Equimultiplicity and Blowing Up

1. Front Matter

   Pages I-XVII

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2. Review of Multiplicity Theory

   • Manfred Herrmann, Ulrich Orbanz, Shin Ikeda

   Pages 1-43

3. Z-Graded Rings and Modules

   • Manfred Herrmann, Ulrich Orbanz, Shin Ikeda

   Pages 44-116

4. Asymptotic Sequences and Quasi-Unmixed Rings

   • Manfred Herrmann, Ulrich Orbanz, Shin Ikeda

   Pages 117-151

5. Various Notions of Equimultiple and Permissible Ideals

   • Manfred Herrmann, Ulrich Orbanz, Shin Ikeda

   Pages 152-203

6. Equimultiplicity and Cohen-Macaulay Property of Blowing Up Rings

   • Manfred Herrmann, Ulrich Orbanz, Shin Ikeda

   Pages 204-239

7. Certain Inequalities and Equalities of Hilbert Functions and Multiplicities

   • Manfred Herrmann, Ulrich Orbanz, Shin Ikeda

   Pages 240-269

8. Local Cohomology and Duality of Graded Rings

   • Manfred Herrmann, Ulrich Orbanz, Shin Ikeda

   Pages 270-325

9. Generalized Cohen-Macaulay Rings and Blowing Up

   • Manfred Herrmann, Ulrich Orbanz, Shin Ikeda

   Pages 326-396

10. Applications of Local Cohomology to the Cohen-Macaulay-behaviour of Blowing Up Rings

    • Manfred Herrmann, Ulrich Orbanz, Shin Ikeda

    Pages 397-446

11. Back Matter

    Pages 447-629

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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.

• Blowing up
• Dimension
• Divisor
• Equivalence
• Grad
• Grothendieck topology
• algebra
• algebraic varieties
• cohomology
• commutative property
• duality
• knowledge
• proof
• sheaves
• tool

• Mathematisches Institut der Universität zu Köln, Köln 41, Germany

  Manfred Herrmann, Ulrich Orbanz

• Mathematical Department Gifu College of Education, Gifu, Japan

  Shin Ikeda

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