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Content and Subject Matter: This research monograph deals with two main subjects, namely the notion of equimultiplicity and the algebraic study of various graded rings in relation to blowing ups. Both subjects are clearly motivated by their use in resolving singularities of algebraic varieties, for which one of the main tools consists in blowing up the variety along an equimultiple subvariety. For equimultiplicity a unified and self-contained treatment of earlier results of two of the authors is given, establishing a notion of equimultiplicity for situations other than the classical ones. For blowing up, new results are presented on the connection with generalized Cohen-Macaulay rings. To keep this part self-contained too, a section on local cohomology and local duality for graded rings and modules is included with detailed proofs. Finally, in an appendix, the notion of equimultiplicity for complex analytic spaces is given a geometric interpretation and its equivalence to the algebraic notion is explained. The book is primarily addressed to specialists in the subject but the self-contained and unified presentation of numerous earlier results make it accessible to graduate students with basic knowledge in commutative algebra.
Contents
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 andDuality 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. 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.
