome.

Contributions in editing the chapters are also welcome. To do this, simply downloadthe source, edit the files, and email the modifications to the same address.

Acknowledgments

We thank Aise Johan de Jong for helpful advice and a blog link.

Version

This file was last updated October 8, 2011.
http://people.fas.harvard.edu/~amathew/contrib.htmlhttp://people.fas.harvard.edu/~amathew/contrib.htmlhttp://people.fas.harvard.edu/~amathew/contrib.htmlhttp://people.fas.harvard.edu/~amathew/contrib.htmlhttp://people.fas.harvard.edu/~amathew/contrib.htmlhttp://people.fas.harvard.edu/~amathew/contrib.html


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Contents

I Fundamentals 1

0 Categories 3

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The language of commutative diagrams . . . . . . . . . . . . . . . . 6

1.3 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Covariant functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Contravariant functors . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Functors and isomorphisms . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Equivalences of categories . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Various universal constructions . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Initial and terminal objects . . . . . . . . . . . . . . . . . . . . . . . 173.3 Push-outs and pull-backs . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Filtered colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.7 The initial object theorem . . . . . . . . . . . . . . . . . . . . . . . . 27

3.8 Completeness and cocompleteness . . . . . . . . . . . . . . . . . . . 28

3.9 Continuous and cocontinuous functors . . . . . . . . . . . . . . . . . 29

3.10 Monomorphisms and epimorphisms . . . . . . . . . . . . . . . . . . . 29

4 Yonedas lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.1 The functors hX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.2 The Yoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Representable functors . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4 Limits as representable functors . . . . . . . . . . . . . . . . . . . . . 32

4.5 Criteria for representability . . . . . . . . . . . . . . . . . . . . . . . 32

5 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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5.3 Adjoints and (co)limits . . . . . . . . . . . . . . . . . . . . . . . . . 36

1 Foundations 371 Commutative rings and their ideals . . . . . . . . . . . . . . . . . . . . . . . 37

1.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.2 The category of rings . . . . . . . . . . . . . . . . . . . . . . . . . . 381.3 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.4 Operations on ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.5 Quotient rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.6 Zerodivisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.1 Rings of holomorphic functions . . . . . . . . . . . . . . . . . . . . . 442.2 Ideals and varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Modules over a commutative ring . . . . . . . . . . . . . . . . . . . . . . . . 473.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 The categorical structure on modules . . . . . . . . . . . . . . . . . . 483.3 Exactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.4 Split exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.5 The five lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Prime and maximal ideals . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Fields and integral domains . . . . . . . . . . . . . . . . . . . . . . . 564.3 Prime avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 The Chinese remainder theorem . . . . . . . . . . . . . . . . . . . . 58

5 Some special classes of domains . . . . . . . . . . . . . . . . . . . . . . . . . 595.1 Principal ideal domains . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Unique factorization domains . . . . . . . . . . . . . . . . . . . . . . 605.3 Euclidean domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 Basic properties of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.1 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.2 Finitely generated modules . . . . . . . . . . . . . . . . . . . . . . . 656.3 Finitely presented modules . . . . . . . . . . . . . . . . . . . . . . . 666.4 Modules of finite length . . . . . . . . . . . . . . . . . . . . . . . . . 68

2 Fields and Extensions 711 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711.2 The characteristic of a field . . . . . . . . . . . . . . . . . . . . . . . 73

2 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.2 Finite extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.3 Algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.4 Minimal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.5 Algebraic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3 Separability and normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


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3.1 Separable extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.2 Purely inseparable extensions . . . . . . . . . . . . . . . . . . . . . . 83

4 Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.4 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5 Transcendental Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.1 Linearly Disjoint Field Extensions . . . . . . . . . . . . . . . . . . . 91

3 Three important functors 931 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

1.1 Geometric intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . 931.2 Localization at a multiplicative subset . . . . . . . . . . . . . . . . . 931.3 Local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

1.4 Localization is exact . . . . . . . . . . . . . . . . . . . . . . . . . . . 971.5 Nakayamas lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

2 The functor Hom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.1 Left-exactness of Hom . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.2 Projective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032.3 Example: the Serre-Swan theorem . . . . . . . . . . . . . . . . . . . 1052.4 Injective modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052.5 The small object argument . . . . . . . . . . . . . . . . . . . . . . . 1082.6 Split exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3 The tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.1 Bilinear maps and the tensor product . . . . . . . . . . . . . . . . . 113

3.2 Basic properties of the tensor product . . . . . . . . . . . . . . . . . 1153.3 The adjoint property . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.4 The tensor product as base-change . . . . . . . . . . . . . . . . . . . 1173.5 Some concrete examples . . . . . . . . . . . . . . . . . . . . . . . . . 1193.6 Tensor products of algebras . . . . . . . . . . . . . . . . . . . . . . . 121

4 Exactness properties of the tensor product . . . . . . . . . . . . . . . . . . . 1234.1 Right-exactness of the tensor product . . . . . . . . . . . . . . . . . 1234.2 A characterization of right-exact functors . . . . . . . . . . . . . . . 1254.3 Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.4 Finitely presented flat modules . . . . . . . . . . . . . . . . . . . . . 129

II Commutative algebra 131

4 The Spec of a ring 1331 The spectrum of a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

1.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . 1341.2 The radical ideal-closed subset correspondence . . . . . . . . . . . . 1361.3 A meta-observation about prime ideals . . . . . . . . . . . . . . . . . 138


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1.4 Functoriality of Spec . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401.5 A basis for the Zariski topology . . . . . . . . . . . . . . . . . . . . . 141

2 Nilpotent elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1442.1 The radical of a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

2.2 Lifting idempotents . . . . . . . . . . . . . . . . . . . . . . . . . . . 1452.3 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

3 Vista: sheaves on Spec R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.1 Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1493.2 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.3 Sheaves on Spec A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5 Noetherian rings and modules 1571 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

1.1 The noetherian condition . . . . . . . . . . . . . . . . . . . . . . . . 1571.2 Stability properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

1.3 The basis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1601.4 Noetherian induction . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

2 Associated primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1622.1 The support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1632.2 Associated primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1632.3 Localization and Ass(M) . . . . . . . . . . . . . . . . . . . . . . . . 1672.4 Associated primes determine the support . . . . . . . . . . . . . . . 1682.5 Primary modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

3 Primary decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1723.1 Irreducible and coprimary modules . . . . . . . . . . . . . . . . . . . 1723.2 Irreducible and primary decompositions . . . . . . . . . . . . . . . . 173

3.3 Uniqueness questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1744 Artinian rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.2 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1774.3 Vista: zero-dimensional non-noetherian rings . . . . . . . . . . . . . 180

6 Graded and filtered rings 1831 Graded rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1841.2 Homogeneous ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1851.3 Finiteness conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1871.4 Localization of graded rings . . . . . . . . . . . . . . . . . . . . . . . 191

1.5 The Proj of a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1922 Filtered rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1942.2 The associated graded . . . . . . . . . . . . . . . . . . . . . . . . . . 1952.3 Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

3 The Artin-Rees Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1973.1 The Artin-Rees Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 197


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3.2 The Krull intersection theorem . . . . . . . . . . . . . . . . . . . . . 198

7 Integrality and valuation rings 201

1 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

1.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2011.2 Le sorite for integral extensions . . . . . . . . . . . . . . . . . . . . . 205

1.3 Integral closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

1.4 Geometric examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

2 Lying over and going up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

2.1 Lying over . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

2.2 Going up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

3 Valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

3.2 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

3.3 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

3.4 Back to the goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

4 The Hilbert Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

4.1 Statement and initial proof of the Nullstellensatz . . . . . . . . . . . 219

4.2 The normalization lemma . . . . . . . . . . . . . . . . . . . . . . . . 220

4.3 Back to the Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . 222

4.4 A little affine algebraic geometry . . . . . . . . . . . . . . . . . . . . 224

5 Serres criterion and its variants . . . . . . . . . . . . . . . . . . . . . . . . . 225

5.1 Reducedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

5.2 The image ofM S1M . . . . . . . . . . . . . . . . . . . . . . . . 2295.3 Serres criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

8 Unique factorization and the class group 233

1 Unique factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

1.2 Gausss lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

1.3 Factoriality and height one primes . . . . . . . . . . . . . . . . . . . 237

1.4 Factoriality and normality . . . . . . . . . . . . . . . . . . . . . . . . 238

2 Weil divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

2.2 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

2.3 Nagatas lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

3 Locally factorial domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2393.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

3.2 The Picard group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

3.3 Cartier divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

3.4 Weil divisors and Cartier divisors . . . . . . . . . . . . . . . . . . . . 243

3.5 Recap and a loose end . . . . . . . . . . . . . . . . . . . . . . . . . . 245

3.6 Further remarks on Weil(R) and Cart(R) . . . . . . . . . . . . . . . 246


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9 Dedekind domains 2491 Discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2491.2 Another approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

2 Dedekind rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2522.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2522.2 A more elementary approach . . . . . . . . . . . . . . . . . . . . . . 2542.3 Modules over Dedekind domains . . . . . . . . . . . . . . . . . . . . 255

3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2573.1 Integral closure in a finite separable extension . . . . . . . . . . . . . 2583.2 The Krull-Akizuki theorem . . . . . . . . . . . . . . . . . . . . . . . 2603.3 Extensions of discrete valuations . . . . . . . . . . . . . . . . . . . . 262

4 Action of the Galois group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2624.1 The orbits of the Galois group . . . . . . . . . . . . . . . . . . . . . 2634.2 The decomposition and inertia groups . . . . . . . . . . . . . . . . . 263

10 Dimension theory 2651 The Hilbert function and the dimension of a local ring . . . . . . . . . . . . 265

1.1 Integer-valued polynomials . . . . . . . . . . . . . . . . . . . . . . . 2651.2 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . 2661.3 The Hilbert function is a polynomial . . . . . . . . . . . . . . . . . . 2681.4 The dimension of a module . . . . . . . . . . . . . . . . . . . . . . . 2711.5 Dimension depends only on the support . . . . . . . . . . . . . . . . 273

2 Other definitions and characterizations of dimension . . . . . . . . . . . . . 2742.1 The topological characterization of dimension . . . . . . . . . . . . . 2742.2 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

2.3 Krull dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2772.4 Yet another definition . . . . . . . . . . . . . . . . . . . . . . . . . . 2782.5 Krulls Hauptidealsatz . . . . . . . . . . . . . . . . . . . . . . . . . . 2802.6 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

3 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2823.1 Change of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2823.2 The dimension of a polynomial ring . . . . . . . . . . . . . . . . . . 2843.3 A refined fiber dimension theorem . . . . . . . . . . . . . . . . . . . 2853.4 An infinite-dimensional noetherian ring . . . . . . . . . . . . . . . . 2863.5 Catenary rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2883.6 Dimension theory for topological spaces . . . . . . . . . . . . . . . . 2903.7 The dimension of a tensor product of fields . . . . . . . . . . . . . . 291

11 Completions 2931 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2931.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2941.3 Classical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2951.4 Noetherianness and completions . . . . . . . . . . . . . . . . . . . . 295


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2 Exactness properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2972.1 Generalities on inverse limits . . . . . . . . . . . . . . . . . . . . . . 2982.2 Completions and flatness . . . . . . . . . . . . . . . . . . . . . . . . 300

3 Hensels lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

3.1 The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3023.2 The classification of complete DVRs (characteristic zero) . . . . . . 3 0 4

4 Henselian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3054.1 Semilocal rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3054.2 Henselian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3064.3 Hensels lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3084.4 Example: Puiseuxs theorem . . . . . . . . . . . . . . . . . . . . . . 310

12 Regularity, differentials, and smoothness 313

1 Regular local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3131.1 Regular local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

1.2 Quotients of regular local rings . . . . . . . . . . . . . . . . . . . . . 3161.3 Regularity and smoothness . . . . . . . . . . . . . . . . . . . . . . . 3181.4 Regular local rings look alike . . . . . . . . . . . . . . . . . . . . . . 319

2 Kahler differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.1 Derivations and Kahler differentials . . . . . . . . . . . . . . . . . . 3212.2 Relative differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 3222.3 The case of a polynomial ring . . . . . . . . . . . . . . . . . . . . . . 3232.4 Exact sequences of Kahler differentials . . . . . . . . . . . . . . . . . 3242.5 Kahler differentials and base change . . . . . . . . . . . . . . . . . . 3272.6 Differentials and localization . . . . . . . . . . . . . . . . . . . . . . 3282.7 Another construction of B/A . . . . . . . . . . . . . . . . . . . . . . 329

3 Introduction to smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . 3313.1 Kahler differentials for fields . . . . . . . . . . . . . . . . . . . . . . . 3313.2 Regularity, smoothness, and Kahler differentials . . . . . . . . . . . . 333

III Topics 337

13 Various topics 3391 Linear algebra over rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

1.1 The determinant trick . . . . . . . . . . . . . . . . . . . . . . . . . . 3391.2 Determinantal ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 3401.3 Lecture 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

2 Finite presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3422.1 Compact objects in a category . . . . . . . . . . . . . . . . . . . . . 3422.2 Finitely presented modules . . . . . . . . . . . . . . . . . . . . . . . 3432.3 Finitely presented algebras . . . . . . . . . . . . . . . . . . . . . . . 345

3 Inductive limits of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3473.1 Prologue: fixed points of polynomial involutions over C . . . . . . . 3473.2 The inductive limit of categories . . . . . . . . . . . . . . . . . . . . 349


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3.3 The category of finitely presented modules . . . . . . . . . . . . . . . 349

3.4 The category of finitely presented algebras . . . . . . . . . . . . . . . 351

3.5 Spec and inductive limits . . . . . . . . . . . . . . . . . . . . . . . . 351

14 Homological Algebra 353

1 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

1.1 Chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

1.2 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

1.3 Long exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 355

1.4 Cochain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

1.5 Long exact sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

1.6 Chain Homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

1.7 Topological remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

2 Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

2.1 Projective resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 3592.2 Injective resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

2.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

2.4 Ext functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

2.5 Application: Modules over DVRs . . . . . . . . . . . . . . . . . . . . 366

15 Flatness revisited 369

1 Faithful flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

1.1 Faithfully flat modules . . . . . . . . . . . . . . . . . . . . . . . . . . 369

1.2 Faithfully flat algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 372

1.3 Descent of properties under faithfully flat base change . . . . . . . . 3731.4 Topological consequences . . . . . . . . . . . . . . . . . . . . . . . . 373

2 Faithfully flat descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

2.1 The Amitsur complex . . . . . . . . . . . . . . . . . . . . . . . . . . 375

2.2 Descent for modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

2.3 Example: Galois descent . . . . . . . . . . . . . . . . . . . . . . . . . 380

3 The Tor functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

3.2 Tor and flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

4 Flatness over noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . 382

4.1 Flatness over a noetherian local ring . . . . . . . . . . . . . . . . . . 383

4.2 The infinitesimal criterion for flatness . . . . . . . . . . . . . . . . . 384

4.3 Generalizations of the local and infinitesimal criteria . . . . . . . . . 3 8 5

4.4 The final statement of the flatness criterion . . . . . . . . . . . . . . 387

4.5 Flatness over regular local rings . . . . . . . . . . . . . . . . . . . . . 389

4.6 Example: construction of flat extensions . . . . . . . . . . . . . . . . 390

4.7 Generic flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393


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16 Homological theory of local rings 395

1 Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

1.1 Depth over local rings . . . . . . . . . . . . . . . . . . . . . . . . . . 395

1.2 Regular sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

1.3 Powers of regular sequences . . . . . . . . . . . . . . . . . . . . . . . 401

1.4 Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

1.5 Depth and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 405

2 Cohen-Macaulayness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

2.1 Cohen-Macaualay modules over a local ring . . . . . . . . . . . . . . 406

2.2 The non-local case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409

2.3 Reformulation of Serres criterion . . . . . . . . . . . . . . . . . . . . 410

3 Projective dimension and free resolutions . . . . . . . . . . . . . . . . . . . 411

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

3.2 Tor and projective dimension . . . . . . . . . . . . . . . . . . . . . . 414

3.3 Minimal projective resolutions . . . . . . . . . . . . . . . . . . . . . 415

3.4 The Auslander-Buchsbaum formula . . . . . . . . . . . . . . . . . . 417

4 Serres criterion and its consequences . . . . . . . . . . . . . . . . . . . . . . 421

4.1 First consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

4.2 Regular local rings are factorial . . . . . . . . . . . . . . . . . . . . . 422

17 Etale, unramified, and smooth morphisms 425

1 Unramified morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

1.2 Unramified extensions of a field . . . . . . . . . . . . . . . . . . . . . 429

1.3 Conormal modules and universal thickenings . . . . . . . . . . . . . 4302 Smooth morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

2.2 Quotients of formally smooth rings . . . . . . . . . . . . . . . . . . . 435

2.3 The Jacobian criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 439

2.4 The fiberwise criterion for smoothness . . . . . . . . . . . . . . . . . 440

2.5 Formal smoothness and regularity . . . . . . . . . . . . . . . . . . . 444

2.6 A counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

3 Etale morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

3.2 The local structure theory . . . . . . . . . . . . . . . . . . . . . . . . 4483.3 Permanence properties of etale morphisms . . . . . . . . . . . . . . . 452

3.4 Application to smooth morphisms . . . . . . . . . . . . . . . . . . . 456

3.5 Lifting under nilpotent extensions . . . . . . . . . . . . . . . . . . . 457

18 Complete local rings 459

1 The Cohen structure theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 459


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19 Homotopical algebra 4611 Model categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4611.2 The retract argument . . . . . . . . . . . . . . . . . . . . . . . . . . 463

20 GNU Free Documentation License 4691 APPLICABILITY AND DEFINITIONS . . . . . . . . . . . . . . . . . . . . 4692 VERBATIM COPYING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713 COPYING IN QUANTITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714 MODIFICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4725 COMBINING DOCUMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . 4746 COLLECTIONS OF DOCUMENTS . . . . . . . . . . . . . . . . . . . . . . 4747 AGGREGATION WITH INDEPENDENT WORKS . . . . . . . . . . . . . 4748 TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4759 TERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

10 FUTURE REVISIONS OF THIS LICENSE . . . . . . . . . . . . . . . . . . 47511 ADDENDUM: How to use this License for your documents . . . . . . . . . 4 7 5


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Part I

Fundamentals

1


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Chapter 0

Categories

The language of categories is not strictly necessary to understand the basics of commuta-tive algebra. Nonetheless, it is extremely convenient and powerful. It will clarify many ofthe constructions made in the future when we can freely use terms like universal prop-erty or adjoint functor. As a result, we begin this book with a brief introduction to

category theory. We only scratch the surface; the interested reader can pursue furtherstudy in [ML98] or [KS06].

Nonetheless, the reader is advised not to take the present chapter too seriously; skip-ping it for the moment to chapter 1 and returning here as a reference could be quitereasonable.

1 Introduction

1.1 Definitions

Categories are supposed to be places where mathematical objects live. Intuitively, to any

given type of structure (e.g. groups, rings, etc.), there should be a category of objectswith that structure. These are not, of course, the only type of categories, but they will bethe primary ones of concern to us in this book.

The basic idea of a category is that there should be objects, and that one should beable to map between objects. These mappings could be functions, and they often are, butthey dont have to be. Next, one has to be able to compose mappings, and associativityand unit conditions are required. Nothing else is required.

Definition 1.1 A category C consists of:1. A collection of objects, ob C.

2. For each pair of objects X, Y ob C, a set of morphisms HomC(X, Y) (abbreviatedHom(X, Y)).

3. For each object X ob C, there is an identity morphism 1X HomC(X, X) (oftenjust abbreviated to 1).

4. There is a composition law : HomC(X, Y)HomC(Y, Z) HomC(X, Z), (g, f) g f for every triple X,Y,Z of objects.

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The CRing Project, 0.1.

5. The composition law is unital and associative. In other words, if f HomC(X, Y),then 1Y f = f 1X = f. Moreover, if g HomC(Y, Z) and h HomC(Z, W) forobjects Z,Y,W, then

h (g f) = (h g) f HomC(X, W).We shall write f : X Y to denote an element of HomC(X, Y). In practice, C will

often be the storehouse for mathematical objects: groups, Lie algebras, rings, etc., inwhich case these morphisms will just be ordinary functions.

Here is a simple list of examples.

Example 1.2 (Categories of structured sets) 1. C = Sets; the objects are sets,and the morphisms are functions.

2. C = Grps; the objects are groups, and the morphisms are maps of groups (i.e.homomorphisms).

3. C = LieAlg; the objects are Lie algebras, and the morphisms are maps of Liealgebras (i.e. homomorphisms).1

4. C = Vectk; the objects are vector spaces over a field k, and the morphisms are linearmaps.

5. C = Top; the objects are topological spaces, and the morphisms are continuousmaps.

6. This example is slightly more subtle. Here the category C has objects consisting oftopological spaces, but the morphisms between two topological spaces X, Y are thehomotopy classes of maps X

Y. Since composition respects homotopy classes,

this is well-defined.

In general, the objects of a category do not have to form a set; they can be too largefor that. For instance, the collection of objects in Sets does not form a set.

Definition 1.3 A category is small if the collection of objects is a set.

The standard examples of categories are the ones above: structured sets together withstructure-preserving maps. Nonetheless, one can easily give other examples that are notof this form.

Example 1.4 (Groups as categories) Let G be a finite group. Then we can make a

category BG where the objects just consist of one element and the maps are theelements g G. The identity is the identity of G and composition is multiplication in thegroup.

In this case, the category does not represent much of a class of objects, but insteadwe think of the composition law as the key thing. So a group is a special kind of (small)category.

1Feel free to omit if the notion of Lie algebra is unfamiliar.

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Example 1.5 (Monoids as categories) A monoid is precisely a category with one ob-ject. Recall that a monoid is a set together with an associative and unital multiplication(but which need not have inverses).

Example 1.6 (Posets as categories) Let (P, ) be a partially ordered (or even pre-ordered) set (i.e. poset). Then P can be regarded as a (small) category, where the objectsare the elements p P, and

HomP(p, q) =

if p q otherwise

There is, however, a major difference between category theory and set theory. Thereis nothing in the language of categories that lets one look inside an object. We thinkof vector spaces having elements, spaces having points, etc. By contrast, categories treatthese kinds of things as invisible. There is nothing inside of an object X

C; the only

way to understand X is to understand the ways one can map into and out of X. Evenif one is working with a category of structured sets, the underlying set of an object inthis category is not part of the categorical data. However, there are instances in whichthe underlying set can be recovered as a Hom-set.

Example 1.7 In the category Top of topological spaces, one can in fact recover theunderlying set of a topological space via the hom-sets. Namely, for each topologicalspace, the points ofX are the same thing as the mappings from a one-point space into X.That is, we have

|X| = HomTop(, X),where

is the one-point space.

Later we will say that the functor assigning to each space its underlying set is corep-resentable.

Example 1.8 Let Ab be the category of abelian groups and group-homomorphisms.Again, the claim is that using only this category, one can recover the underlying set of agiven abelian group A. This is because the elements of A can be canonically identifiedwith morphisms Z A (based on where 1 Z maps).

Definition 1.9 We say that C is a subcategory of the category D if the collection ofobjects ofC is a subclass of the collection of objects of D, and if whenever X, Y C, wehave

HomC(X, Y) HomD(X, Y)with the laws of composition in C induced by that in D.

C is called a full subcategory if HomC(X, Y) = HomD(X, Y) whenever X, Y C.

Example 1.10 The category of abelian groups is a full subcategory of the category ofgroups.

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1.2 The language of commutative diagrams

While the language of categories is, of course, purely algebraic, it will be convenientfor psychological reasons to visualize categorical arguments through diagrams. We shall

introduce this notation here.Let C be a category, and let X, Y be objects in C. Iff Hom(X, Y), we shall sometimes

write f as an arrow

f : X Yor

Xf Y

as if f were an actual function. If Xf Y and Y g Z are morphisms, composition

g f : X Z can be visualized by the picture

X

f

Yg

Z.Finally, when we work with several objects, we shall often draw collections of mor-

phisms into diagrams, where arrows indicate morphisms between two objects.

Definition 1.11 A diagram will be said to commute if whenever one goes from oneobject in the diagram to another by following the arrows in the right order, one obtainsthe same morphism. For instance, the commutativity of the diagram

X

f

f GG W

g

Y

g GG Z

is equivalent to the assertion that

g f = g f Hom(X, Z).

As an example, the assertion that the associative law holds in a category C can bestated as follows. For every quadruple X,Y,Z,W C , the following diagram (of sets)commutes:

Hom(X, Y)

Hom(Y, Z)

Hom(Z, W) GG

Hom(X, Z)

Hom(Z, W)

Hom(X, Y) Hom(Y, W) GG Hom(X, W).

Here the maps are all given by the composition laws in C. For instance, the downwardmap to the left is the product of the identity on Hom( X, Y) with the composition lawHom(Y, Z) Hom(Z, W) Hom(Y, W).

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1.3 Isomorphisms

Classically, one can define an isomorphism of groups as a bijection that preserves thegroup structure. This does not generalize well to categories, as we do not have a notion

of bijection, as there is no way (in general) to talk about the underlying set of anobject. Moreover, this definition does not generalize well to topological spaces: there, anisomorphism should not just be a bijection, but something which preserves the topology(in a strong sense), i.e. a homeomorphism.

Thus we make:

Definition 1.12 An isomorphism between objects X, Y in a category C is a map f :X Y such that there exists g : Y X with

g f = 1X , f g = 1Y.Such a g is called an inverse to f.

Remark It is easy to check that the inverse g is unique. Indeed, suppose g, g both wereinverses to f. Then

g = g 1Y = g (f g) = (g f) g = 1X g = g.

This notion is isomorphism is more correct than the idea of being one-to-one and onto.A bijection of topological spaces is not necessarily a homeomorphism.

Example 1.13 It is easy to check that an isomorphism in the category Grp is an iso-morphism of groups, that an isomorphism in the category Set is a bijection, and so on.

We are supposed to be able to identify isomorphic objects. In the categorical sense,

this means mapping into X should be the same as mapping into Y, ifX, Y are isomorphic,via an isomorphism f : X Y. Indeed, let Z be another object ofC. Then we can definea map

HomC(Z, X) HomC(Z, Y)given by post-composition with f. This is a bijection if f is an isomorphism (the inverseis given by postcomposition with the inverse to f). Similarly, one can easily see thatmapping out of X is essentially the same as mapping out of Y. Anything in generalcategory theory that is true for X should be true for Y (as general category theory canonly try to understand X in terms of maps into or out of it!).

Exercise 0.1 The relation X, Y are isomorphic is an equivalence relation on the class

of objects of a category C.Exercise 0.2 Let P be a preordered set, and make P into a category as in Example 1.6.Then P is a poset if and only if two isomorphic objects are equal.

For the next exercise, we need:

Definition 1.14 A groupoid is a category where every morphism is an isomorphism.

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Exercise 0.3 The sets HomC(A, A) are groups ifC is a groupoid and A C. A group isessentially the same as a groupoid with one object.

Exercise 0.4 Show that the following is a groupoid. Let X be a topological space, and let

1(X) be the category defined as follows: the objects are elements of X, and morphismsx y (for x, y X) are homotopy classes of maps [0, 1] X (i.e. paths) that send0 x, 1 y. Composition of maps is given by concatenation of paths. (Check that,because one is working with homotopy classes of paths, composition is associative.)

1(X) is called the fundamental groupoid of X. Note that Hom1(X)(x, x) is thefundamental group 1(X, x).

2 Functors

A functor is a way of mapping from one category to another: each object is sent to anotherobject, and each morphism is sent to another morphism. We shall study many functorsin the sequel: localization, the tensor product, Hom, and fancier ones like Tor, Ext, andlocal cohomology functors. The main benefit of a functor is that it doesnt simply sendobjects to other objects, but also morphisms to morphisms: this allows one to get newcommutative diagrams from old ones. This will turn out to be a powerful tool.

2.1 Covariant functors

Let C, D be categories. IfC, D are categories of structured sets (of possibly different types),there may be a way to associate objects in D to objects in C. For instance, to every groupG we can associate its group ring Z[G] (which we do not define here); to each topologicalspace we can associate its singular chain complex, and so on. In many cases, given a map

between objects in C preserving the relevant structure, there will be an induced map onthe corresponding objects in D. It is from here that we define a functor.Definition 2.1 A functor F : C D consists of a function F : C D (that is, a rulethat assigns to each object in C an object of D) and, for each pair X, Y C, a mapF : HomC(X, Y) HomD(F X , F Y ), which preserves the identity maps and composition.

In detail, the last two conditions state the following.

1. IfX C, then F(1X) is the identity morphism 1F(X) : F(X) F(X).

2. If Af B g C are morphisms in C, then F(g f) = F(g) F(f) as morphisms

F(A)

F(C). Alternatively, we can say that F preserves commutative diagrams.

In the last statement of the definition, note that if

Xh

22eee

eeeef GG Y

g

Z

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is a commutative diagram in C, then the diagram obtained by applying the functor F,namely

F(X)

F(h)

55rrr

rrrrrr

F(f)GG F(Y)

F(g)

F(Z)

also commutes. It follows that applying F to more complicated commutative diagramsalso yields new commutative diagrams.

Let us give a few examples of functors.

Example 2.2 There is a functor from Sets AbelianGrp sending a set S to the freeabelian group on the set. (For the definition of a free abelian group, or more generally afree R-module over a ring R, see Definition 6.1.)

Example 2.3 Let X be a topological space. Then to it we can associate the set 0(X)of connected components of X.Recall that the continuous image of a connected set is connected, so if f : X Y is

a continuous map and X X connected, f(X) is contained in a connected componentofY. It follows that 0 is a functor Top Sets. In fact, it is a functor on the homotopycategory as well, because homotopic maps induce the same maps on 0.

Example 2.4 Fix n. There is a functor from Top AbGrp (categories of topologicalspaces and abelian groups) sending a space X to its nth homology group Hn(X). We knowthat given a map of spaces f : X Y, we get a map of abelian groups f : Hn(X) Hn(Y). See [Hat02], for instance.

We shall often need to compose functors. For instance, we will want to see, for instance,that the tensor product (to be defined later, see Section 3) is associative, which is reallya statement about composing functors. The following (mostly self-explanatory) definitionelucidates this.

Definition 2.5 If C, D, Eare categories, F : C D, G : D E are covariant functors,then one can define a composite functor

F G : C E

This sends an object X C to G(F(X)). Similarly, a morphism f : X Y is sent toG(F(f)) : G(F(X))

G(F(Y)). We leave the reader to check that this is well-defined.

Example 2.6 In fact, because we can compose functors, there is a category of categories.Let Cat have objects as the small categories, and morphisms as functors. Composition isdefined as in Definition 2.5.

Example 2.7 (Group actions) Fix a group G. Let us understand what a functor BG Sets is. Here BG is the category of Example 1.4.

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The unique object of BG goes to some set X. For each element g G, we get amap g : and thus a map X X. This is supposed to preserve the composition law(which in G is just multiplication), as well as identities.

In particular, we get maps ig : X

X corresponding to each g

G, such that the

following diagram commutes for each g1, g2 G:

Xig1 GG

ig1g2 22fff

ffff

f X

ig2

X.

Moreover, if e G is the identity, then ie = 1X . So a functor BG Sets is just a leftG-action on a set X.

An important example of functors is given by the following. Let C be a category ofstructured sets. Then, there is a functor F : C Sets that sends a structured set tothe underlying set. For instance, there is a functor from groups to sets that forgets thegroup structure. More generally, suppose given two categories C, D, such that C can beregarded as structured objects in D. Then there is a functor C D that forgets thestructure. Such examples are called forgetful functors.

2.2 Contravariant functors

Sometimes what we have described above are called covariant functors. Indeed, we shallalso be interested in similar objects that reverse the arrows, such as duality functors:

Definition 2.8 A contravariant functor C F D (between categories C, D) is similardata as in Definition 2.1 except that now a map X

f

Y now goes to a map F(Y)

F(f)

F(X). Composites are required to be preserved, albeit in the other direction. In otherwords, if X

f Y, Y g Z are morphisms, then we requireF(g f) = F(f) F(g) : F(Z) F(X).

We shall sometimes say just functor for covariant functor. When we are dealingwith a contravariant functor, we will always say the word contravariant.

A contravariant functor also preserves commutative diagrams, except that the arrowshave to be reversed. For instance, if F : C D is contravariant and the diagram

A

GG C

B

bb~~~~~~~

is commutative in C, then the diagramF(A) F(C)oo

{{vvvvvvvvv

F(B)

yy

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commutes in D.One can, of course, compose contravariant functors as in Definition 2.5. But the

composition of two contravariant functors will be covariant. So there is no category ofcategories where the morphisms between categories are contravariant functors.

Similarly as in Example 2.7, we have:

Example 2.9 A contravariant functor from BG (defined as in Example 1.4) to Setscorresponds to a set with a right G-action.

Example 2.10 (Singular cohomology) In algebraic topology, one encounters contravari-ant functors on the homotopy category of topological spaces via the singular cohomologyfunctors X Hn(X;Z). Given a continuous map f : X Y, there is a homomorphismof groups

f : Hn(Y;Z) Hn(X;Z).

Example 2.11 (Duality for vector spaces) On the category Vect of vector spacesover a field k, we have the contravariant functor

V V.

sending a vector space to its dual V = Hom(V, k). Given a map V W of vector spaces,there is an induced map

W V

given by the transpose.

Example 2.12 If we map BG BG sending and g g1, we get a contravariantfunctor.

We now give a useful (linguistic) device for translating between covariance and con-travariance.

Definition 2.13 (The opposite category) Let C be a category. Define the oppositecategory Cop of C to have the same objects as C but such that the morphisms betweenX, Y in Cop are those between Y and X in C.

There is a contravariant functor C Cop. In fact, contravariant functors out ofC arethe same as covariant functors out of Cop.

As a result, when results are often stated for both covariant and contravariant functors,for instance, we can often reduce to the covariant case by using the opposite category.

Exercise 0.5 A map that is an isomorphism in C corresponds to an isomorphism in Cop.

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2.3 Functors and isomorphisms

Now we want to prove a simple and intuitive fact: if isomorphisms allow one to say thatone object in a category is essentially the same as another, functors should be expected

to preserve this.

Proposition 2.14 If f : X Y is a map in C, and F : C D is a functor, thenF(f) : F X F Y is an isomorphism.

The proof is quite straightforward, though there is an important point here. Note thatthe analogous result holds for contravariant functors too.

Proof. If we have maps f : X Y and g : Y X such that the composites both waysare identities, then we can apply the functor F to this, and we find that since

f g = 1Y, g f = 1X ,

it must hold that

F(f) F(g) = 1F(Y), F(g) F(f) = 1F(X).We have used the fact that functors preserve composition and identities. This implies thatF(f) is an isomorphism, with inverse F(g).

Categories have a way of making things so general that are trivial. Hence, this materialis called general abstract nonsense. Moreover, there is another philosophical point aboutcategory theory to be made here: often, it is the definitions, and not the proofs, thatmatter. For instance, what matters here is not the theorem, but the definition of anisomorphism. It is a categorical one, and much more general than the usual notion via

injectivity and surjectivity.

Example 2.15 As a simple example, {0, 1} and [0, 1] are not isomorphic in the homotopycategory of topological spaces (i.e. are not homotopy equivalent) because 0([0, 1]) = while 0({0, 1}) has two elements.

Example 2.16 More generally, the higher homotopy group functors n (see [Hat02]) canbe used to show that the n-sphere Sn is not homotopy equivalent to a point. For thenn(S

n, ) would be trivial, and it is not.

There is room, nevertheless, for something else. Instead of having something that

sends objects to other objects, one could have something that sends an object to a map.

2.4 Natural transformations

Suppose F, G : C D are functors.

Definition 2.17 A natural transformation T : F G consists of the following data.For each X C, there is a morphism T X : F X GX satisfying the following condition.

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Whenever f : X Y is a morphism, the following diagram must commute:

F X

TX

F(f)GG F Y

TY

GXG(f)GG GY

.

If T X is an isomorphism for each X, then we shall say that T is a natural isomor-phism.

It is similarly possible to define the notion of a natural transformation between con-travariant functors.

When we say that things are natural in the future, we will mean that the trans-formation between functors is natural in this sense. We shall use this language to statetheorems conveniently.

Example 2.18 (The double dual) Here is the canonical example of naturality. LetC be the category of finite-dimensional vector spaces over a given field k. Let us furtherrestrict the category such that the only morphisms are the isomorphisms of vector spaces.For each V C, we know that there is an isomorphism

V V = Homk(V, k),

because both have the same dimension.Moreover, the maps V V, V V are both covariant functors on C.2 The first is

the identity functor; for the second, iff : V W is an isomorphism, then there is induceda transpose map ft : W V (defined by sending a map W k to the precompositionV

f

W k), which is an isomorphism; we can take its inverse. So we have two functorsfrom C to itself, the identity and the dual, and we know that V V for each V (thoughwe have not chosen any particular set of isomorphisms).

However, the isomorphism V V cannot be made natural. That is, there is no wayof choosing isomorphisms

TV : V V

such that, whenever f : V W is an isomorphism of vector spaces, the following diagramcommutes:

Vf GG

TV

W

TW

V (ft

)1

GG W.

Indeed, fix d > 1, and choose V = kd. Identify V with kd, and so the map TV is a d-by-dmatrix M with coefficients in k. The requirement is that for each invertible d-by-d matrixN, we have

(Nt)1M = M N,2Note that the dual was defined as a contravariant functor in Example 2.11.

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by considering the above diagram with V = W = kd, and f corresponding to the matrixN. This is impossible unless M = 0, by elementary linear algebra.

Nonetheless, it is possible to choose a natural isomorphism

V V.To do this, given V, recall that V is the collection of maps V k. To give a mapV V is thus the same as giving linear functions lv, v V such that lv : V k islinear in v. We can do this by letting lv be evaluation at v. That is, lv sends a linearfunctional : V k to (v) k. We leave it to the reader to check (easily) that thisdefines a homomorphism V V, and that everything is natural.

Exercise 0.6 Suppose there are two functors BG Sets, i.e. G-sets. What is a naturaltransformation between them?

Natural transformations can be composed. Suppose given functors F,G,H : C D anatural transformation T : F G and a natural transformation U : G H. Then, foreach X C, we have maps T X : F X GX,UX : GX HY. We can compose U withT to get a natural transformation U T : F H.

In fact, we can thus define a category of functors Fun(C, D) (at least ifC, D are small).The objects of this category are the functors F : C D. The morphisms are naturaltransformations between functors. Composition of morphisms is as above.

2.5 Equivalences of categories

Often we want to say that two categories C, D are essentially the same. One way offormulating this precisely is to say that

C,

Dare isomorphic in the category of categories.

Unwinding the definitions, this means that there exist functors

F : C D, G : D C

such that F G = 1D, G F = 1C. This notion, of isomorphism of categories, is generallyfar too restrictive.

For instance, we could consider the category of all finite-dimensional vector spacesover a given field k, and we could consider the full subcategory of vector spaces of theform kn. Clearly both categories encode essentially the same mathematics, in some sense,but they are not isomorphic: one has a countable set of objects, while the other has anuncountable set of objects. Thus, we need a more refined way of saying that two categories

are essentially the same.

Definition 2.19 Two categories C, D are called equivalent if there are functors

F : C D, G : D C

and natural isomorphisms

F G 1D, GF 1C.

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For instance, the category of all vector spaces of the form kn is equivalent to thecategory of all finite-dimensional vector spaces. One functor is the inclusion from vectorspaces of the form kn; the other functor maps a finite-dimensional vector space V to kdimV.Defining the second functor properly is, however, a little more subtle. The next criterion

will be useful.

Definition 2.20 A functor F : C D is fully faithful ifF : HomC(X, Y) HomD(F X , F Y )is a bijection for each pair of objects X, Y C. F is called essentially surjective if everyelement ofD is isomorphic to an object in the image of F.

So, for instance, the inclusion of a full subcategory is fully faithful (by definition). Theforgetful functor from groups to sets is not fully faithful, because not all functions betweengroups are automatically homomorphisms.

Proposition 2.21 A functor F : C D induces an equivalence of categories if and onlyif it is fully faithful and essentially surjective.

Proof. TO BE ADDED: this proof, and the definitions in the statement.

3 Various universal constructions

Now that we have introduced the idea of a category and showed that a functor takesisomorphisms to isomorphisms, we shall take various steps to characterize objects in termsof maps (the most complete of which is the Yoneda lemma, Theorem 4.2). In generalcategory theory, this is generally all we can do, since this is all the data we are given. Weshall describe objects satisfying certain universal properties here.

As motivation, we first discuss the concept of the product in terms of a universal

property.

3.1 Products

Recall that if we have two sets X and Y, the product XY is the set of all elements of theform (x, y) where x X and y Y. The product is also equipped with natural projectionsp1 : X Y X and p2 : X Y Y that take (x, y) to x and y respectively. Thus anyelement of X Y is uniquely determined by where they project to on X and Y. In fact,this is the case more generally; if we have an index set I and a product X =

iIXi, then

an element x X determined uniquely by where where the projections pi(x) land in Xi.To get into the categorical spirit, we should speak not of elements but of maps to X.

Here is the general observation: if we have any other set S with maps fi : S Xi thenthere is a unique map S X =iIXi given by sending s S to the element {fi(s)}iI.This leads to the following characterization of a product using only mapping properties.

Definition 3.1 Let {Xi}iI be a collection of objects in some category C. Then an objectP C with projections pi : P Xi is said to be the product

iIXi if the following

universal property holds: let S be any other object in C with maps fi : S Xi. Thenthere is a unique morphism f : S P such that pif = fi.

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In other words, to map into X is the same as mapping into all the {Xi} at once. Wehave thus given a precise description of how to map into X. Note that, however, theproduct need not exist! If it does, however, we can express the above formalism by thefollowing natural isomorphism of contravariant functors

Hom(,I

Xi) I

Hom(, Xi).

This is precisely the meaning of the last part of the definition. Note that this observationshows that products in the category of sets are really fundamental to the idea of productsin any category.

Example 3.2 One of the benefits of this construction is that an actual category is notspecified; thus when we take C to be Sets, we recover the cartesian product notion of sets,but if we take C to be Grp, we achieve the regular notion of the product of groups (thereader is invited to check these statements).

The categorical product is not unique, but it is as close to being so as possible.

Proposition 3.3 (Uniqueness of products) Any two products of the collection {Xi}in C are isomorphic by a unique isomorphism commuting with the projections.

This is a special case of a general abstract nonsense type result that we shall seemany more of in the sequel. The precise statement is the following: let X be a productof the {Xi} with projections pi : X Xi, and let Y be a product of them too, withprojections qi : Y Xi. Then the claim is that there is a unique isomorphism

f : X

Y

such that the diagrams below commute for each i I:

Xpi

33fff

ffff

ff GG Y

qi

~~||||||||

Xi.

(1)

Proof. This is a trivial result, and is part of a general fact that objects with the sameuniversal property are always canonically isomorphic. Indeed, note that the projectionspi : X

Xi and the fact that mapping into Y is the same as mapping into all the Xi

gives a unique map f : X Y making the diagrams (1) commute. The same reasoning(applied to the qi : Y Xi) gives a map g : Y X making the diagrams

Yqi

22eee

eeee

g GG Xpi

~~}}}}}}}}

Xi

(2)

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commute. By piecing the two diagrams together, it follows that the composite g f makesthe diagram

X

pi22eeeeee

eegf GG X

pi~~}}}}}}}

}

Xi

(3)

commute. But the identity 1X : X X also would make (3) commute, and the uniquenessassertion in the definition of the product shows that g f = 1X . Similarly, f g = 1Y.We are done.

Remark If we reverse the arrows in the above construction, the universal property ob-tained (known as the coproduct) characterizes disjoint unions in the category of setsand free products in the category of groups. That is, to map out of a coproduct of objects{Xi} is the same as mapping out of each of these. We shall later study this constructionmore generally.

Exercise 0.7 Let P be a poset, and make P into a category as in Example 1.6. Fixx, y P. Show that the product of x, y is the greatest lower bound of {x, y} (if it exists).This claim holds more generally for arbitrary subsets of P.

In particular, consider the poset of subsets of a given set S. Then the product inthis category corresponds to the intersection of subsets.

We shall, in this section, investigate this notion of universality more thoroughly.

3.2 Initial and terminal objects

We now introduce another example of universality, which is simpler but more abstractthan the products introduced in the previous section.

Definition 3.4 Let C be a category. An initial object in C is an object X C with theproperty that HomC(X, Y) has one element for all Y C.

So there is a unique map out of X into each Y C. Note that this idea is faithful tothe categorical spirit of describing objects in terms of their mapping properties. Initialobjects are very easy to map out of.

Example 3.5 If C is Sets, then the empty set is an initial object. There is a uniquemap from the empty set into any other set; one has to make no decisions about where

elements are to map when constructing a map X.

Example 3.6 In the category Grp of groups, the group consisting of one element is aninitial object.

Note that the initial object in Grp is not that in Sets. This should not be toosurprising, because cannot be a group.

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Example 3.7 Let P be a poset, and make it into a category as in Example 1.6. Thenit is easy to see that an initial object of P is the smallest object in P (if it exists). Notethat this is equivalently the product of all the objects in P. In general, the initial objectof a category is not the product of all objects in

C(this does not even make sense for a

large category).

There is a dual notion, called a terminal object, where every object can map into it inprecisely one way.

Definition 3.8 A terminal object in a category C is an object Y C such thatHomC(X, Y) = for each X C.

Note that an initial object in C is the same as a terminal object in Cop, and vice versa.As a result, it suffices to prove results about initial objects, and the corresponding resultsfor terminal objects will follow formally. But there is a fundamental difference betweeninitial and terminal objects. Initial objects are characterized by how one maps out ofthem, while terminal objects are characterized by how one maps into them.

Example 3.9 The one point set is a terminal object in Sets.

The important thing about the next theorems is the conceptual framework.

Proposition 3.10 (Uniqueness of the initial (or terminal) object) Any two initial(resp. terminal) objects in C are isomorphic by a unique isomorphism.Proof. The proof is easy. We do it for terminal objects. Say Y, Y are terminal ob-jects. Then Hom(Y, Y) and Hom(Y, Y) are one point sets. So there are unique mapsf : Y Y, g : Y Y, whose composites must be the identities: we know thatHom(Y, Y), Hom(Y, Y) are one-point sets, so the composites have no other choice tobe the identities. This means that the maps f : Y Y, g : Y Y are isomorphisms.

There is a philosophical point to be made here. We have characterized an objectuniquely in terms of mapping properties. We have characterized it uniquely up to uniqueisomorphism, which is really the best one can do in mathematics. Two sets are notgenerally the same, but they may be isomorphic up to unique isomorphism. Theyare different, but the sets are isomorphic up to unique isomorphism. Note also that theargument was essentially similar to that of Proposition 3.3.

In fact, we could interpret Proposition 3.3 as a special case of Proposition 3.10. If Cis a category and {Xi}iI is a family of objects in C, then we can define a category D as

follows. An object ofD is the data of an object Y C and morphisms fi : Y Xi for alli I. A morphism between objects (Y, {fi : Y Xi}) and (Z, {gi : Z Xi}) is a mapY Z making the obvious diagrams commute. Then a product Xi in C is the samething as a terminal object in D, as one easily checks from the definitions.

3.3 Push-outs and pull-backs

Let C be a category.

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Now we are going to talk about more examples of universal constructions, which canall be phrased via initial or terminal objects in some category. This, therefore, is the prooffor the uniqueness up to unique isomorphism ofeverything we will do in this section. Laterwe will present these in more generality.

Suppose we have objects A ,B,C ,X C.Definition 3.11 A commutative square

A

GG B

C GG X

.

is a pushout square (and X is called the push-out) if, given a commutative diagram

A GG

B

C GG Y

there is a unique map X Y making the following diagram commute:A

GG B

$$HHHHHHHHHHHHHH

C GG

99

X

22eee

eeee

e

Y

Sometimes push-outs are also called fibered coproducts. We shall also write X =CA B.

In other words, to map out of X = C A B into some object Y is to give mapsB Y, C Y whose restrictions to A are the same.

The next few examples will rely on notions to be introduced later.

Example 3.12 The following is a pushout square in the category of abelian groups:

Z/2 GG

Z/4

Z/6G

G Z/12In the category of groups, the push-out is actually SL2(Z), though we do not prove it.The point is that the property of a squares being a push-out is actually dependent on thecategory.

In general, to construct a push-out of groups CA B, one constructs the direct sumC B and quotients by the subgroup generated by (a, a) (where a A is identified withits image in C B). We shall discuss this later, more thoroughly, for modules over a ring.

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Example 3.13 Let R be a commutative ring and let S and Q be two commutative R-algebras. In other words, suppose we have two maps of rings s : R S and q : R Q.Then we can fit this information together into a pushout square:

R GG

S

Q GG X

It turns out that the pushout in this case is the tensor product of algebras SR Q (seeSection 3.6 for the construction). This is particularly important in algebraic geometryas the dual construction will give the correct notion of products in the category ofschemes over a field.

Proposition 3.14 LetC be any category. If the push-out of the diagram

A

GG B

C

exists, it is unique up to unique isomorphism.

Proof. We can prove this in two ways. One is to suppose that there were two pushoutsquares:

A

GG B

C GG X

A

GG B

C GG X

Then there are unique maps f : X X, g : X X from the universal property. Indetail, these maps fit into commutative diagrams

A

GG B

$$IIIIIIIIIIIIIII

C GG

99

X

f

22fff

ffff

f

X

A

GG B

$$HHHHHHHHHHHHHHH

C GG

@@

X

g

22fff

ffff

f

X

Then g

f and f

g are the identities of X, X again by uniqueness of the map in the

definition of the push-out.Alternatively, we can phrase push-outs in terms of initial objects. We could consider

the category of all diagrams as above,

A

GG B

C GG D

,

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where A B, A C are fixed and D varies. The morphisms in this category of diagramsconsist of commutative diagrams. Then the initial object in this category is the push-out,as one easily checks.

Often when studying categorical constructions, one can create a kind of dualconstructionby reversing the direction of the arrows. This is exactly the relationship between the push-out construction and the pull-back construction to be described below. So suppose we havetwo morphisms A C and B C, forming a diagram

B

A GG C.

Definition 3.15 The pull-back or fibered product of the above diagram is an objectP with two morphisms P B and P C such that the following diagram commutes:

P

GG B

A GG C

Moreover, the object P is required to be universal in the following sense: given any P andmaps P A and P B making the square commute, there is a unique map P Pmaking the following diagram commute:

P

22eee

eeee

99

##HHHHHHHHHHHHHH

P

GG B

A GG C

We shall also write P = B CA.Example 3.16 In the category Set of sets, if we have sets A ,B,C with maps f : A C, g : B C, then the fibered product A CB consists of pairs (a, b) A B such thatf(a) = g(b).

Example 3.17 (Requires prerequisites not developed yet) The next example maybe omitted without loss of continuity.

As said above, the fact that the tensor product of algebras is a push-out in the categoryof commutative R-algebras allows for the correct notion of the product of schemes. Wenow elaborate on this example: naively one would think that we could pick the underlyingspace of the product scheme to just be the topological product of two Zariski topologies.

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7/31/2019 A Collabo

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