Spis treści: Foundations of Grothendieck Duality for Diagrams of Schemes


Part I Joseph Lipman: Notes on Derived Functors and Grothendieck Duality

Abstract, s. 3
Introduction, s. 5
Part II Mitsuyasu Hashimoto: Equivariant Twisted Inverses

1. Derived and Triangulated Categories, s. 11

1.1    The Homotopy Category K, s. 12
1.2    The Derived Category D, s. 13
1.3    Mapping Cones, s. 15
1.4    Triangulated Categories (Δ-Categories), s. 16
1.5    Triangle-Preserving Functors (Δ-Functors), s. 25
1.6    A-Subcategories, s. 29
1.7    Localizing Subcategories of K; Δ-Equivalent Categories, s. 30
1.8    Examples, s. 33
1.9    Complexes with Homology in a Plump Subcategory, s. 35
1.10   Truncation Functors, s. 36
1.11   Bounded Functors; Way-Out Lemma, s. 38

2. Derived Functors, s. 43

2.1    Definition of Derived Functors, s. 43
2.2    Existence of Derived Functors, s. 45
2.3    Right-Derived Functors via Injective Resolutions, s. 52
2.4    Derived Homomorphism Functors, s. 56
2.5    Derived Tensor Product, s. 60
2.6    Adjoint Associativity, s. 65
2.7    Acyclic Objects; Finite-Dimensional Derived Functors, s. 71

3. Derived Direct and Inverse Image, s. 83

3.1    Preliminaries, s. 85
3.3    A-Adjoint Functors, s. 97
3.4    Adjoint Functors between Monoidal Categories, s. 101
3.5    Adjoint Functors between Closed Categories, s. 110
3.6    Adjoint Monoidal A-Pseudofunctors, s. 118
3.7    More Formal Consequences: Projection, Base Change, s. 124
3.8    Direct Sums, s. 131
3.9    Concentrated Scheme-Maps, s. 132
3.10   Independent Squares; Kiinneth Isomorphism, s. 144

4. Abstract Grothendieck Duality for Schemes, s. 159

4.1    Global Duality, s. 160
4.2    Sheafified Duality—Preliminary Form, s. 169
4.3    Pseudo-Coherence and Quasi-Properness, s. 171
4.4    Sheafified Duality, Base Change, s. 177
4.5    Proof of Duality and Base Change: Outline, s. 179
4.6    Steps in the Proof, s. 179
4.7    Quasi-Perfect Maps, s. 190
4.8    Two Fundamental Theorems, s. 203
4.9    Perfect Maps of Noetherian Schemes, s. 230
4.10   Appendix: Dualizing Complexe, s. 239
References, s. 253
Index, s. 257

Part II Mitsuyasu Hashimoto: Equivariant Twisted Inverses

Introduction, s. 267

1   Commutativity of Diagrams Constructed from a Monoidal Pair of Pseudofunctors, s. 271
2   Sheaves on Ringed Sites, s. 287
3   Derived Categories and Derived Functors of Sheaves on Ringed Sites, s. 311
4   Sheaves over a Diagram of S-Schemes, s. 321
5   The Left and Right Inductions and the Direct and Inverse Images, s. 327
6   Operations on Sheaves Via the Structure Data, s. 331
7   Quasi-Coherent Sheaves Over a Diagram of Schemes, s. 345
8   Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes, s. 351
9   Simplicial Objects, s. 359
10   Descent Theory, s. 363
11   Local Noetherian Property, s. 371
12   Groupoid of Schemes, s. 375
13   Bokstedt-Neeman Resolutions and HyperExt Sheaves, s. 381
14   The Right Adjoint of the Derived Direct Image Functor, s. 385
15   Comparison of Local Ext Sheaves, s. 393
16   The Composition of Two Almost-Pseudofunctors, s. 395
17   The Right Adjoint of the Derived Direct Image Functor of a Morphism of Diagrams, s. 401
18   Commutativity of Twisted Inverse with Restrictions, s. 405
19   Open Immersion Base Change, s. 413
20   The Existence of Compactification and Composition Data for Diagrams of Schemes Over an Ordered Finite Category, s. 415
21   Flat Base Change, s. 419
22   Preservation of Quasi-Coherent Cohomology, s. 423
23   Compatibility with Derived Direct Images, s. 425
24   Compatibility with Derived Right Inductions, s. 427
25   Equivariant Grothendieck's Duality, s. 429
26   Morphisms of Finite Flat Dimension, s. 431
27   Cartesian Finite Morphisms, s. 435
28   Cartesian Regular Embeddings and Cartesian Smooth Morphisms, s. 439
29   Group Schemes Flat of Finite Type, s. 445
30   Compatibility with Derived G-Invariance, s. 449
31   Equivariant Dualizing Complexes and Canonical Modules, s. 451
32   A Generalization of Watanabe's Theorem, s. 457
33   Other Examples of Diagrams of Schemes, s. 463

Glossary, s. 467
References, s. 473
Index, s. 477

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