| Preface |
| Contents |
| Introduction |
| Prime Ideals and Localization / I： |
| Notation and definitions / 1： |
| Nakayama's lemma / 2： |
| Localization / 3： |
| Noetherian rings and modules / 4： |
| Spectrum / 5： |
| The noetherian case / 6： |
| Associated prime ideals / 7： |
| Primary decompositions / 8： |
| Tools / II： |
| Filtrations and Gradings / A： |
| Filtered rings and modules |
| Topology defined by a filtration |
| Completion of filtered modules |
| Graded rings and modules |
| Where everything becomes noetherian again - <$>\mathfr {q}<$> -adic filtrations |
| Hilbert-Samuel Polynomials / B： |
| Review on integer-valued polynomials |
| Polynomial-like functions |
| The Hilbert polynomial |
| The Samuel polynomial |
| Dimension Theory / III： |
| Dimension of Integral Extensions |
| Definitions |
| Cohen-Seidenberg first theorem |
| Cohen-Seidenberg second theorem |
| Dimension in Noetherian Rings |
| Dimension of a module |
| The case of noetherian local rings |
| Systems of parameters |
| Normal Rings / C： |
| Characterization of normal rings |
| Properties of normal rings |
| Integral closure |
| Polynomial Rings / D： |
| Dimension of the ring A[X1, ..., Xn] |
| The normalization lemma |
| Applications. I. Dimension in polynomial algebras |
| Applications. II. Integral closure of a finitely generated algebra |
| Applications. III. Dimension of an intersection in affine space |
| Homological Dimension and Depth / IV： |
| The Koszul Complex |
| The simple case |
| Acyclicity and functorial properties of the Koszul complex |
| Filtration of a Koszul complex |
| The depth of a module over a noetherian local ring |
| Cohen-Macaulay Modules |
| Definition of Cohen-Macaulay modules |
| Several characterizations of Cohen-Macaulay modules |
| The support of a Cohen-Macaulay module |
| Prime ideals and completion |
| Homological Dimension and Noetherian Modules |
| The homological dimension of a module |
| The local case |
| Regular Rings |
| Properties and characterizations of regular local rings |
| Permanence properties of regular local rings |
| Delocalization |
| A criterion for normality |
| Regularity in ring extensions |
| Minimal Resolutions / Appendix I： |
| Definition of minimal resolutions |
| Application |
| The case of the Koszul complex |
| Positivity of Higher Euler-Poincare Characteristics / Appendix II： |
| Graded-polynomial Algebras / Appendix III： |
| Notation |
| Graded-polynomial algebras |
| A characterization of graded-polynomial algebras |
| Ring extensions |
| Application: the Shephard-Todd theorem |
| Multiplicities / V： |
| Multiplicity of a Module |
| The group of cycles of a ring |
| Multiplicity of a module |
| Intersection Multiplicity of Two Modules |
| Reduction to the diagonal |
| Completed tensor products |
| Regular rings of equal characteristic |
| Conjectures |
| Regular rings of unequal characteristic (unramified case) |
| Arbitrary regular rings |
| Connection with Algebraic Geometry |
| Tor-formula |
| Cycles on a non-singular affine variety |
| Basic formulae |
| Proof of theorem 1 |
| Rationality of intersections |
| Direct images |
| Pull-backs |
| Extensions of intersection theory |
| Bibliography |
| Index |
| Index of Notation |
図書
