Ebook: Equimultiplicity and Blowing Up: An Algebraic Study

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.

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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.

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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.
Content:
Front Matter....Pages I-XVII
Review of Multiplicity Theory....Pages 1-43
Z-Graded Rings and Modules....Pages 44-116
Asymptotic Sequences and Quasi-Unmixed Rings....Pages 117-151
Various Notions of Equimultiple and Permissible Ideals....Pages 152-203
Equimultiplicity and Cohen-Macaulay Property of Blowing Up Rings....Pages 204-239
Certain Inequalities and Equalities of Hilbert Functions and Multiplicities....Pages 240-269
Local Cohomology and Duality of Graded Rings....Pages 270-325
Generalized Cohen-Macaulay Rings and Blowing Up....Pages 326-396
Applications of Local Cohomology to the Cohen-Macaulay-behaviour of Blowing Up Rings....Pages 397-446
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.
Content:
Front Matter....Pages I-XVII
Review of Multiplicity Theory....Pages 1-43
Z-Graded Rings and Modules....Pages 44-116
Asymptotic Sequences and Quasi-Unmixed Rings....Pages 117-151
Various Notions of Equimultiple and Permissible Ideals....Pages 152-203
Equimultiplicity and Cohen-Macaulay Property of Blowing Up Rings....Pages 204-239
Certain Inequalities and Equalities of Hilbert Functions and Multiplicities....Pages 240-269
Local Cohomology and Duality of Graded Rings....Pages 270-325
Generalized Cohen-Macaulay Rings and Blowing Up....Pages 326-396
Applications of Local Cohomology to the Cohen-Macaulay-behaviour of Blowing Up Rings....Pages 397-446
Back Matter....Pages 447-629
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