Equimultiplicity and Blowing Up

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. Geometricequimultiplicity 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.