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AuthorHerrmann, Manfred. author
TitleEquimultiplicity and Blowing Up [electronic resource] : An Algebraic Study / by Manfred Herrmann, Ulrich Orbanz, Shin Ikeda
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1988
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Descript XVII, 629 p. online resource


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


I โ{128}{148} Review of Multiplicity Theory -- ยง1 The multiplicity symbol -- ยง2 Hilbert functions -- ยง3 Generalized multiplicities and Hilbert functions -- ยง4 Reductions and integral closure of ideals -- ยง5 Faithfully flat extensions -- ยง6 Projection formula and criterion for multiplicity one -- ยง7 Examples -- II โ{128}{148} Z-Graded Rings and Modules -- ยง8 Associated graded rings and Rees algebras -- ยง9 Dimension -- ยง10 Homogeneous parameters -- ยง11 Regular sequences on graded modules -- ยง12 Review on blowing up -- ยง13 Standard bases -- ยง14 Examples -- Appendix โ{128}{148} Homogeneous subrings of a homogeneous ring -- III โ{128}{148} Asymptotic Sequences and Quasi-Unmixed Rings -- ยง15 Auxiliary results on integral dependence of ideals -- ยง16 Associated primes of the integral closure of powers of an ideal -- ยง17 Asymptotic sequences -- ยง18 Quasi-unmixed rings -- ยง19 The theorem of Rees-Bรถger -- IV โ{128}{148} Various Notions of Equimultiple and Permissible Ideals -- ยง20 Reinterpretation of the theorem of Rees-Bรถger -- ยง21 Hironaka-Grothendieck homomorphism -- ยง22 Projective normal flatness and numerical characterization of permissibility -- ยง23 Hierarchy of equimultiplicity and permissibility -- ยง24 Open conditions and transitivity properties -- V โ{128}{148} Equimultiplicity and Cohen-Macaulay Property of Blowing Up Rings -- ยง25 Graded Cohen-Macaulay rings -- ยง26 The case of hypersurfaces -- ยง27 Transitivity of Cohen-Macaulayness of Rees rings -- Appendix (K. Yamagishi and U. Orbanz) โ{128}{148} Homogeneous domains of minimal multiplicity -- VI โ{128}{148} Certain Inequalities and Equalities of Hilbert Functions and Multiplicities -- ยง28 Hyperplane sections -- ยง29 Quadratic transformations -- ยง30 Semicontinuity -- ยง31 Permissibility and blowing up of ideals -- ยง32 Transversal ideals and flat families -- VII โ{128}{148} Local Cohomology and Duality of Graded Rings -- ยง33 Review on graded modules -- ยง34 Matlis duality -- I: Local case -- II: Graded case -- ยง35 Local cohomology -- ยง36 Local duality for graded rings -- Appendix โ{128}{148} Characterization of local Gorenstein-rings by its injective dimension -- VIII โ{128}{148} Generalized Cohen-Macaulay Rings and Blowing Up -- ยง37 Finiteness of local cohomology -- ยง38 Standard system of parameters -- ยง39 The computation of local cohomology of generalized Cohen-Macaulay rings -- ยง40 Blowing up of a standard system of parameters -- ยง41 Standard ideals on Buchsbaum rings -- ยง42 Examples -- IX โ{128}{148} Applications of Local Cohomology to the Cohen-Macaulay Behaviour of Blowing Up Rings -- ยง43 Generalized Cohen-Macaulay rings with respect to an ideal -- ยง44 The Cohen-Macaulay property of Rees algebras -- ยง45 Rees algebras of m-primary ideals -- ยง46 The Rees algebra of parameter ideals -- ยง47 The Rees algebra of powers of parameter ideals -- ยง48 Applications to rings of low multiplicity -- Examples -- Appendix (B. Moonen) โ{128}{148} Geometric Equimultiplicity -- I. Local Complex Analytic Geometry -- ยง 1. Local analytic algebras -- 1.1. Formal power series -- 1.2. Convergent power series -- 1.3. Local analytic k-algebras -- ยง 2. Local Weierstraร{159} 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 Weierstraร{159} Preparation Theorem and the Krull Intersection Theorem -- 2.6. The classical Weierstraร{159} Theorems -- ยง 3. Complex spaces and the Equivalence Theorem -- 3.1. Complex spaces -- 3.3. The Equivalence Theorem -- 3.4. The analytic spectrum -- ยง 4. Local Weierstraร{159} Theory II: Finite morphisms -- 4.1. Finite morphisms -- 4.2. Weierstraร{159} maps -- 4.3. The Finite Mapping Theorem -- 4.4. The Integrality Theorem -- ยง 5. Dimension and Nullstellensatz -- 5.1. Local dimension -- 5.2. Active elements and the Active Lemma -- 5.3. The Rรผckert Nullstellensatz -- 5.4. Analytic sets and local decomposition -- ยง 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 -- ยง 7. Coherence -- 7.1. Coherent sheaves -- 7.2. Nonzerodivisors -- 7.3. Purity of dimension and local decomposition -- 7.4. Reduction -- II. Geometric Multiplicity -- ยง 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 -- ยง 2. Local mapping degree -- 2.1. Local decomposition revisited -- 2.2. Local mapping degree -- ยง 3. Geometric multiplicity -- 3.1. The tangent cone -- 3.2. Multiplicity -- ยง 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 -- ยง 5. Algebraic multiplicity -- 5.1. Algebraic degree -- 5.2. Algebraic multiplicity -- III. Geometric Equimultiplicity -- ยง 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 -- ยง 2. Geometric equimultiplicity along a smooth subspace -- 2.1. Zariski equimultiplicity -- 2.2. The Hironaka-Schickhoff Theorem -- ยง 3. Geometric equimultiplicity along a general subspace -- 3.1. Zariski equimultiplicity -- 3.2. Normal pseudoflatness -- References -- References โ{128}{148} Chapter I -- References โ{128}{148} Chapter II -- References โ{128}{148} Appendix Chapter II -- References โ{128}{148} Chapter III -- References โ{128}{148} Chapter IV -- References โ{128}{148} Chapter V -- References โ{128}{148} Appendix Chapter V -- References โ{128}{148} Chapter VI -- References โ{128}{148} Chapter VII -- References โ{128}{148} Chapter VIII -- References โ{128}{148} Chapter IX -- Bibliography to the Appendix Geometric Equimultiplicity -- General Index

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