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Commutative Algebra: Constructive Methods: Finite Projective Modules 1st ed. 2015 [Hardback]

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  • Formāts: Hardback, 996 pages, height x width: 235x155 mm, 80 Illustrations, black and white; XLIX, 996 p. 80 illus., 1 Hardback
  • Sērija : Algebra and Applications 20
  • Izdošanas datums: 04-Aug-2015
  • Izdevniecība: Springer
  • ISBN-10: 9401799431
  • ISBN-13: 9789401799430
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Commutative Algebra: Constructive Methods: Finite Projective Modules 1st ed. 2015Palielināt
  • Formāts: Hardback, 996 pages, height x width: 235x155 mm, 80 Illustrations, black and white; XLIX, 996 p. 80 illus., 1 Hardback
  • Sērija : Algebra and Applications 20
  • Izdošanas datums: 04-Aug-2015
  • Izdevniecība: Springer
  • ISBN-10: 9401799431
  • ISBN-13: 9789401799430
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9789401799430.html
Keywords:
Translated from the popular French edition, this book offers a detailed introduction to various basic concepts, methods, principles, and results of commutative algebra. It takes a constructive viewpoint in commutative algebra and studies algorithmic approaches alongside several abstract classical theories. Indeed, it revisits these traditional topics with a new and simplifying manner, making the subject both accessible and innovative.

The algorithmic aspects of such naturally abstract topics as Galois theory, Dedekind rings, Prüfer rings, finitely generated projective modules, dimension theory of commutative rings, and others in the current treatise, are all analysed in the spirit of the great developers of constructive algebra in the nineteenth century.

This updated and revised edition contains over 350 well-arranged exercises, together with their helpful hints for solution. A basic knowledge of linear algebra, group theory, elementary number theory as well as the fundamentals of ring and module theory is required. Commutative Algebra: Constructive Methods will be useful for graduate students, and also researchers, instructors and theoretical computer scientists.

Recenzijas

The book treats many of the standard topics from commutative algebra and projective modules in great detail, all using constructive methods. The exhaustive bibliography draws both from classical and constructive mathematics. This will be a great addition to the bookshelves of the community of constructive mathematicians. (N. Mohan Kumar, Mathematical Reviews, October, 2016)

The book has over 350 well-arranged exercises, together with helpful hints for solution. Many methods, like lazy evaluation and dynamic evaluation, are discussed in great detail. There is a detailed flow chart on chapter dependencies and the order in which chapters should be read. The book will be useful for graduate students, as well as researchers, instructors, and theoretical computer scientists. (Naga Narayanaswamy, Computing Reviews, February, 2016)

The book under review is the faithful English translation of the original French edition published in 2011 under the same title. No doubt, the English edition will certainly increase both its international significance and its wider popularity among graduate students, researchers, instructors, and interested scientists in general. (Werner Kleinert, zbMATH 1327.13001, 2016)

I Examples
1(14)
Introduction
1(1)
1 Vector Bundles on a Smooth Compact Manifold
2(6)
Some Localizations of the Algebra of Continuous Functions
2(2)
Vector Bundles and Finitely Generated Projective Modules
4(1)
Tangent Vectors and Derivations
5(1)
Differentials and Cotangent Bundle
6(1)
The Smooth Algebraic Compact Manifolds Case
6(1)
The Differential Module and the Module of Derivations of a Finitely Presented Algebra
7(1)
2 Differential Forms with Polynomial Coefficients on a Smooth Affine Manifold
8(7)
The Module of Differential Forms with Polynomial Coefficients on the Sphere
8(1)
The Module of Differential Forms with Polynomial Coefficients on a Smooth Algebraic Manifold
9(1)
The Smooth Hypersurface Case
9(1)
The Smooth Complete Intersection Case
10(1)
The General Case
11(4)
II The Basic Local-Global Principle and Systems of Linear Equations
15(62)
Introduction
15(1)
1 Some Facts Concerning Quotients and Localizations
15(3)
2 The Basic Local-Global Principle
18(8)
Comaximal Localizations and the Local-Global Principle
18(5)
Finite Character Properties
23(2)
Forcing Comaximality
25(1)
3 Coherent Rings and Modules
26(5)
A Fundamental Notion
26(3)
Local Character of Coherence
29(1)
About the Equality and the Membership Tests
30(1)
Strongly Discrete Coherent Rings and Modules
31(1)
4 Fundamental Systems of Orthogonal Idempotents
31(4)
5 A Little Exterior Algebra
35(21)
Free Submodules as Direct Summands (Splitting Off)
35(1)
The Rank of a Free Module
36(1)
Exterior Powers of a Module
37(1)
Determinantal Ideals
38(1)
The Rank of a Matrix
39(1)
Generalized Pivot Method
40(2)
Generalized Cramer Formula
42(1)
A Magic Formula
43(1)
Generalized Inverses and Locally Simple Maps
44(2)
Grassmannians
46(1)
Injectivity and Surjectivity Criteria
47(1)
Characterization of Locally Simple Maps
48(2)
Trace, Norm, Discriminant, Transitivity
50(6)
6 Basic Local-Global Principle for Modules
56(21)
Complexes and Exact Sequences
56(2)
Localization and Exact Sequences
58(1)
Local-Global Principle for Exact Sequences of Modules
58(2)
Exercises and Problems
60(8)
Some Solutions, or Sketches of Solutions
68(7)
Bibliographic Comments
75(2)
III The Method of Undetermined Coefficients
77(96)
Introduction
77(1)
A Few Words on Finite Sets
78(1)
1 Polynomial Rings
79(4)
Partial Factorization Algorithm
79(1)
Universal Property of Polynomial Rings
79(1)
Algebraic Identities
80(2)
Symmetric Polynomials
82(1)
2 Dedekind-Mertens Lemma
83(2)
3 One of Kronecker's Theorems
85(4)
A-algebras and Integral Elements
85(1)
The Theorem
86(3)
4 The Universal Splitting Algebra for a Monic Polynomial Over a Commutative Ring (1)
89(3)
5 Discriminant, Diagonalization
92(6)
Definition of the Discriminant of a Monic Polynomial
92(1)
Diagonalization of the Matrices on a Ring
92(2)
The Generic Matrix is Diagonalizable
94(1)
An Identity Concerning Characteristic Polynomials
94(1)
An Identity Concerning Exterior Powers
95(1)
Tschirnhaus Transformation
95(1)
New Version of the Discriminant
96(1)
Discriminant of a Universal Splitting Algebra
97(1)
6 Basic Galois Theory (1)
98(11)
Factorization and Zeros
98(1)
Strictly Finite Algebras over a Discrete Field
99(2)
The Elementary Case of Galois Theory
101(5)
Construction of a Splitting Field by Means of a Galois Resolvent, Basic Galois Theory
106(3)
7 The Resultant
109(8)
Elimination Theory
109(2)
The Sylvester Matrix
111(4)
Revisiting the Discriminant
115(2)
8 Algebraic Number Theory, First Steps
117(12)
Integral Algebras
118(4)
Number Fields
122(1)
Rings of Integers of a Number Field
123(6)
9 Hilbert's Nullstellensatz
129(8)
The Algebraic Closure of Q and of Finite Fields
129(1)
The Classical Nullstellensatz (Algebraically Closed Case)
130(5)
The Formal Nullstellensatz
135(2)
10 Newton's Method in Algebra
137(36)
Exercises and Problems
140(16)
Some Solutions, or Sketches of Solutions
156(16)
Bibliographic Comments
172(1)
IV Finitely Presented Modules
173(66)
Introduction
173(1)
1 Definition, Changing Generator Set
173(4)
A Digression on the Algebraic Computation
177(1)
2 Finitely Presented Ideals
177(6)
Trivial Syzygies
178(2)
Regular Sequences
180(1)
A Geometry Example
181(2)
3 The Category of Finitely Presented Modules
183(2)
4 Stability Properties
185(11)
Coherence and Finite Presentation
186(1)
Tensor Product, Exterior Powers, Symmetrical Powers
186(5)
Changing the Base Ring
191(2)
Modules of Linear Maps
193(1)
The Local Character of the Finitely Presented Modules
194(1)
Null Tensors
195(1)
5 Classification Problems for Finitely Presented Modules
196(1)
Two Results Concerning Finitely Generated Modules
196(1)
6 Quasi-Integral Rings
197(4)
Equational Definition of pp-rings
198(1)
Elementary Local-Global Machinery no. 1
199(1)
Annihilators of the Finitely Generated Ideals in pp-rings
200(1)
Concrete Local-Global Principle for the pp-rings
200(1)
7 Bezout Rings
201(3)
Finitely Presented Modules over Valuation Rings
201(2)
Finitely Presented Modules over PIDs
203(1)
8 Zero-Dimensional Rings
204(10)
Basic Properties
204(1)
Reduced Zero-Dimensional Rings
205(1)
Characteristic Properties
205(1)
Equational Definition of Reduced Zero-Dimensional Rings
206(1)
Elementary Local-Global Machinery no. 2
207(2)
Finitely Presented Modules over Reduced Zero-Dimensional Rings
209(1)
Zero-Dimensional Polynomial Systems
210(4)
9 Fitting Ideals
214(3)
Fitting Ideals of a Finitely Presented Module
214(2)
Fitting Ideals of a Finitely Generated Module
216(1)
10 Resultant Ideal
217(22)
Exercises and Problems
219(8)
Some Solutions, or Sketches of Solutions
227(11)
Bibliographic Comments
238(1)
V Finitely Generated Projective Modules, 1
239(56)
1 Introduction
239(1)
2 Generalities
240(8)
Characteristic Properties
240(2)
Local-Global Principle
242(1)
Projective Modules and Schanuel's Lemma
243(1)
The Category of Finitely Generated Projective Modules
244(4)
3 Finitely Generated Projective Modules over Zero-Dimensional Rings
248(2)
4 Stably Free Modules
250(4)
When is a Stably Free Module Free?
251(1)
Bass' Stable Range
252(2)
5 Natural Constructions
254(2)
6 Local Structure Theorem
256(1)
7 Locally Cyclic Projective Modules and Finitely Generated Projective Ideals
257(6)
Locally Cyclic Modules
257(4)
Cyclic Projective Modules
261(1)
Locally Cyclic Projective Modules
262(1)
Finitely Generated Projective Ideals
262(1)
8 Determinant, Characteristic Polynomial, Fundamental Polynomial and Rank Polynomial
263(11)
The Determinant, the Characteristic Polynomial and the Cotransposed Endomorphism
264(2)
The Fundamental Polynomial and the Rank Polynomial
266(3)
Some Explicit Computations
269(2)
The Annihilator of a Finitely Generated Projective Module
271(1)
Canonical Decomposition of a Projective Module
272(1)
Rank Polynomial and Fitting Ideals
273(1)
9 Properties of Finite Character
274(21)
Exercises and Problems
276(8)
Some Solutions, or Sketches of Solutions
284(9)
Bibliographic Comments
293(2)
VI Strictly Finite Algebras and Galois Algebras
295(84)
Introduction
295(1)
1 Etale Algebras over a Discrete Field
296(9)
Structure Theorem for Etale Algebras
296(5)
Etale Algebras over a Separably Factorial Field
301(1)
Perfect Fields, Separable Closure and Algebraic Closure
302(3)
2 Basic Galois Theory (2)
305(2)
3 Finitely Presented Algebras
307(11)
Generalities
307(2)
The Zeros of a Polynomial System
309(2)
The Tensor Product of Two k-algebras
311(2)
Integral Algebras
313(1)
The Lying Over Lemma
313(1)
Algebras Integral over Zero-Dimensional Rings
313(1)
A Weak Nullstellensatz
314(1)
Integral Algebras over a pp-ring
315(1)
Algebras that are Finitely Presented Modules
316(1)
Integral Algebra over an Integrally Closed Ring
317(1)
4 Strictly Finite Algebras
318(3)
The Dual Module and the Trace
318(1)
Norm and Cotransposed Element
319(1)
Transitivity and Rank
320(1)
5 Dualizing Linear Forms, Strictly Finite Algebras
321(7)
Dualizing Forms
321(2)
Strictly Etale Algebras
323(2)
Tensor Products
325(1)
Integral Elements, Idempotents, Diagonalization
325(3)
6 Separable Algebras, Separability Idempotent
328(13)
Towards the Separability Idempotent
329(3)
Derivations
332(2)
Separability Idempotent of a Strictly Etale Algebra
334(2)
Separable Algebras
336(5)
7 Galois Algebras, General Theory
341(38)
Galois Correspondence, Obvious Facts
342(1)
A Natural Definition
342(2)
Dedekind's Lemma
344(1)
Artin's Theorem and First Consequences
345(10)
The Galois Correspondence When A Is Connected
355(1)
Quotients of Galois Algebras
356(1)
Exercises and Problems
357(9)
Some Solutions, or Sketches of Solutions
366(11)
Bibliographic Comments
377(2)
VII The Dynamic Method
379(56)
Introduction
379(1)
1 The Nullstellensatz Without Algebraic Closure
380(8)
The Case of an Infinite Basis Field
380(3)
Changing Variables
383(1)
The General Case
384(1)
The Actual Nullstellensatz
385(2)
Syzygies
387(1)
2 The Dynamic Method
388(3)
Splitting Fields and Galois Theory in Classical Mathematics
389(1)
Lazily Bypassing the Obstacle
390(1)
3 Introduction to Boolean Algebras
391(7)
Discrete Boolean Algebras
392(1)
Boolean Algebra of the Idempotents of a Commutative Ring
392(1)
Galoisian Elements in a Boolean Algebra
393(5)
4 The Universal Splitting Algebra (2)
398(11)
Galois Quotients of Pre-Galois Algebras
398(3)
Case Where the Boolean Algebra of a Universal Decomposition Algebra is Discrete
401(1)
Discriminant
402(2)
Fixed Points
404(1)
Separability
405(2)
Triangular Structure of Galoisian Ideals
407(2)
5 Splitting Field of a Polynomial over a Discrete Field
409(3)
"Reduced" Galois Quotients of the Universal Splitting Algebra
409(3)
Uniqueness of the Splitting Field
412(1)
6 Galois Theory of a Separable Polynomial over a Discrete Field
412(23)
Existence and Uniqueness of the Dynamic and Static Splitting Field
412(1)
Structure of the Galois Quotients of the Universal Splitting Algebra
413(1)
Where the Computations Take Place
414(1)
Changing the Base Ring, Modular Method
415(1)
Lazy Galois Theory
416(1)
The Basic Algorithm
417(1)
When a Relative Resolvent Factorizes
418(2)
When the Triangular Structure Is Missing
420(1)
Exercises and Problems
420(5)
Some Solutions, or Sketches of Solutions
425(7)
Bibliographic Comments
432(3)
VIII Flat Modules
435(42)
Introduction
435(1)
1 First Properties
435(9)
Definition and Basic Properties
435(3)
Local-Global Principle
438(1)
Other Characterization of Flatness
439(3)
Flat Quotients
442(2)
2 Finitely Generated Flat Modules
444(3)
3 Flat Principal Ideals
447(2)
4 Finitely Generated Flat Ideals
449(4)
Arithmetic Rings and Prufer Rings
451(1)
Local-Global Principle
451(1)
Local-Global Machinery
452(1)
5 Flat Algebras
453(3)
6 Faithfully Flat Algebras
456(21)
Exercises and Problems
462(3)
Some Solutions, or Sketches of Solutions
465(9)
Bibliographic Comments
474(3)
IX Local Rings, or Just About
477(46)
1 A Few Constructive Definitions
477(5)
The Jacobson Radical, Local Rings, Fields
477(3)
Prime and Maximal Ideals
480(1)
The Jacobson Radical and Units in an Integral Extension
481(1)
2 Four Important Lemmas
482(4)
3 Localization at 1 + a
486(3)
4 Examples of Local Rings in Algebraic Geometry
489(11)
Local Algebra at a Zero
489(5)
Local Ring at an Isolated Point
494(2)
Local Ring at a Non-Singular Point of a Complete Intersection Curve
496(4)
5 Decomposable Rings
500(3)
Decomposable Elements
500(2)
Lifting Idempotents
502(1)
6 Local-Global Rings
503(20)
Definitions and the Concrete Local-Global Principle
503(3)
Remarkable Local-Global Properties
506(3)
Congruential Systems
509(1)
Stability by Integral Extension
510(2)
Exercises and Problems
512(6)
Some Solutions, or Sketches of Solutions
518(4)
Bibliographic Comments
522(1)
X Finitely Generated Projective Modules, 2
523(86)
Introduction
523(1)
1 The Finitely Generated Projective Modules are Locally Free
523(7)
Complements on Exterior Powers of a Finitely Generated Projective Module
524(2)
Case of the Modules of Constant Rank
526(1)
General Case
526(2)
Modules of Constant Rank: Some Precisions
528(2)
Generic Case
530(1)
2 The Semiring H0+(A), and the Ring of Generalized Ranks H0(A)
530(5)
3 Some Applications of the Local Structure Theorem
535(5)
Trace of an Endomorphism and New Expression for the Fundamental Polynomial
535(1)
Tensor Product
536(1)
Ranks and Linear Maps
537(1)
Transitivity Formulas
537(2)
Projective Modules of Rank 1
539(1)
4 Grassmannians
540(15)
The Generic Rings Gn and Gn,k
540(5)
Affine Schemes, Tangent Spaces
545(1)
Nullstellensatz and Equivalence of Two Categories
545(2)
Affine Schemes
547(1)
Tangent Space at a Point of a Functor
548(3)
Tangent Spaces to the Grassmannians
551(1)
Projectors and Ranks
551(1)
Affine Grassmannian
551(2)
Projective Grassmannian
553(2)
5 Grothendieck and Picard Groups
555(10)
When the Projective Modules of Constant Rank are Free
556(1)
GK0(A), K0(A) and KT0(A)
557(1)
The Picard Group
558(3)
The Semirings GK0(A), GK0(Ared) and GK0(A/Rad A)
561(1)
The Milnor Square
562(3)
6 A Nontrivial Example: Identification of Points on the Affine Line
565(44)
Preliminaries
565(1)
Identification of Points Without Multiplicities
566(2)
Exercises and Problems
568(20)
Some Solutions, or Sketches of Solutions
588(20)
Bibliographic Comments
608(1)
XI Distributive Lattices Lattice-Groups
609(60)
Introduction
609(1)
1 Distributive Lattices and Boolean Algebras
610(7)
Quotient Lattices, Ideals, Filters
612(2)
Boolean Algebras
614(1)
Boolean Algebra Generated by a Distributive Lattice
615(2)
2 Lattice-Groups
617(11)
First Steps
617(2)
Remarkable Identities in the l-Groups
619(1)
Simultaneous Congruences, Covering Principle by Quotients
620(4)
Partial Decomposition, Complete Decomposition
624(4)
3 GCD-Monoids, GCD-Domains
628(6)
Non-negative Submonoid of an l-group
628(1)
GCD-Monoids
629(1)
GCD-Rings
630(1)
GCD-Domains of Dimension at Most 1
631(1)
GCD in a Polynomial Ring
632(2)
4 Zariski Lattice of a Commutative Ring
634(13)
Generalities
634(1)
Duality in the Commutative Rings
635(1)
Annihilating and Inverting Simultaneously
635(1)
Dual Definitions
636(1)
Saturated Pairs
637(2)
Closed Covering Principles
639(2)
Reduced Zero-Dimensional Closure of a Commutative Ring
641(6)
5 Entailment Relations and Heyting Algebras
647(22)
A New Look at Distributive Lattices
647(2)
Duality Between Finite Distributive Lattices and Finite Ordered Sets
649(1)
Heyting Algebras
650(2)
Exercises and Problems
652(7)
Some Solutions, or Sketches of Solutions
659(9)
Bibliographic Comments
668(1)
XII Prufer and Dedekind Rings
669(66)
Introduction
669(1)
1 Arithmetic Rings
670(7)
Locally Principal Ideals, Principal Localization Matrix
670(2)
First Properties
672(3)
Multiplicative Structure of Finitely Generated Ideals
675(2)
2 Integral Elements and Localization
677(4)
3 Prufer Rings
681(5)
Extensions of Prufer Rings
684(2)
4 Coherent Prufer Rings
686(6)
First Properties
686(1)
Kernel, Image and Cokernel of a Matrix
687(2)
Extensions of Coherent Prufer Rings
689(3)
5 pp-Rings of Dimension at Most 1
692(3)
6 Coherent Prufer Rings of Dimension ≤ 1
695(3)
When a Prufer Rings Is a Bezout Ring
695(1)
An Important Characterization
695(1)
The Structure of Finitely Presented Modules
696(2)
7 Factorization of Finitely Generated Ideals
698(37)
General Factorizations
698(1)
Factorizations in Dimension 1
699(1)
Prufer Rings Admitting Partial Factorizations
699(1)
Dedekind Rings
700(4)
Exercises and Problems
704(11)
Some Solutions, or Sketches of Solutions
715(17)
Bibliographic Comments
732(3)
XIII Krull Dimension
735(62)
Introduction
735(1)
1 Spectral Spaces
735(3)
The Zariski Lattice and the Zariski Spectrum
736(1)
Spectrum of a Distributive Lattice
736(1)
Spectral Subspaces
737(1)
A Heuristic Approach to the Krull Dimension
738(1)
2 Constructive Definition and First Consequences
738(9)
Iterated Boundaries, Singular Sequences, Complementary Sequences
742(4)
A Regular Sequence "Is Not" Singular
746(1)
Lower Bounds of the Krull Dimension
747(1)
3 A Few Elementary Properties of the Krull Dimension
747(3)
4 Integral Extensions
750(1)
5 Dimension of Geometric Rings
751(3)
Polynomial Rings over a Discrete Field
751(2)
An Interesting Corollary
753(1)
Geometric Rings
754(1)
6 Krull Dimension of Distributive Lattices
754(3)
7 Dimension of Morphisms
757(8)
Definition and First Properties
757(2)
The Minimal pp-ring Closure of a Reduced Ring
759(4)
Application
763(2)
8 Valuative Dimension
765(8)
Dimension of Valuation Rings
765(3)
Valuative Dimension of a Commutative Ring
768(1)
Valuative Dimension of a Polynomial Ring
769(4)
9 Lying Over, Going Up and Going Down
773(24)
Exercises and problems
778(8)
Some Solutions, or Sketches of Solutions
786(8)
Bibliographic Comments
794(3)
XIV The Number of Generators of a Module
797(38)
Introduction
797(1)
1 Kronecker's Theorem and Bass' Stable Range (Non-Noetherian Versions of Heitmann)
797(4)
Kronecker's Theorem
797(2)
Bass' "Stable Range" Theorem, 1
799(1)
The Local Kronecker Theorem
800(1)
2 Heitmann Dimension and Bass' Theorem
801(5)
Bass' "Stable Range" Theorem, 2
803(1)
"Improved" Variant of Kronecker's Theorem
804(2)
3 Serre's Splitting Off Theorem, The Forster-Swan Theorem, and Bass' Cancellation Theorem
806(9)
Serre's Splitting Off theorem
808(1)
The Forster-Swan Theorem
808(3)
Bass' Cancellation Theorem
811(2)
A Simple Characteristic Property for Gdim A < n
813(2)
4 Supports and n-stability
815(7)
Supports, Dimension, Stability
815(4)
Constructions and Patchings of Supports
819(3)
5 Elementary Column Operations
822(13)
With the Stability of a Support
823(2)
With the Heitmann Dimension
825(1)
Exercises and Problems
826(2)
Some Solutions, or Sketches of Solutions
828(4)
Bibliographic Comments
832(3)
XV The Local-Global Principle
835(50)
Introduction
835(1)
1 Comaximal Monoids, Coverings
836(3)
2 A Few Concrete Local-Global Principles
839(5)
Linear Systems
839(2)
Finiteness Properties for Modules
841(1)
Properties of Commutative Rings
842(1)
Concrete Local-Global Principles for Algebras
842(2)
3 A Few Abstract Local-Global Principles
844(3)
4 Concrete Patching of Objects
847(10)
Glue and Scissors
847(2)
A Simple Case
849(1)
Patching of Objects in Modules
850(2)
Patching of Modules
852(4)
Patching of Homomorphisms Between Rings
856(1)
5 The Basic Constructive Local-Global Machinery
857(5)
Decryption of Classical Proofs Using Localization at All Primes
857(2)
Examples of the Basic Local-Global Machinery
859(1)
First Example
859(1)
Second Example: A Quasi-Global Result Obtained from a Given Proof for a Local Ring
860(2)
6 Quotienting by All the Maximal Ideals
862(4)
7 Localizing at All the Minimal Prime Ideals
866(1)
8 Local-Global Principles in Depth 1
866(4)
McCoy's Theorem
868(2)
9 Local-Global Principles in Depth 2
870(15)
Patchings in Depth 2
872(3)
Exercises and Problems
875(3)
Some Solutions, or Sketches of Solutions
878(5)
Bibliographic Comments
883(2)
XVI Extended Projective Modules
885(44)
Introduction
885(1)
1 Extended Modules
885(2)
The Problem of the Extension
885(1)
The Case of the Polynomial Rings
886(1)
2 The Traverso-Swan's Theorem, Seminormal Rings
887(8)
Preliminaries
888(1)
Seminormal Rings
889(1)
The Case of Integral Rings
890(4)
General Case
894(1)
3 Patching a la Quillen-Vaserstein
895(4)
A Roitman Theorem
897(2)
4 Horrocks' Theorem
899(4)
5 Solution to Serre's Problem
903(9)
A la Quillen
903(3)
A la Suslin, Vaserstein or Rao
906(6)
6 Projective Modules Extended from Valuation or Arithmetic Rings
912(17)
The Univariate Case
912(5)
The Multivariate Case
917(5)
Conclusion: A Few Conjectures
922(1)
Exercises and Problems
923(2)
Some Solutions, or Sketches of Solutions
925(2)
Bibliographic Comments
927(2)
XVII Suslin's Stability Theorem, the Field Case
929(44)
Introduction
929(1)
1 The Elementary Group
929(4)
Transvections
929(2)
Special Matrices
931(2)
2 The Mennicke Symbol
933(2)
3 Unimodular Polynomial Vectors
935(2)
4 Suslin's and Rao's Local-Global Principles
937(36)
Exercises and Problems
941(1)
Some Solutions, or Sketches of Solutions
942(3)
Bibliographic Comments
945(2)
Annex
947(16)
Tables of Theorems
963(10)
Bibliography 973(12)
Index 985
Henri Lombardi is a researcher in constructive mathematics, real algebra and algorithmic complexity. Since 2003, with Marie-Franēoise Roy and Thierry Coquand, he has developed the international group MAP (Mathematics, Algorithms, Proofs). He has published Epistemologie mathematique, (Ellipse, 2011), and Methodes matricielles. Introduction a la complexite algebrique, (Springer, 2003, in collaboration with Jounaidi Abdeljaoued).

Claude Quitté is a researcher in effective commutative algebra, computer algebra and computer science. He has published Algorithmique algebrique, (Masson, 1991) in collaboration with Patrice Naudin.

Henri Lombardi and Claude Quitté also published together with Maria-Gema Dķaz-Toca the book Modules sur les anneaux commutatifs (Calvage & Mounet, 2014).