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 |