Equimultiplicity and Blowing up: An Algebraic Study

Equimultiplicity and Blowing up......Page 1
Preface......Page 5
Table of contents......Page 10
§1. The multiplicity symbol......Page 15
§2. Hilbert functions......Page 20
§3. Generalized multiplicities and Hilbert functions......Page 24
§4. Reductions and integral closure of ideals......Page 30
§5. Faithfully flat extensions......Page 39
§6. Projection formula and criterion for multiplicity one......Page 41
§7. Examples......Page 48
References......Page 57
§8. Associated graded rings and Rees algebras......Page 58
§9. Dimension......Page 63
§10. Homogeneous parameters......Page 69
§11. Regular sequences on graded modules......Page 82
§12. Review on blowing up......Page 91
§13. Standard bases......Page 102
§14. Examples......Page 114
Appendix: Homogeneous subrings of a homogeneous ring......Page 126
References......Page 129
§15. Auxiliary results on integral dependence of ideals......Page 131
§16. Associated primes of the integral closure of powers of an ideal......Page 136
§17. Asymptotic sequences......Page 147
§18. Quasi-unmixed rings......Page 151
§19. The theorem of Rees-Böger......Page 160
References......Page 165
§20. Reinterpretation of the theorem of Rees-Böger......Page 166
§21. Hironaka-Grothendieck homomorphism......Page 173
§22. Projective normal flatness and numerical characterization of permissibility......Page 180
§23. Hierarchy of equimultiplicity and permissibility......Page 196
§24. Open conditions and transitivity properties......Page 208
References......Page 217
V. Equimultiplicity and Cohen-Macaulay property of blowing up rings......Page 218
§25. Graded Cohen-Macaulay rings......Page 219
§26. The case of hypersurfaces......Page 226
§27. Transitivity of Cohen-Macaulayness of Rees rings......Page 237
References......Page 243
Appendix: Homogeneous domains of minimal multiplicity......Page 244
References......Page 253
§28. Hyperplane sections......Page 254
§29. Quadratic transformations......Page 257
§30. Semicontinuity......Page 264
§31. Permissibility and blowing up of ideals......Page 267
§32. Transversal ideals and flat modules......Page 272
References......Page 283
§33. Review on graded modules......Page 284
§34. Matlis duality......Page 303
§35. Local cohomology......Page 309
§36. Local duality for graded rings......Page 324
Appendix: Characterization of local Gorenstein-rings by its injective dimension......Page 334
References......Page 338
§37. Finiteness of local cohomology......Page 340
§38. Standard system of parameters......Page 349
§39. The computation of local cohomology of generalized Cohen-Macaulay rings......Page 364
§40. Blowing up of a standard system of parameters......Page 367
§41. Standard ideals and Buchsbaum rings......Page 381
§42. Examples......Page 404
References......Page 409
§43. Generalized Cohen-Macaulay-rings with respect to an ideal......Page 411
§44. The Cohen-Macaulay property of Rees algebras......Page 414
§45. Rees algebras of π-primary ideals......Page 418
§46. The Rees algebra of parameter ideals......Page 429
§47. The Rees algebra of powers of parameter ideals......Page 432
§48. Applications to rings of low multiplicity. Examples.......Page 435
References......Page 459
Appendix: Geometric equimultiplicity......Page 461
Introduction......Page 462
1.1. Formal power series......Page 466
1.2. Convergent power series......Page 468
1.3. Local analytic K-algebras......Page 470
2.1. Ordering the monomials......Page 472
2.2. Monomial ideals and leitideals......Page 473
2.3. The division theorem......Page 475
2.4. Division with respect to an ideal; standard bases......Page 480
2.5. Applications of standard bases: The general Weierstrass preparation theorem and the Krull intersection theorem......Page 481
2.6. The classical Weierstrass theorems......Page 482
§3. Complex spaces and the equivalence theorem......Page 483
3.1. Complex spaces......Page 484
3.2. Constructions in cpl......Page 486
3.3. The equivalence theorem......Page 491
3.4. The analytic spectrum......Page 494
§4. Local Weierstrass theory II: Finite morphisms......Page 495
4.2. Weierstrass maps......Page 496
4.3. The finite mapping theorem......Page 498
4.4. The integrality theorem......Page 502
§5. Dimensions and Nullstellensatz......Page 505
5.1. Local dimension......Page 506
5.2. Active elements and the active lemma......Page 507
5.3. The Rückert Nullstellensatz......Page 508
5.4. Analytic sets and local decomposition......Page 510
6.1. Openness and dimension......Page 512
6.2. Geometric interpretation of the local dimension and of a system of parameters; algebraic Noether normalization......Page 513
6.3. The local representation theorem; geometric Noether normalization......Page 515
7.1. Coherent sheaves......Page 520
7.3. Purity of dimension and local decomposition......Page 521
7.4. Reduction......Page 522
II. Geometric multiplicity......Page 524
1.2. Stein subsets......Page 528
1.3. Compact Stein subsets and the flatness theorem......Page 529
1.4. Existence of compact Stein neighbourhoods......Page 530
2.1. Local decomposition revisited......Page 534
2.2. Local mapping degree......Page 537
§3. Geometric multiplicity......Page 542
3.1. The tangent cone......Page 543
3.2. Multiplicity......Page 545
4.1. Degree of a projective variety......Page 550
4.2. Hilbert functions......Page 559
4.3. A generalization......Page 562
5.1. Algebraic degree......Page 563
5.2. Algebraic multiplicity......Page 569
III. Geometric equimultiplicity......Page 571
1.1. Generalities from complex analytic geometry......Page 572
1.2. The analytic and projective analytic spectrum......Page 576
1.3. Flatness of admissible graded algebras......Page 581
1.4. The normal cone, normal flatness, and normal pseudoflatness......Page 584
§2. Geometric equimultiplicity along a smooth subspace......Page 591
2.1. Zariski-equimultiplicity......Page 592
2.2. The Hironaka-Schickhoff-theorem......Page 595
§3. Geometric equimultiplicity along a general subspace......Page 620
3.1. Zariski-equimultiplicity......Page 621
3.2. Normal pseudoflatness......Page 622
Bibliography......Page 630
General Index......Page 635