Author | Herrmann, Manfred. author |
---|---|
Title | Equimultiplicity and Blowing Up [electronic resource] : An Algebraic Study / by Manfred Herrmann, Ulrich Orbanz, Shin Ikeda |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1988 |
Connect to | http://dx.doi.org/10.1007/978-3-642-61349-4 |
Descript | XVII, 629 p. online resource |
I โ 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 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 Homogeneous subrings of a homogeneous ring -- III โ 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 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 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 Homogeneous domains of minimal multiplicity -- VI โ 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 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 Characterization of local Gorenstein-rings by its injective dimension -- VIII โ 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 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 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 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 Preparation Theorem and the Krull Intersection Theorem -- 2.6. The classical Weierstraร 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 Theory II: Finite morphisms -- 4.1. Finite morphisms -- 4.2. Weierstraร 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 Chapter I -- References โ Chapter II -- References โ Appendix Chapter II -- References โ Chapter III -- References โ Chapter IV -- References โ Chapter V -- References โ Appendix Chapter V -- References โ Chapter VI -- References โ Chapter VII -- References โ Chapter VIII -- References โ Chapter IX -- Bibliography to the Appendix Geometric Equimultiplicity -- General Index