| Preface |
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ix | (5) |
| Notation and conventions |
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xiv | |
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1 The local cohomology functors |
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1 | (16) |
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1 | (2) |
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1.2 Local cohomology modules |
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3 | (7) |
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1.3 Connected sequences of functors |
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10 | (7) |
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2 Torsion modules and ideal transforms |
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17 | (36) |
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18 | (4) |
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22 | (16) |
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2.3 Geometrical significance |
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38 | (9) |
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3 The Mayer-Vietoris Sequence |
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47 | (19) |
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3.1 Comparison of systems of ideals |
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48 | (3) |
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3.2 Construction of the sequence |
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51 | (4) |
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55 | (4) |
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59 | (7) |
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66 | (16) |
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67 | (4) |
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4.2 The Independence Theorem |
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71 | (4) |
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4.3 The Flat Base Change Theorem |
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75 | (7) |
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82 | (20) |
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5.1 Use of Cech complexes |
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83 | (11) |
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5.2 Use of Koszul complexes |
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94 | (8) |
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6 Fundamental vanishing theorems |
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102 | (21) |
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6.1 Grothendieck's Vanishing Theorem |
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103 | (4) |
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6.2 Connections with grade |
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107 | (5) |
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6.3 Exactness of ideal transforms |
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112 | (5) |
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6.4 An Affineness Criterion due to Serre |
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117 | (6) |
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7 Artinian local cohomology modules |
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123 | (13) |
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123 | (4) |
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7.2 Secondary representation |
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127 | (4) |
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7.3 The Non-vanishing Theorem again |
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131 | (5) |
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8 The Lichtenbaum-Hartshorne Theorem |
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136 | (16) |
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137 | (7) |
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144 | (8) |
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9 The Annihilator and Finiteness Theorems |
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152 | (27) |
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9.1 Finiteness dimensions |
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152 | (3) |
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155 | (4) |
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159 | (4) |
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9.4 The second inequality |
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163 | (7) |
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170 | (5) |
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175 | (4) |
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179 | (18) |
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10.1 Indecomposable injective modules |
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179 | (6) |
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185 | (12) |
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197 | (19) |
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11.1 Minimal injective resolutions |
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198 | (3) |
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11.2 Local Duality Theorems |
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201 | (6) |
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207 | (9) |
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12 Foundations in the graded case |
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216 | (21) |
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12.1 (*)Injective modules |
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217 | (4) |
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12.2 The (*)restriction property |
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221 | (4) |
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225 | (4) |
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12.4 Some examples and applications |
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229 | (8) |
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13 Graded versions of basic theorems |
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237 | (28) |
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13.1 Fundamental theorems |
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237 | (9) |
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13.2 (*)Indecomposable *injective modules |
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246 | (7) |
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13.3 (*)Canonical modules |
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253 | (5) |
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13.4 Graded local duality |
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258 | (7) |
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14 Links with projective varieties |
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265 | (12) |
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14.1 Affine algebraic cones |
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265 | (4) |
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14.2 Projective varieties |
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269 | (8) |
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15 Castelnuovo regularity |
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277 | (17) |
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15.1 Finitely generated components |
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277 | (4) |
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15.2 The basics of Castelnuovo regularity |
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281 | (8) |
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15.3 Degrees of generators |
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289 | (5) |
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16 Bounds of diagonal type |
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294 | (18) |
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295 | (4) |
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16.2 The right bounding functions |
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299 | (6) |
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16.3 Polynomial representations |
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305 | (4) |
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16.4 Bounding systems for numerical invariants |
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309 | (3) |
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312 | (13) |
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17.1 The characteristic function |
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313 | (6) |
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17.2 Bounds in terms of Hilbert coefficients |
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319 | (6) |
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18 Applications to reductions of ideals |
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325 | (17) |
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18.1 Reductions and integral closures |
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325 | (5) |
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330 | (3) |
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18.3 Links with Castelnuovo regularity |
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333 | (9) |
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19 Connectivity in algebraic varieties |
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342 | (32) |
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19.1 The connectedness dimension |
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342 | (5) |
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19.2 Complete local rings and connectivity |
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347 | (5) |
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19.3 Some local dimensions |
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352 | (7) |
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19.4 Connectivity of affine algebraic cones |
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359 | (1) |
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19.5 Connectivity of projective varieties |
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360 | (3) |
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19.6 Connectivity of intersections |
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363 | (5) |
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19.7 The projective spectrum and connectedness |
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368 | (6) |
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20 Links with sheaf cohomology |
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374 | (33) |
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20.1 The Deligne Isomorphism |
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375 | (11) |
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20.2 The graded Deligne Isomorphism |
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386 | (3) |
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20.3 Links with sheaf theory |
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389 | (9) |
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20.4 Applications to projective schemes |
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398 | (9) |
| Bibliography |
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407 | (3) |
| Index |
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410 | |
