| Preface |
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vii | |
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Rings, Ideals and Standard Bases |
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1 | (88) |
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Rings, Polynomials and Ring Maps |
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1 | (8) |
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9 | (10) |
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Ideals and Quotient Rings |
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19 | (11) |
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Local Rings and Localization |
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30 | (8) |
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Rings Associated to Monomial Orderings |
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38 | (6) |
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Normal Forms and Standard Bases |
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44 | (9) |
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The Standard Basis Algorithm |
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53 | (14) |
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Operations on Ideals and Their Computation |
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67 | (22) |
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67 | (2) |
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Intersection with Subrings |
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69 | (2) |
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Zariski Closure of the Image |
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71 | (3) |
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Solvability of Polynomial Equations |
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74 | (1) |
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Solving Polynomial Equations |
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74 | (3) |
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77 | (2) |
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79 | (1) |
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79 | (2) |
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81 | (3) |
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84 | (1) |
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Algebraic Dependence and Subalgebra Membership |
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85 | (4) |
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89 | (102) |
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Modules, Submodules and Homomorphisms |
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89 | (23) |
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112 | (4) |
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Standard Bases for Modules |
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116 | (10) |
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Exact Sequences and free Resolutions |
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126 | (11) |
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Computing Resolutions and the Syzygy Theorem |
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137 | (14) |
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Modules over Principal Ideal Domains |
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151 | (14) |
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165 | (10) |
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Operations on Modules and Their Computation |
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175 | (16) |
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Module Membership Problem |
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175 | (2) |
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Intersection with free Submodules |
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177 | (1) |
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Intersection of Submodules |
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178 | (1) |
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179 | (2) |
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Radical and Zerodivisors of Modules |
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181 | (2) |
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183 | (1) |
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Kernel of a Module Homomorphism |
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184 | (1) |
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Solving Systems of Linear Equations |
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185 | (6) |
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Noether Normalization and Applications |
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191 | (48) |
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Finite and Integral Extensions |
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191 | (7) |
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198 | (7) |
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205 | (5) |
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210 | (5) |
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215 | (9) |
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An Algorithm to Compute the Normalization |
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224 | (7) |
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231 | (8) |
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Primary Decomposition and Related Topics |
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239 | (36) |
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The Theory of Primary Decomposition |
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239 | (5) |
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Zero--dimensional Primary Decomposition |
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244 | (9) |
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Higher Dimensional Primary Decomposition |
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253 | (5) |
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The Equidimensional Part of an Ideal |
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258 | (3) |
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261 | (4) |
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265 | (10) |
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Hilbert Function and Dimension |
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275 | (38) |
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The Hilbert Function and the Hilbert Polynomial |
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275 | (4) |
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Computation of the Hilbert-Poincare Series |
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279 | (4) |
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Properties of the Hilbert Polynomial |
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283 | (7) |
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Filtrations and the Lemma of Artin--Rees |
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290 | (2) |
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The Hilbert-Samuel Function |
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292 | (7) |
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Characterization of the Dimension of Local Rings |
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299 | (6) |
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305 | (8) |
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313 | (22) |
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Formal Power Series Rings |
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313 | (4) |
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Weierstraß Preparation Theorem |
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317 | (8) |
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325 | (6) |
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331 | (4) |
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335 | (66) |
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335 | (6) |
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341 | (5) |
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346 | (11) |
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Local Criteria for Flatness |
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357 | (5) |
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Flatness and Standard Bases |
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362 | (7) |
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369 | (13) |
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382 | (6) |
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Further Characterization of Cohen--Macaulayness |
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388 | (8) |
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Homological Characterization of Regular Rings |
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396 | (5) |
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401 | (152) |
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401 | (94) |
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A.1 Introduction by Pictures |
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401 | (9) |
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A.2 Affine Algebraic Varieties |
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410 | (11) |
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A.3 Spectrum and Affine Schemes |
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421 | (8) |
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429 | (12) |
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A.5 Projective Schemes and Varieties |
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441 | (5) |
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A.6 Morphisms Between Varieties |
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446 | (8) |
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A.7 Projective Morphisms and Elimination |
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454 | (14) |
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A.8 Local Versus Global Properties |
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468 | (14) |
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482 | (13) |
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B. SINGULAR --- A Short Introduction |
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495 | (58) |
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B.1 Downloading Instructions |
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495 | (3) |
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498 | (4) |
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B.3 Procedures and Libraries |
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502 | (2) |
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504 | (6) |
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510 | (16) |
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526 | (1) |
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527 | (1) |
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528 | (16) |
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544 | (4) |
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B.10 Singular and Mathematica |
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548 | (1) |
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549 | (4) |
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553 | (10) |
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563 | (4) |
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567 | (18) |
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585 | (2) |
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587 | |
Lisainfo e-raamatute kohta