| Preface | p. v |
| Contents | p. vii |
| Introduction | p. xi |
| Prime Ideals and Localization | p. 1 |
| Notation and definitions | p. 1 |
| Nakayama's lemma | p. 1 |
| Localization | p. 2 |
| Noetherian rings and modules | p. 4 |
| Spectrum | p. 4 |
| The noetherian case | p. 5 |
| Associated prime ideals | p. 6 |
| Primary decompositions | p. 10 |
| Tools | p. 11 |
| Filtrations and Gradings | p. 11 |
| Filtered rings and modules | p. 11 |
| Topology defined by a filtration | p. 12 |
| Completion of filtered modules | p. 13 |
| Graded rings and modules | p. 14 |
| Where everything becomes noetherian again - <$>mathfr {q}<$> -adic filtrations | p. 17 |
| Hilbert-Samuel Polynomials | p. 19 |
| Review on integer-valued polynomials | p. 19 |
| Polynomial-like functions | p. 21 |
| The Hilbert polynomial | p. 21 |
| The Samuel polynomial | p. 24 |
| Dimension Theory | p. 29 |
| Dimension of Integral Extensions | p. 29 |
| Definitions | p. 29 |
| Cohen-Seidenberg first theorem | p. 30 |
| Cohen-Seidenberg second theorem | p. 32 |
| Dimension in Noetherian Rings | p. 33 |
| Dimension of a module | p. 33 |
| The case of noetherian local rings | p. 33 |
| Systems of parameters | p. 36 |
| Normal Rings | p. 37 |
| Characterization of normal rings | p. 37 |
| Properties of normal rings | p. 38 |
| Integral closure | p. 40 |
| Polynomial Rings | p. 40 |
| Dimension of the ring A[X1, ..., Xn] | p. 40 |
| The normalization lemma | p. 42 |
| Applications. I. Dimension in polynomial algebras | p. 44 |
| Applications. II. Integral closure of a finitely generated algebra | p. 46 |
| Applications. III. Dimension of an intersection in affine space | p. 47 |
| Homological Dimension and Depth | p. 51 |
| The Koszul Complex | p. 51 |
| The simple case | p. 51 |
| Acyclicity and functorial properties of the Koszul complex | p. 53 |
| Filtration of a Koszul complex | p. 56 |
| The depth of a module over a noetherian local ring | p. 59 |
| Cohen-Macaulay Modules | p. 62 |
| Definition of Cohen-Macaulay modules | p. 63 |
| Several characterizations of Cohen-Macaulay modules | p. 64 |
| The support of a Cohen-Macaulay module | p. 66 |
| Prime ideals and completion | p. 68 |
| Homological Dimension and Noetherian Modules | p. 70 |
| The homological dimension of a module | p. 70 |
| The noetherian case | p. 71 |
| The local case | p. 73 |
| Regular Rings | p. 75 |
| Properties and characterizations of regular local rings | p. 75 |
| Permanence properties of regular local rings | p. 78 |
| Delocalization | p. 80 |
| A criterion for normality | p. 82 |
| Regularity in ring extensions | p. 83 |
| Minimal Resolutions | p. 84 |
| Definition of minimal resolutions | p. 84 |
| Application | p. 85 |
| The case of the Koszul complex | p. 86 |
| Positivity of Higher Euler-Poincare Characteristics | p. 88 |
| Graded-polynomial Algebras | p. 91 |
| Notation | p. 91 |
| Graded-polynomial algebras | p. 92 |
| A characterization of graded-polynomial algebras | p. 93 |
| Ring extensions | p. 93 |
| Application: the Shephard-Todd theorem | p. 95 |
| Multiplicities | p. 99 |
| Multiplicity of a Module | p. 99 |
| The group of cycles of a ring | p. 99 |
| Multiplicity of a module | p. 100 |
| Intersection Multiplicity of Two Modules | p. 101 |
| Reduction to the diagonal | p. 101 |
| Completed tensor products | p. 102 |
| Regular rings of equal characteristic | p. 106 |
| Conjectures | p. 107 |
| Regular rings of unequal characteristic (unramified case) | p. 108 |
| Arbitrary regular rings | p. 110 |
| Connection with Algebraic Geometry | p. 112 |
| Tor-formula | p. 112 |
| Cycles on a non-singular affine variety | p. 113 |
| Basic formulae | p. 114 |
| Proof of theorem 1 | p. 116 |
| Rationality of intersections | p. 116 |
| Direct images | p. 117 |
| Pull-backs | p. 117 |
| Extensions of intersection theory | p. 119 |
| Bibliography | p. 123 |
| Index | p. 127 |
| Index of Notation | p. 129 |
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