I - Review of Multiplicity Theory -- §1 The multiplicity symbol -- §2 Hilbert functions -- §3 Generalized multiplicities and Hilbert functions -- §4 Reductions and integral closure of ideals -- §5 Faithfully flat extensions -- §6 Projection formula and criterion for multiplicity one -- §7 Examples -- II - Z-Graded Rings and Modules -- §8 Associated graded rings and Rees algebras -- §9 Dimension -- §10 Homogeneous parameters -- §11 Regular sequences on graded modules -- §12 Review on blowing up -- §13 Standard bases -- §14 Examples -- Appendix - Homogeneous subrings of a homogeneous ring -- III - Asymptotic Sequences and Quasi-Unmixed Rings -- §15 Auxiliary results on integral dependence of ideals -- §16 Associated primes of the integral closure of powers of an ideal -- §17 Asymptotic sequences -- §18 Quasi-unmixed rings -- §19 The theorem of Rees-Böger -- IV - Various Notions of Equimultiple and Permissible Ideals -- §20 Reinterpretation of the theorem of Rees-Böger -- §21 Hironaka-Grothendieck homomorphism -- §22 Projective normal flatness and numerical characterization of permissibility -- §23 Hierarchy of equimultiplicity and permissibility -- §24 Open conditions and transitivity properties -- V - Equimultiplicity and Cohen-Macaulay Property of Blowing Up Rings -- §25 Graded Cohen-Macaulay rings -- §26 The case of hypersurfaces -- §27 Transitivity of Cohen-Macaulayness of Rees rings -- Appendix (K. Yamagishi and U. Orbanz) - Homogeneous domains of minimal multiplicity -- VI - Certain Inequalities and Equalities of Hilbert Functions and Multiplicities -- §28 Hyperplane sections -- §29 Quadratic transformations -- §30 Semicontinuity -- §31 Permissibility and blowing up of ideals -- §32 Transversal ideals and flat families -- VII - Local Cohomology and Duality of Graded Rings -- §33 Review on graded modules -- §34 Matlis duality -- I: Local case -- II: Graded case -- §35 Local cohomology -- §36 Local duality for graded rings -- Appendix - Characterization of local Gorenstein-rings by its injective dimension -- VIII - Generalized Cohen-Macaulay Rings and Blowing Up -- §37 Finiteness of local cohomology -- §38 Standard system of parameters -- §39 The computation of local cohomology of generalized Cohen-Macaulay rings -- §40 Blowing up of a standard system of parameters -- §41 Standard ideals on Buchsbaum rings -- §42 Examples -- IX - Applications of Local Cohomology to the Cohen-Macaulay Behaviour of Blowing Up Rings -- §43 Generalized Cohen-Macaulay rings with respect to an ideal -- §44 The Cohen-Macaulay property of Rees algebras -- §45 Rees algebras of m-primary ideals -- §46 The Rees algebra of parameter ideals -- §47 The Rees algebra of powers of parameter ideals -- §48 Applications to rings of low multiplicity -- Examples -- Appendix (B. Moonen) - Geometric Equimultiplicity -- I. Local Complex Analytic Geometry -- § 1. Local analytic algebras -- 1.1. Formal power series -- 1.2. Convergent power series -- 1.3. Local analytic k-algebras -- § 2. Local Weierstraß Theory I: The Division Theorem -- 2.1. Ordering the monomials -- 2.2. Monomial ideals and leitideals -- 2.3. The Division Theorem -- 2.4. Division with respect to an ideal; standard bases -- 2.5. Applications of standard bases: the General Weierstraß Preparation Theorem and the Krull Intersection Theorem -- 2.6. The classical Weierstraß Theorems -- § 3. Complex spaces and the Equivalence Theorem -- 3.1. Complex spaces -- 3.3. The Equivalence Theorem -- 3.4. The analytic spectrum -- § 4. Local Weierstraß Theory II: Finite morphisms -- 4.1. Finite morphisms -- 4.2. Weierstraß maps -- 4.3. The Finite Mapping Theorem -- 4.4. The Integrality Theorem -- § 5. Dimension and Nullstellensatz -- 5.1. Local dimension -- 5.2. Active elements and the Active Lemma -- 5.3. The Rückert Nullstellensatz -- 5.4. Analytic sets and local decomposition -- § 6. The Local Representation Theorem for comple space-germs (Noether normalization) -- 6.1. Openness and dimension -- 6.2. Geometric interpretation of the local dimension and of a system of parameters; algebraic Noether normalization -- 6.3. The Local Representation Theorem; geometric Noether normalization -- § 7. Coherence -- 7.1. Coherent sheaves -- 7.2. Nonzerodivisors -- 7.3. Purity of dimension and local decomposition -- 7.4. Reduction -- II. Geometric Multiplicity -- § 1. Compact Stein neighbourhoods -- 1.1. Coherent sheaves on closed subsets -- 1.2. Stein subsets -- 1.3. Compact Stein subsets and the Flatness Theorem -- 1.4. Existence of compact Stein neighbourhoods -- § 2. Local mapping degree -- 2.1. Local decomposition revisited -- 2.2. Local mapping degree -- § 3. Geometric multiplicity -- 3.1. The tangent cone -- 3.2. Multiplicity -- § 4. The geometry of Samuel multiplicity -- 4.1. Degree of a projective variety -- 4.2. Hilbert functions -- 4.3. A generalization -- 4.4. Samuel multiplicity -- § 5. Algebraic multiplicity -- 5.1. Algebraic degree -- 5.2. Algebraic multiplicity -- III. Geometric Equimultiplicity -- § 1. Normal flatness and pseudoflatness -- 1.1. Generalities from Complex Analytic Geometry -- 1.2. The analytic and projective analytic spectrum -- 1.3. Flatness of admissible graded algebras -- 1.4 The normal cone, normal flatness, and normal pseudoflatness -- § 2. Geometric equimultiplicity along a smooth subspace -- 2.1. Zariski equimultiplicity -- 2.2. The Hironaka-Schickhoff Theorem -- § 3. Geometric equimultiplicity along a general subspace -- 3.1. Zariski equimultiplicity -- 3.2. Normal pseudoflatness -- References -- References - Chapter I -- References - Chapter II -- References - Appendix Chapter II -- References - Chapter III -- References - Chapter IV -- References - Chapter V -- References - Appendix Chapter V -- References - Chapter VI -- References - Chapter VII -- References - Chapter VIII -- References - Chapter IX -- Bibliography to the Appendix Geometric Equimultiplicity -- General Index.