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Oxford Graduate Texts in Mathematics
Series Editors
R. Cohen S. K. Donaldson S. Hildebrandt
T. J. Lyons M. J. Taylor
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OXFORD GRADUATE TEXTS IN MATHEMATICS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Keith Hannabuss: An Introduction to Quantum Theory
Reinhold Meise and Dietmar Vogt: Introduction to Functional Analysis
James G. Oxley: Matroid Theory
N.J. Hitchin, G.B. Segal, and R.S. Ward: Integrable Systems: Twistors,
Loop Groups, and Riemann Surfaces
Wulf Rossmann: Lie groups: An Introduction Through Linear Groups
Q. Liu: Algebraic Geometry and Arithmetic Curves
Martin R. Bridson and Simon M, Salamon (eds): Invitations to Geometry
and Topology
Shmuel Kantorovitz: Introduction to Modern Analysis
Terry Lawson: Topology: A Geometric Approach
Meinolf Geck: An Introduction to Algebraic Geometry and Algebraic
Groups
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Algebraic Geometry and
Arithmetic Curves
Qing Liu
Professor
CNRS Laboratoire de Th´eorie des Nombres et d’Algorithmique Arithm´
etique
Universit´e Bordeaux 1
Translated by
Reinie Ern´
e
Institut de Recherche Math´ematique de Rennes
Universit´e Rennes 1
3
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1
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c Qing Liu, 2002
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First published 2002
First published in paperback 2006
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Typeset by Newgen Imaging Systems (P) Ltd, Chennai, India
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ISBN 0–19–850284–2 978–0–19–850284–5
ISBN 0–19–920249–4 (Pbk.) 978–0–19–920249–2 (Pbk.)
1 3 5 7 9 10 8 6 4 2
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To my mother
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Preface
This book begins with an introduction to algebraic geometry in the language of
schemes. Then, the general theory is illustrated through the study of arithmetic
surfaces and the reduction of algebraic curves. The origin of this work is notes
distributed to the participants of a course on arithmetic surfaces for graduate
students. The aim of the course was to describe the foundation of the geometry
of arithmetic surfaces as presented in [56] and [90], and the theory of stable
reduction [26]. In spite of the importance of recent developments in these subjects
and of their growing implications in number theory, unfortunately there does not
exist any book in the literature that treats these subjects in a systematic manner,
and at a level that is accessible to a student or to a mathematician who is not a
specialist in the field. The aim of this book is therefore to gather together these
results, now classical and indispensable in arithmetic geometry, in order to make
them more easily accessible to a larger audience.
The first part of the book presents general aspects of the theory of schemes.
It can be useful to a student of algebraic geometry, even if a thorough examination of the subjects treated in the second part is not required. Let us briefly
present the contents of the first seven chapters that make up this first part.
I believe that we cannot separate the learning of algebraic geometry from the
study of commutative algebra. That is the reason why the book starts with a
chapter on the tensor product, flatness, and formal completion. These notions
will frequently recur throughout the book. In the second chapter, we begin with
Hilbert’s Nullstellensatz, in order to give an intuitive basis for the theory of
schemes. Next, schemes and morphisms of schemes, as well as other basic notions,
are defined. In Chapter 3, we study the fibered product of schemes and the fundamental concept of base change. We examine the behavior of algebraic varieties
with respect to base change, before going on to proper morphisms and to projective morphisms. Chapter 4 treats local properties of schemes and of morphisms
such as normality and smoothness. We conclude with an elementary proof of
Zariski’s Main Theorem. The global aspect of schemes is approached through
the theory of coherent sheaves in Chapter 5. After studying coherent sheaves
ˇ
on projective schemes, we define the Cech
cohomology of sheaves, and we look
at some fundamental theorems such as Serre’s finiteness theorem, the theorem
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viii
Preface
of formal functions, and as an application, Zariski’s connectedness principle.
Chapter 6 studies particular coherent sheaves: the sheaf of differentials, and, in
certain favorable cases (local complete intersections), the relative dualizing sheaf.
At the end of that chapter, we present Grothendieck’s duality theory. Chapter 7
starts with a rather general study of divisors, which is then restricted to the
case of projective curves over a field. The theorem of Riemann–Roch, as well
as Hurwitz’s theorem, are proven with the help of duality theory. The chapter
concludes with a detailed study of the Picard group of a not necessarily reduced
projective curve over an algebraically closed field. The necessity of studying singular curves arises, among other things, from the fact that an arithmetic (hence
regular) surface in general has fibers that are singular. These seven chapters can
be used for a basic course on algebraic geometry.
The second part of the book is made up of three chapters. Chapter 8 begins
with the study of blowing-ups. An intermediate section digresses towards commutative algebra by giving, often without proof, some principal results concerning Cohen–Macaulay, Nagata, and excellent rings. Next, we present the general
aspects of fibered surfaces over a Dedekind ring and the theory of desingularization of surfaces. Chapter 9 studies intersection theory on an arithmetic surface,
and its applications. In particular, we show the adjunction formula, the factorization theorem, Castelnuovo’s criterion, and the existence of the minimal regular
model. The last chapter treats the reduction theory of algebraic curves. After
discussing general properties that essentially follow from the study of arithmetic
surfaces, we treat the different types of reduction of elliptic curves in detail. The
end of the chapter is devoted to stable curves and stable reduction. We describe
the proof of the stable reduction theorem of Deligne–Mumford by Artin–Winters,
and we give some concrete examples of computations of the stable reduction.
From the outset, the book was written with arithmetic geometry in mind. In
particular, we almost never suppose that the base field is algebraically closed,
nor of characteristic zero, nor even perfect. Likewise, for the arithmetic surfaces,
in general we do not impose any hypothesis on the base (Dedekind) rings. In
fact, it does not demand much effort to work in general conditions, and does not
affect the presentation in an unreasonable way. The advantage is that it lets us
acquire good reflexes right from the beginning.
As far as possible, the treatment is self-contained. The prerequisites for reading this book are therefore rather few. A good undergraduate student, and in
any case a graduate student, possesses, in principle, the background necessary
to begin reading the book. In addressing beginners, I have found it necessary to
render concepts explicit with examples, and above all exercises. In this spirit,
all sections end with a list of exercises. Some are simple applications of already
proven results, others are statements of results which did not fit in the main
text. All are sufficiently detailed to be solved with a minimum of effort. This
book should therefore allow the reader to approach more specialized works such
as [25] and [15] with more ease.
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Preface
ix
Acknowledgements
It is my great pleasure to thank Michel Matignon and Martin Taylor, who
encouraged me to write up my lecture notes. Reinie Ern´e combined her linguistic
and mathematical talents to translate this book from French to English. I thank
her for her patience and generous help. I thank Philippe Cassou-Nogu`es, Reinie
Ern´e, Arnaud Lacoume, Thierry Sageaux, Alain Thi´ery, and especially Dino
Lorenzini and Sylvain Maugeais for their careful reading of the manuscript. It
is due to their vigilance that many errors were found and corrected. My thanks
also go to Jean Fresnel, Dino Lorenzini, and Michel Matignon for mathematical discussions during the preparation of the book. I thank the Laboratoire
de Math´ematiques Pures de Bordeaux for providing me with such an agreeable
environment for the greatest part of the writing of this book.
I cannot thank my friends and family enough for their constant encouragement and their understanding. I apologize for not being able to name them
individually. Finally, special thanks to Isabelle, who supported me and who put
up with me during the long period of writing. Without her sacrifices and the
encouragement that she gave me in moments of doubt, this book would probably be far from being finished today.
Numbering style
The book is organized by chapter/section/subsection. Each section ends with
a series of exercises. The statements and exercises are numbered within each
section. References to results and definitions consist of the chapter number followed by the section number and the reference number within the section. The
first one is omitted when the reference is to a result within the same chapter. Thus
a reference to Proposition 2.7; 3.2.7; means, respectively, Section 2, Proposition
2.7 of the same chapter; and Chapter 3, Section 2, Proposition 2.7. On the
contrary, we always refer to sections and subsections with the chapter number
followed by the section number, and followed by the subsection number for subsections: e.g., Section 3.2 and Subsection 3.2.4.
Errata
Future errata will be listed at
/>Q.L.
Bordeaux
June 2001
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x
Preface
Preface to the paper-back edition
I am very much indebted to many people who have contributed comments
and corrections since this book was first published in 2002. My hearty thanks
to Robert Ash, Michael Brunnbauer, Oliver Dodane, R´emy Eupherte, Xander
Faber, Anton Geraschenko, Yves Laszlo, Yogesh More, and especially to Lars
Halvard Halle, Carlos Ivorra, Dino Lorenzini and Ren´e Schmidt.
The list of all changes made from the first edition is found on my web page
liu/Book/errata.html
This web page will also include the list of errata for the present edition.
Q.L.
Bordeaux
March 2006
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Contents
1
2
Some topics in commutative algebra
1.1 Tensor products
1.1.1 Tensor product of modules
1.1.2 Right-exactness of the tensor product
1.1.3 Tensor product of algebras
1
1
1
4
5
1.2
Flatness
1.2.1 Left-exactness: flatness
1.2.2 Local nature of flatness
1.2.3 Faithful flatness
6
6
9
12
1.3
Formal completion
1.3.1 Inverse limits and completions
1.3.2 The Artin–Rees lemma and applications
1.3.3 The case of Noetherian local rings
15
15
20
22
General properties of schemes
2.1 Spectrum of a ring
2.1.1 Zariski topology
2.1.2 Algebraic sets
26
26
26
29
2.2
Ringed topological spaces
2.2.1 Sheaves
2.2.2 Ringed topological spaces
33
33
37
2.3
Schemes
2.3.1 Definition of schemes and examples
2.3.2 Morphisms of schemes
2.3.3 Projective schemes
2.3.4 Noetherian schemes, algebraic varieties
41
42
45
50
55
2.4
Reduced schemes and integral schemes
2.4.1 Reduced schemes
59
59
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xii
Contents
2.5
3
4
5
2.4.2 Irreducible components
2.4.3 Integral schemes
61
64
Dimension
2.5.1 Dimension of schemes
2.5.2 The case of Noetherian schemes
2.5.3 Dimension of algebraic varieties
67
68
70
73
Morphisms and base change
3.1 The technique of base change
3.1.1 Fibered product
3.1.2 Base change
78
78
78
81
3.2
Applications to algebraic varieties
3.2.1 Morphisms of finite type
3.2.2 Algebraic varieties and extension of the base field
3.2.3 Points with values in an extension of the base field
3.2.4 Frobenius
87
87
89
92
94
3.3
Some global properties of morphisms
3.3.1 Separated morphisms
3.3.2 Proper morphisms
3.3.3 Projective morphisms
99
99
103
107
Some local properties
115
4.1 Normal schemes
115
4.1.1 Normal schemes and extensions of regular functions 115
4.1.2 Normalization
119
4.2
Regular schemes
4.2.1 Tangent space to a scheme
4.2.2 Regular schemes and the Jacobian criterion
126
126
128
4.3
Flat morphisms and smooth morphisms
4.3.1 Flat morphisms
´
4.3.2 Etale
morphisms
4.3.3 Smooth morphisms
135
136
139
141
4.4
Zariski’s ‘Main Theorem’ and applications
149
ˇ
Coherent sheaves and Cech
cohomology
5.1 Coherent sheaves on a scheme
5.1.1 Sheaves of modules
5.1.2 Quasi-coherent sheaves on an affine scheme
5.1.3 Coherent sheaves
5.1.4 Quasi-coherent sheaves on a projective scheme
157
157
157
159
161
164
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Contents
5.2
5.3
6
7
xiii
ˇ
Cech
cohomology
5.2.1 Differential modules and cohomology with values
in a sheaf
ˇ
5.2.2 Cech
cohomology on a separated scheme
5.2.3 Higher direct image and flat base change
178
185
188
Cohomology of projective schemes
5.3.1 Direct image theorem
5.3.2 Connectedness principle
5.3.3 Cohomology of the bers
195
195
198
201
178
Sheaves of dierentials
210
6.1
Kăahler dierentials
6.1.1 Modules of relative dierential forms
6.1.2 Sheaves of relative differentials (of degree 1)
210
210
215
6.2
Differential study of smooth morphisms
6.2.1 Smoothness criteria
6.2.2 Local structure and lifting of sections
220
220
223
6.3
Local complete intersection
6.3.1 Regular immersions
6.3.2 Local complete intersections
227
228
232
6.4
Duality theory
6.4.1 Determinant
6.4.2 Canonical sheaf
6.4.3 Grothendieck duality
236
236
238
243
Divisors and applications to curves
252
7.1
Cartier divisors
7.1.1 Meromorphic functions
7.1.2 Cartier divisors
7.1.3 Inverse image of Cartier divisors
252
252
256
260
7.2
Weil divisors
7.2.1 Cycles of codimension 1
7.2.2 Van der Waerden’s purity theorem
267
267
272
7.3
Riemann–Roch theorem
7.3.1 Degree of a divisor
7.3.2 Riemann–Roch for projective curves
275
275
278
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8
Contents
7.4
Algebraic curves
7.4.1 Classification of curves of small genus
7.4.2 Hurwitz formula
7.4.3 Hyperelliptic curves
7.4.4 Group schemes and Picard varieties
284
284
289
292
297
7.5
Singular curves, structure of Pic0 (X)
303
Birational geometry of surfaces
8.1 Blowing-ups
8.1.1 Definition and elementary properties
8.1.2 Universal property of blowing-up
8.1.3 Blowing-ups and birational morphisms
8.1.4 Normalization of curves by blowing-up points
317
317
318
323
326
330
8.2
332
8.3
9
Excellent schemes
8.2.1 Universally catenary schemes and the
dimension formula
8.2.2 Cohen–Macaulay rings
8.2.3 Excellent schemes
332
335
341
Fibered surfaces
347
8.3.1 Properties of the fibers
347
8.3.2 Valuations and birational classes of fibered surfaces 353
8.3.3 Contraction
356
8.3.4 Desingularization
361
Regular surfaces
9.1 Intersection theory on a regular surface
9.1.1 Local intersection
9.1.2 Intersection on a fibered surface
9.1.3 Intersection with a horizontal divisor,
adjunction formula
375
376
376
381
9.2
Intersection and morphisms
9.2.1 Factorization theorem
9.2.2 Projection formula
9.2.3 Birational morphisms and Picard groups
9.2.4 Embedded resolutions
394
394
397
401
404
9.3
Minimal surfaces
9.3.1 Exceptional divisors and Castelnuovo’s criterion
9.3.2 Relatively minimal surfaces
9.3.3 Existence of the minimal regular model
9.3.4 Minimal desingularization and minimal
embedded resolution
411
412
418
421
388
424
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9.4
xv
Applications to contraction; canonical model
9.4.1 Artin’s contractability criterion
9.4.2 Determination of the tangent spaces
9.4.3 Canonical models
9.4.4 Weierstrass models and regular models of
elliptic curves
10 Reduction of algebraic curves
10.1 Models and reductions
10.1.1 Models of algebraic curves
10.1.2 Reduction
10.1.3 Reduction map
10.1.4 Graphs
429
430
434
438
442
454
454
455
462
467
471
10.2 Reduction of elliptic curves
10.2.1 Reduction of the minimal regular model
10.2.2 N´eron models of elliptic curves
10.2.3 Potential semi-stable reduction
483
484
489
498
10.3 Stable
10.3.1
10.3.2
10.3.3
505
505
511
reduction of algebraic curves
Stable curves
Stable reduction
Some sufficient conditions for the existence of
the stable model
10.4 Deligne–Mumford theorem
10.4.1 Simplifications on the base scheme
10.4.2 Proof of Artin–Winters
10.4.3 Examples of computations of the potential
stable reduction
521
532
533
537
543
Bibliography
557
Index
562
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1
Some topics in commutative
algebra
Unless otherwise specified, all rings in this book will be supposed commutative
and with unit.
In this chapter, we introduce some indispensable basic notions of commutative algebra such as the tensor product, localization, and flatness. Other, more
elaborate notions will be dealt with later, as they are needed. We assume that
the reader is familiar with linear algebra over a commutative ring, and with
Noetherian rings and modules.
1.1
Tensor products
In the theory of schemes, the fibered product plays an important role (in particular the technique of base change). The corresponding notion in commutative
algebra is the tensor product of modules over a ring.
1.1.1
Tensor product of modules
Definition 1.1. Let A be a commutative ring with unit. Let M , N be two
A-modules. The tensor product of M and N over A is defined to be an A-module
H, together with a bilinear map φ : M ×N → H satisfying the following universal
property:
For every A-module L and every bilinear map f : M × N → L, there
exists a unique homomorphism of A-modules f˜ : H → L making the
following diagram commutative:
f
M ×N
φ
P
N
f˜NN
N
u N
H
wL
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2
1. Some topics in commutative algebra
Proposition 1.2. Let A be a ring, and let M , N be A-modules. The tensor
product (H, φ) exists, and is unique up to isomorphism.
Proof As the solution of a universal problem, the uniqueness is automatic, and
its proof is standard. We give it here as an example. Let (H, φ) and (H , φ ) be
˜
two solutions. By the universal property, φ and φ factor respectively as φ = φ◦φ
˜
˜
˜
and φ = φ ◦ φ. It follows that φ = (φ ◦ φ ) ◦ φ. As φ = Id ◦ φ, it follows from
the uniqueness of the decomposition of φ that (φ˜ ◦ φ˜ ) = Id. Thus we see that
φ˜ : H → H is an isomorphism.
Let us now show existence. Consider the free A-module A(M ×N ) with basis
M × N . Let {ex,y }(x,y)∈M ×N denote its canonical basis. Let L be the submodule
of A(M ×N ) generated by the elements having one of the following forms:
ex1 +x2 ,y − ex1 ,y − ex2 ,y
ex,y1 +y2 − ex,y1 − ex,y2
eax,y − ex,ay , aex,y − eax,y , a ∈ A.
Let H = A(M ×N ) /L, and φ : M × N → H be the map defined by φ(x, y) =
the image of ex,y in H. One immediately verifies that the pair (H, φ) verifies the
universal property mentioned above.
Notation. We denote the tensor product of M and N by (M ⊗A N, φ). In general,
the map φ is omitted in the notation. For any (x, y) ∈ M ×N , we let x⊗y denote
its image by φ. By the bilinearity of φ, we have a(x ⊗ y) = (ax) ⊗ y = x ⊗ (ay)
for every a ∈ A.
Remark 1.3. By construction, M ⊗A N is generated as an A-module by its
elements of the form x ⊗ y. Thus every element of M ⊗A N can be written
(though not in a unique manner) as a finite sum i xi ⊗ yi , with xi ∈ M and
yi ∈ N . In general, an element of M ⊗A N cannot be written x ⊗ y.
Example 1.4. Let A = Z, M = A/2A, and N = A/3A. Then M ⊗A N = 0.
In fact, for every (x, y) ∈ M × N , we have x ⊗ y = 3(x ⊗ y) − 2(x ⊗ y) =
x ⊗ (3y) − (2x) ⊗ y = 0.
Proposition 1.5. Let A be a ring, and let M , N , Mi be A-modules. We have
the following canonical isomorphisms of A-modules:
(a) M ⊗A A
M;
(b) (commutativity) M ⊗A N
N ⊗A M ;
(c) (associativity) (L ⊗A M ) ⊗A N
L ⊗A (M ⊗A N );
(d) (distributivity) (⊕i∈I Mi ) ⊗A N
⊕i∈I (Mi ⊗A N ).
Proof Everything follows from the universal property. Let us, for example,
show (a) and (d).
(a) Let φ : M × A → M be the bilinear map defined by (x, a) → ax. For any
bilinear map f : M × A → L, set f˜ : M → L, x → f (x, 1). Then f = f˜ ◦ φ, and
f˜ is the unique linear map M → L having this property. Hence (M, φ) is the
tensor product of M and A.
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1.1. Tensor products
3
(d) Let φ : (⊕i∈I Mi ) × N → ⊕i∈I (Mi ⊗A N ) be the map defined by
xi , y →
φ:
i
(xi ⊗ y).
i
Let f : (⊕i∈I Mi )×N → L be a bilinear map. For every i ∈ I, f induces a bilinear
map fi : Mi × N → L which factors through fi : Mi ⊗A N → L. One verifies
that f factors uniquely as f = f˜ ◦ ψ, where ψ : (⊕i∈I Mi ) × N → (⊕i∈I Mi ) ⊗ N
is the canonical map and f˜ = ⊕i fi . Hence ⊕i∈I (Mi ⊗A N ) is the tensor product
of (⊕i∈I Mi ) with N .
Corollary 1.6. Let M be a free A-module with basis {ei }i∈I . Then every element of M ⊗A N can be written uniquely as a finite sum i ei ⊗ yi , with yi ∈ N .
In particular, if A is a field and {ei }i∈I (resp. {dj }j∈J ) is a basis of M (resp. of
N ), then {ei ⊗ dj }(i,j)∈I×J is a basis of M ⊗A N .
Remark 1.7. The associativity of the tensor product allows us to define the
tensor product M1 ⊗A · · · ⊗A Mn of a finite number of A-modules. This tensor
product has a universal property analogous to that of the tensor product of two
modules, with the bilinear maps replaced by multilinear ones.
Definition 1.8. Let u : M → M , v : N → N be linear maps of A-modules.
By the universal property of the tensor product, there exists a unique A-linear
map u ⊗ v : M ⊗A N → M ⊗A N such that (u ⊗ v)(x ⊗ y) = u(x) ⊗ v(y). In
fact, the map g : M × N → M ⊗A N defined by g(x, y) = u(x) ⊗ v(y) is clearly
bilinear, and hence factors uniquely as (u ⊗ v) ◦ φ, where φ is the canonical map
M × N → M ⊗ N . The map u ⊗ v is called the tensor product of u and v. The
notation is justified by Exercise 1.2.
Let ρ : A → B be a ring homomorphism, and N a B-module. Then ρ induces,
in a natural way, the structure of an A-module on N : for any a ∈ A and y ∈ N ,
we set a · y = ρ(a)y. We denote this A-module by ρ∗ N , or simply by N .
Definition 1.9. Let M be an A-module. We can endow M ⊗A N with the
structure of a B-module as follows. Let b ∈ B. Let tb : N → N denote the
multiplication by b, and for any z ∈ M ⊗A N , set b · z := (IdM ⊗tb )(z). One
easily verifies that this defines the structure of a B-module. We denote the Bmodule M ⊗A B by ρ∗ M . This is called the extension of scalars of M by B. By
construction, we have b(x ⊗ y) = x ⊗ (by) for every b ∈ B, x ∈ M , and y ∈ N .
Proposition 1.10. Let ρ : A → B be a ring homomorphism, M an A-module,
and let N , P be B-modules. Then there exists a canonical isomorphism of Bmodules
M ⊗A (N ⊗B P ) (M ⊗A N ) ⊗B P.
Proof Let us show that there exist A-linear maps
f : M ⊗A (N ⊗B P ) → (M ⊗A N ) ⊗B P,
g : (M ⊗A N ) ⊗B P → M ⊗A (N ⊗B P )
such that for every x ∈ M , y ∈ N , and z ∈ P , we have f (x⊗(y ⊗z)) = (x⊗y)⊗z
and g((x ⊗ y) ⊗ z) = x ⊗ (y ⊗ z). This will imply that f is an isomorphism, with
inverse g. The B-linearity of f follows from this identity.
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4
1. Some topics in commutative algebra
Let us fix x ∈ M . Let tx : N → M ⊗A N denote the A-linear map defined
by tx (y) = x ⊗ y. Consider the map h : M × (N ⊗B P ) → (M ⊗A N ) ⊗B P
defined by h(x, u) = (tx ⊗ IdP )(u). This map is A-bilinear, and hence induces an
A-linear map f as desired. The construction of g is similar.
Taking N = B in the proposition above, we obtain:
Corollary 1.11. Let ρ : A → B be a ring homomorphism, let M be an Amodule, and N a B-module. There exists a canonical isomorphism of B-modules
(M ⊗A B) ⊗B N
1.1.2
M ⊗A N
(simplification by B).
Right-exactness of the tensor product
Let us recall that a complex of A-modules consists of a (finite or infinite) sequence
of A-modules Mi , together with linear maps fi : Mi → Mi+1 , such that fi+1 ◦fi =
0. A complex is written more visually as
fi+1
fi
· · · → Mi −→ Mi+1 −−−→ Mi+2 → · · · .
The complex is called exact if Ker(fi+1 ) = Im(fi ) for all i. An exact complex is
also called an exact sequence. For example, a sequence
f
→N
0→M −
f
(resp. M −
→ N → 0)
is exact if and only if f is injective (resp. surjective).
Let f : N → N be a linear map of A-modules. For simplicity, for any Amodule M , we denote the linear map f ⊗ IdM : N ⊗A M → N ⊗A M by fM .
Proposition 1.12. Let A be a ring, and let
f
g
→N −
→N →0
N −
be an exact sequence of A-modules. Then for any A-module M , the sequence
fM
gM
N ⊗A M −−→ N ⊗A M −−→ N ⊗A M → 0
is exact.
Proof The surjectivity of gM follows from that of g (use Remark 1.3). It
remains to show that Ker(gM ) = Im fM ; in other words, that the canonical
homomorphism g : (N ⊗A M )/(Im fM ) → N ⊗A M is an isomorphism. Let
h : N × M → (N ⊗A M )/(Im fM ) be defined by h(x, z) = the image of y ⊗ z in
the quotient, where y ∈ g −1 (x). The map h is well defined and moreover bilinear.
It therefore induces a linear map h : N ⊗A M → (N ⊗A M )/(Im fM ), and it is
easy to see that this is the inverse of g.
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1.1. Tensor products
5
Corollary 1.13. Let A be a ring. Let N , M be A-modules, and i : N → N a
submodule of N . There exists a canonical isomorphism
(N ⊗A M )/(Im iM )
(N/N ) ⊗A M.
In particular, if I is an ideal of A, then we have M ⊗A (A/I)
M/(IM ).
Proof It suffices to apply Proposition 1.12 to the exact sequence N → N →
(N/N ) → 0. If N = A and N = I, we also use Proposition 1.5(a).
1.1.3
Tensor product of algebras
An A-algebra is a commutative ring B with unit, endowed with a ring homomorphism A → B. Let B and C be A-algebras. We can canonically endow B ⊗A C
with the structure of an A-algebra, as follows. Let g : B × C × B × C → B ⊗A C
be defined by g(b, c, b , c ) = (bb ) ⊗ (cc ). This is a multilinear map, and hence
factors through g : B ⊗A C ⊗A B ⊗A C → B ⊗A C (see Remark 1.7). We can then
define the product on B ⊗A C using the composition of (B ⊗A C) × (B ⊗A C) →
(B ⊗A C) ⊗A (B ⊗A C) with g. More precisely, we set i (bi ⊗ ci ) · j (bj ⊗ cj ) =
i,j (bi bj ) ⊗ (ci cj ). The point was to see that this is well defined.
We have homomorphisms of A-algebras p1 : B → B ⊗A C, p2 : C → B ⊗A C
defined by p1 (b) = b ⊗ 1 and p2 (c) = 1 ⊗ c. The following proposition can be
verified immediately:
Proposition 1.14. Let us keep the notation above. The triplet (B ⊗A C, p1 , p2 )
satisfies the following universal property:
For every A-algebra D, and for every pair of homomorphisms of
A-algebras q1 : B → D, q2 : C → D, there exists a unique homomorphism of A-algebras q : B ⊗A C → D such that qi = q ◦ pi .
Example 1.15. Let A[T1 , . . . , Tn ] be the polynomial ring in n variables over A.
Let B be an A-algebra. One easily verifies (either with the universal property
given above or using Proposition 1.5(d)) that the homomorphism of B-algebras
A[T1 , . . . , Tn ] ⊗A B → B[T1 , . . . , Tn ]
defined by ( ν aν T )⊗b → ν (baν T ν ) is an isomorphism. In particular, taking
a polynomial ring for B, we obtain
ν
A[T1 , . . . , Tn ] ⊗A A[S1 , . . . , Sm ]
A[T1 , . . . , Tn , S1 , . . . , Sm ].
Exercises
1.1. Let {Mi }i∈I , {Nj }j∈J be two families of modules over a ring A. Show that
(⊕i∈I Mi ) ⊗A (⊕j∈J Nj )
⊕(i,j)∈I×J (Mi ⊗A Nj ).
1.2. Show that there exists a unique A-linear map
f : HomA (M, M ) ⊗A HomA (N, N ) → HomA (M ⊗A N, M ⊗A N )
such that f (u ⊗ v) = u ⊗ v.
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6
1. Some topics in commutative algebra
1.3. Let M , N be A-modules, and i : M → M , j : N → N submodules of M
and N , respectively. Then there exists a canonical isomorphism
(M/M ) ⊗A (N/N )
(M ⊗A N )/(Im iN + Im jM ).
Show that (Z/nZ) ⊗Z (Z/mZ) = Z/lZ, where l = gcd(m, n).
1.4. Let M , N be A-modules, and let B, C be A-algebras.
(a) If M and N are finitely generated over A, then so is M ⊗A N .
(b) If B and C are finitely generated over A, then so is B ⊗A C.
(c) Taking A = Z, M = B = Z/2Z, and N = C = Q, show that the
converse of (a) and (b) is false.
1.5. Let ρ : A → B be a ring homomorphism, M an A-module, and N a
B-module. Show that there exists a canonical isomorphism of A-modules
HomA (M, ρ∗ N )
HomB (ρ∗ M, N ).
1.6. Let (Ni )i∈I be a direct system of A-modules. Then for any A-module M ,
(lim
there exists a canonical isomorphism −→
lim(Ni ⊗A M )
−→ Ni ) ⊗A M .
(Hint: show that −→
lim(Ni ⊗A M ) verifies the universal property of the tensor
product −→
lim(Ni ) ⊗A M .)
1.7. Let B be an A-algebra, and let M , N be B-modules. Show that there exists
a canonical surjective homomorphism M ⊗A N → M ⊗B N .
1.2
Flatness
The flatness of a module over a ring is a property concerning extensions of
scalars. In algebraic geometry, it assures a certain ‘continuity’ behavior. In this
section, we study a few elementary aspects of flatness. We conclude the section
with faithful flatness.
1.2.1
Left-exactness: flatness
Definition 2.1. An A-module M is called flat (over A) if for every injective
homomorphism of A-modules N → N , N ⊗A M → N ⊗A M is injective. An
A-algebra B is called flat if B is flat over A for its A-module structure, and the
canonical homomorphism A → B will be called a flat homomorphism.
It follows from Proposition 1.12 that if 0 → N → N → N → 0 is an exact
sequence of A-modules, and M is flat over A, then
0 → N ⊗A M → N ⊗A M → N ⊗A M → 0
is an exact sequence.
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1.2. Flatness
7
Proposition 2.2. Let A be a ring. We have the following properties:
(a) Every free A-module is flat.
(b) (Product) The tensor product of modules that are flat over A is flat
over A.
(c) (Base change) Let B be an A-algebra. If M is flat over A, then M ⊗A B
is flat over B.
(d) (Transitivity) Let B be a flat A-algebra. Then every B-module that is
flat over B is flat over A.
Proof These easily follow from the definition and the general properties of the
tensor product (Proposition 1.5 and Corollary 1.11).
Example 2.3. Let A = Z, n ≥ 2, and M = A/nA. Then M is not flat over A.
In fact, if we tensor the canonical injection nA → A by M , the image of nA⊗A M
in A ⊗A M = M is equal to nM = 0, while nA ⊗A M A ⊗A M = M = 0.
Theorem 2.4. Let M be an A-module. Then M is flat if and only if for every
ideal I of A, the canonical homomorphism I ⊗A M → IM is an isomorphism.
Proof If M is flat over A, then for any ideal I of A, we can tensor the canonical
injection I → A by M . This shows that I ⊗A M → A ⊗A M = M is injective.
The image of this map is clearly IM , whence the isomorphism I ⊗A M IM .
Conversely, let us suppose that we have this isomorphism for every ideal I,
and let us show that M is flat. Let N → N be an injective homomorphism of
A-modules. We need to show that N ⊗ M → N ⊗ M is injective.
Let us first suppose that N is free of finite rank, and let us show the injectivity
by induction on the rank n of N . The case n = 1 follows from the hypothesis.
Let us suppose that n ≥ 2 and that the result holds for every free module of rank
< n. The module N is a direct sum of two free submodules N1 and N2 , different
from N . Let N1 = N1 ∩ N , and let N2 be the image of N in N2 = N/N1 . We
then have the following commutative diagram
N1
wN
w N2
u
N1
u
wN
u
w N2
whose horizontal lines are exact, and whose vertical arrows are injective. Tensoring by M gives the commutative diagram
N1 ⊗ M
w N ⊗M
γ
β
u
N1 ⊗ M
w N2 ⊗ M
α
u
w N ⊗M
u
w N2 ⊗ M
whose horizontal lines are still exact (Proposition 1.12). The map α is injective
because N1 is a direct factor of N (Proposition 1.5(d)), and β, γ are injective by
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8
1. Some topics in commutative algebra
the induction hypothesis applied to the Ni . It follows that N ⊗ M → N ⊗ M is
injective.
Let us now suppose N is free of arbitrary rank. Let N0 be a direct factor of
N of finite rank. By the above, the map (N ∩ N0 ) ⊗ M → N0 ⊗ M is injective.
Hence so is (N ∩ N0 ) ⊗ M → N ⊗ M , since N0 is a direct factor of N . Because
for every x ∈ N ⊗ M there exists an N0 such that x is contained in the image
of (N ∩ N0 ) ⊗ M → N ⊗ M (use Corollary 1.6), we see that N ⊗ M → N ⊗ M
is injective.
Let N now be an arbitrary A-module. There exist a free A-module L and
a surjective homomorphism p : L → N . Let us set L = p−1 (N ). We have a
commutative diagram
Ker(p)
wL
Ker(p)
u
wL
p
wN
w0
u
wN
w0
whose horizontal lines are exact, whence the commutative diagram
Ker(p) ⊗ M
w L ⊗M
w N ⊗M
w0
Ker(p) ⊗ M
u
w L⊗M
u
w N ⊗M
w0
whose horizontal lines are also exact. As the middle vertical arrow is injective,
this shows the injectivity of N ⊗ M → N ⊗ M .
Let A be an integral domain and M an A-module. An element x ∈ M is
called a torsion element if there exists a non-zero a ∈ A such that ax = 0. We
call M torsion-free (over A) if there is no non-zero torsion element in M . When
A is a principal ideal domain, flatness can be expressed in a very simple way.
Corollary 2.5. Let A be a principal ideal domain. An A-module M is flat if
and only if it is torsion-free over A.
Proof Let I = aA = 0 be an ideal of A. Let ta (resp. ua ) denote multiplication
by a in A (resp. in M ). Then ta : A → I is an isomorphism. We have the following
commutative diagram:
M = A ⊗A M
ua
u
IM
ta ⊗IdM
w I ⊗A M
''
''
''f
'
*'
'
where f is the canonical homomorphism. Consequently, f is an isomorphism if
and only if ua is an isomorphism, which is equivalent to saying that ax = 0 for
x ∈ M implies that x = 0. Hence the corollary follows from Theorem 2.4.
