申请加精Springer Undergraduate Mathematics Series

SMM Lie Algebras And Algebraic Groups (2005) Tauvel Yu
Preface
The theory of groups and Lie algebras is interesting for many reasons. In
the mathematical viewpoint, it employs at the same time algebra, analysis
and geometry. On the other hand, it intervenes in other areas of science, in
particular in different branches of physics and chemistry. It is an active domain
of current research.
One of the difficulties that graduate students or mathematicians interested
in the theory come across, is the fact that the theory has very much advanced,
and consequently, they need to read a vast amount of books and articles before
they could tackle interesting problems.
One of the goals we wish to achieve with this book is to assemble in
a single volume the basis of the algebraic aspects of the theory of groups
and Lie algebras. More precisely, we have presented the foundation of the
study of finite-dimensional Lie algebras over an algebraically closed field of
characteristic zero.
Here, the geometrical aspect is fundamental, and consequently, we need to
use the notion of algebraic groups. One of the main differences between this
book and many other books on the subject is that we give complete proofs
for the relationships between algebraic groups and Lie algebras, instead of
admitting them.
We have also given the proofs of certain results on commutative algebra
and algebraic geometry that we needed so as to make this book as selfcontained
as possible. We believe that in this way, the book can be useful for
both graduate students and mathematicians working in this area.
Let us give a brief description of the material treated in this book.
As we have stated earlier, our goal is to study Lie algebras over an algebraically
closed field of characteristic zero. This allows us to avoid, in considering
questions concerning algebraic geometry, the notion of separability, which
simplifies considerably our presentation. In fact, under certain conditions of
separability, the correspondence between Lie algebras and algebraic groups
described in chapter 24 has a very nice generalization when the algebraically
closed base field has prime characteristic.

Contents
1 Results on topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Irreducible sets and spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Noetherian spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Constructible sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Gluing topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Rings and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Prime and maximal ideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Rings of fractions and localization. . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Localizations of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
24 Factorial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Unique factorization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Principal ideal domains and Euclidean domains . . . . . . . . . . . . . 43
4.4 Polynomials and factorial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 Symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 Resultant and discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

16.5 Smooth points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
17 Normal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
17.1 Normal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
17.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
17.3 Products of normal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
17.4 Properties of normal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
18 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
18.1 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
18.2 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
18.3 Root systems and bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . 238
18.4 Passage to the field of real numbers . . . . . . . . . . . . . . . . . . . . . . . 239
18.5 Relations between two roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
18.6 Examples of root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
18.7 Base of a root system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
18.8 Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
18.9 Highest root . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
18.10 Closed subsets of roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
18.11Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
20 Semisimple and reductive Lie algebras . . . . . . . . . . . . . . . . . . . . . 299
20.1 Semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
20.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
20.3 Semisimplicity of representations . . . . . . . . . . . . . . . . . . . . . . . . . . 302
20.4 Semisimple and nilpotent elements . . . . . . . . . . . . . . . . . . . . . . . . 305
20.5 Reductive Lie algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

38 Semisimple symmetric Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 561
38.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
38.2 Iwasawa decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
38.7 Symmetric invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
38.8 Double centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
38.9 Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
38.10 Distinguished elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
39 Sheets of Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
39.1 Jordan classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
39.2 Topology of Jordan classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
39.3 Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
39.4 Dixmier sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
39.5 Jordan classes in the symmetric case . . . . . . . . . . . . . . . . . . . . . . 605
39.6 Sheets in the symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608
40 Index and linear forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
40.1 Stable linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
40.2 Index of a representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
40.3 Some useful inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
40.4 Index and semi-direct products . . . . . . . . . . . . . . . . . . . . . . . . . . . 618
40.5 Heisenberg algebras in semisimple Lie algebras . . . . . . . . . . . . . 621
40.6 Index of Lie subalgebras of Borel subalgebras . . . . . . . . . . . . . . . 625
40.7 Seaweed Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
40.8 An upper bound for the index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 630
40.9 Cases where the bound is exact . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
40.10 On the index of parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . 638
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
List of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

SMM Random Fields and Geometry
Preface
Since the term “random field’’ has a variety of different connotations, ranging from
agriculture to statistical mechanics, let us start by clarifying that, in this book, a
random field is a stochastic process, usually taking values in a Euclidean space, and
defined over a parameter space of dimensionality at least 1.
Consequently, random processes defined on countable parameter spaces will not
appear here. Indeed, even processes on R1 will make only rare appearances and,
from the point of view of this book, are almost trivial. The parameter spaces we like
best are manifolds, although for much of the time we shall require no more than that
they be pseudometric spaces.
With this clarification in hand, the next thing that you should know is that this
book will have a sequel dealing primarily with applications.
In fact, as we complete this book, we have already started, together with KW
(KeithWorsley), on a companion volume [8] tentatively entitled RFG-A, or Random
Fields and Geometry: Applications. The current volume—RFG—concentrates on
the theory and mathematical background of random fields, while RFG-A is intended
to do precisely what its title promises. Once the companion volume is published,
you will find there not only applications of the theory of this book, but of (smooth)
random fields in general.
Making a clear split between theory and practice has both advantages and disadvantages.
It certainly eased the pressure on us to attempt the almost impossible goal
of writing in a style that would be accessible to all. It also, to a large extent, eases the
load on you, the reader, since you can now choose the volume closer to your interests
and so avoid either “irrelevant’’mathematical detail or the “real world,’’depending on
your outlook and tastes. However, these are small gains when compared to the major
loss of creating an apparent dichotomy between two things that should, in principle,
go hand-in-hand: theory and application. What is true in principle is particularly true
of the topic at hand, and, to explain why, we shall indulge ourselves in a paragraph
or two of history.
The precusor to both of the current volumes was the 1981 monograph The Geometry
of Random Fields (GRF) which grew out of RJA’s (i.e., Robert Adler’s) Ph.D.
thesis under Michael Hasofer. The problem that gave birth to the thesis was an applied
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Part I Gaussian Processes
1 Gaussian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Gaussian Variables and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Boundedness and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.1 Fields on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.2 Differentiability on RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.3 The Brownian Family of Processes . . . . . . . . . . . . . . . . . . . . 24
1.4.4 Generalized Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4.5 Set-Indexed Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.4.6 Non-Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Majorizing Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 Gaussian Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.1 Borell–TIS Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 Comparison Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1 The General Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 The Karhunen–Loève Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Excursion Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1 Entropy Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Processes with a Unique Point of Maximal Variance . . . . . . . . . . . . . 86
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
13 Mean Intrinsic Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
13.1 Crofton’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
13.2 Mean Intrinsic Volumes: The Isotropic Case . . . . . . . . . . . . . . . . . . . 333
13.3 A Gaussian Crofton Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
13.4 Mean Intrinsic Volumes: The General Case . . . . . . . . . . . . . . . . . . . . 342
13.5 Two Gaussian Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
14 Excursion Probabilities for Smooth Fields . . . . . . . . . . . . . . . . . . . . . . . . 349
14.1 On Global Suprema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
14.1.1 A First Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
14.1.2 The Problem with the First Representation . . . . . . . . . . . . . . 354
14.1.3 A Second Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
14.1.4 Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
14.1.5 Suprema and Euler Characteristics . . . . . . . . . . . . . . . . . . . . 362
14.2 Some Fine Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
14.3 Gaussian Fields with Constant Variance . . . . . . . . . . . . . . . . . . . . . . . 368
14.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
14.4.1 Stationary Processes on [0, T ] . . . . . . . . . . . . . . . . . . . . . . . . 372
14.4.2 Isotropic Fields with Monotone Covariance . . . . . . . . . . . . . 374
14.4.3 A Geometric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
14.4.4 The Cosine Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
15 Non-Gaussian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
15.1 A Plan of Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
15.2 A Representation for Mean Intrinsic Volumes . . . . . . . . . . . . . . . . . . 391
15.3 Proof of the Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
15.4 Poincaré’s Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
15.5 Kinematic Fundamental Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
15.5.1 The KFF on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
15.5.2 The KFF on Sλ(Rn). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
15.6 A Model Process on the l-Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
15.6.1 The Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
15.6.2 Mean Curvatures for the Model Process . . . . . . . . . . . . . . . . 404
15.7 The Canonical Gaussian Field on the l-Sphere . . . . . . . . . . . . . . . . . . 410
15.7.1 Mean Curvatures for Excursion Sets . . . . . . . . . . . . . . . . . . . 411
15.7.2 Implications for More General Fields . . . . . . . . . . . . . . . . . . 415
15.8 Warped Products of Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . 416
15.8.1 Warped Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
15.8.2 A Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . . 419
15.9 Non-Gaussian Mean Intrinsic Volumes . . . . . . . . . . . . . . . . . . . . . . . . 421

SMM Variational and Potential Methods for a Class of Linear Hyperbolic Evolutionary Processes
Preface
Variational and boundary integral equation techniques are two of the most
useful methods for solving time-dependent problems described by systems of
equations of the form
∂2u
∂t2 = Au,
where u = u(x, t) is a vector-valued function, x is a point in a domain in R2 or
R3, and A is a linear elliptic differential operator. To facilitate a better understanding
of these two types of methods, below we propose to illustrate their
mechanisms in action on a specific mathematical model rather than in a more
impersonal abstract setting. For this purpose, we have chosen the hyperbolic
system of partial differential equations governing the nonstationary bending
of elastic plates with transverse shear deformation. The reason for our choice
is twofold. On the one hand, in a certain sense this is a “hybrid” system, consisting
of three equations for three unknown functions in only two independent
variables, which makes it more unusual—and thereby more interesting to the
analyst—than other systems arising in solid mechanics. On the other hand,
this particular plate model has received very little attention compared to the
so-called classical one, based on Kirchhoff’s simplifying hypotheses, although,
as acknowledged by practitioners, it represents a substantial refinement of the
latter and therefore needs a rigorous discussion of the existence, uniqueness,
and continuous dependence of its solution on the data before any construction
of numerical approximation algorithms can be contemplated.
The first part of our analysis is conducted by means of a procedure that
is close in both nature and detail to a variational method, and which, for this
reason, we also call variational. Once the results have been established in the
general setting of Sobolev spaces, we carry out the second part of the study by
seeking useful, closed-form integral representations of the solutions in terms
of dynamic (retarded) plate potentials.

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Formulation of the Problems and Their Nonstationary
Boundary Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Initial-Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . 1
1.2 A Matrix of Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Time-dependent Plate Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Nonstationary Boundary Integral Equations . . . . . . . . . . . . . . . 16
2 Problems with Dirichlet Boundary Conditions . . . . . . . . . . . . 19
2.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Solvability of the Transformed Problems . . . . . . . . . . . . . . . . . . 21
2.3 Solvability of the Time-dependent Problems . . . . . . . . . . . . . . . 28
3 Problems with Neumann Boundary Conditions . . . . . . . . . . . 37
3.1 The Poincar´e–Steklov Operators . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Solvability of the Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Boundary Integral Equations for Problems with Dirichlet
and Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Time-dependent Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Nonstationary Boundary Integral Equations . . . . . . . . . . . . . . . 51
4.3 The Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Transmission Problems and Multiply Connected Plates . . . 57
5.1 Infinite Plate with a Finite Inclusion . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Multiply Connected Finite Plate . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Finite Plate with an Inclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

XII Contents
6 Plate Weakened by a Crack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.1 Formulation and Solvability of the Problems . . . . . . . . . . . . . . . 81
6.2 The Poincar´e–Steklov Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Time-dependent Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4 Infinite Plate with a Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5 Finite Plate with a Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 Initial-Boundary Value Problems with Other Types of
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.1 Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2 Combined Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3 Elastic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 Boundary Integral Equations for Plates on a Generalized
Elastic Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.1 Formulation and Solvability of the Problems . . . . . . . . . . . . . . . 119
8.2 A Matrix of Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . 121
8.3 Properties of the Boundary Operators. . . . . . . . . . . . . . . . . . . . . 126
8.4 Solvability of the Boundary Equations . . . . . . . . . . . . . . . . . . . . 127
9 Problems with Nonhomogeneous Equations
and Nonhomogeneous Initial Conditions . . . . . . . . . . . . . . . . . 129
9.1 The Time-dependent Area Potential . . . . . . . . . . . . . . . . . . . . . . 129
9.2 The Nonhomogeneous Equation of Motion . . . . . . . . . . . . . . . . . 131
9.3 Initial Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A The Fourier and Laplace Transforms of Distributions . . . . . 139
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

SMM Valued fields (Engler A. Prestel A. )
Preface
The purpose of this book is to give a self-contained and comprehensive introduction
to the theory of general valuations, in contrast to classical absolute
values. In particular, we present some applications of the general theory going
beyond the use of absolute values. The book does not aim for an encyclopaedic
presentation, but rather prefers a streamlined style, leading eventually to deep
results of recent research.
While the classical theory of absolute values can be found in many books,
in particular those on number theory, there are few textbooks devoted to the
general theory of valuations. To our knowledge, these are O. Schilling (1950,
[27]), P. Ribenboim (1965, [23]), and O. Endler (1972, [6]). Besides those, one
can find, however, chapters on general valuation theory in several books, such
as in O. Zariski – P. Samuel (1960, [33]) or Y. Ershov (2001, [8]). Concerning
the history of valuation theory, the reader is referred to P. Roquette [25].
Both authors of this book have been deeply influenced by the late Otto
Endler – the first author as a student, the second as a colleague. It was at the
IMPA in Rio de Janeiro where we all met in the mid seventies. Since then we
became followers of Krull’s development of henselian valued fields, and since
then we have tried to convince other mathematicians of the beauty of this
theory.
The book is based on courses given by the first author in Brazil, and
by the second author in Pisa, Freiburg and Konstanz. We are grateful to
K. Becher, M. Illengo, I. Klep, J. Koenigsmann, J. Schmid and T. Unger for
reading parts of the book and making many valuable comments. We are also
grateful to C. N. Delzell for checking the layout and the use of the English
language.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 Absolute Values – Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Archimedean Complete Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Non-Archimedean Complete Fields . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Ordered Abelian Groups – Valuations . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Constructions of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Rational Function Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 Ordered Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.3 Rigid Elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3 Dependent Valuations – Induced Topology . . . . . . . . . . . . . . . . . . 42
2.4 Approximation – Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Extension of Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 Chevalley’s Extension Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Algebraic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 The Fundamental Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4 Transcendental Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1 Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 p-Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.3 Ordered Henselian Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 The Canonical Henselian Valuation . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5 Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1 Infinite Galois Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 Unramified Extensions – First Exact Sequence . . . . . . . . . . . . . . 120
5.3 Ramified Extensions – Second Exact Sequence . . . . . . . . . . . . . . 126
5.4 Galois Characterization of Henselian Fields . . . . . . . . . . . . . . . . . 136
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6 Applications of Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.1 Artin’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2 p-Adically Closed Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 A Local-Global Principle for Quadratic Forms . . . . . . . . . . . . . . . 163
A Ultraproducts of Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B Classification of V -Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Standard Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

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yadangsmi 发表于 2009-9-10 14:05
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