Foundations of Grothendieck Duality for Diagrams of Schemes
The first part by Joseph Lipman is a full exposition of the abstract foundations of Grothendieck duality theory for schemes (twisted inverse image, tor-independent base change,...), in part without noetherian hypotheses, and with some refinements for maps of finite tor-dimension. The ground is prepared by a lengthy treatment of the rich formalism of relations among the derived functors, for unbounded complexes over ringed spaces, of the sheaf functors tensor, hom, direct and inverse image....
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The first part is a full exposition of the abstract foundations of Grothendieck duality theory for schemes (twisted inverse image, tor-independent base change,...), in part without noetherian hypotheses, and with some refinements for maps of finite tor-dimension. The ground is prepared by a lengthy treatment of the rich formalism of relations among the derived functors, for unbounded complexes over ringed spaces, of the sheaf functors tensor, hom, direct and inverse image. Included are enhancements, for quasi-compact quasi-separated schemes, of classical results such as the projection and Künneth isomorphisms. In the second part, the theory is extended to the context of diagrams of schemes. This includes, as a special case, an equivariant theory for schemes with group actions. In particular, after various basic operations on sheaves such as (derived) direct images and inverse images are set up, Grothendieck duality and flat base change for diagrams of schemes are proved. Also, dualizing complexes are studied in this context. As an application to group actions, we generalize Watanabe's theorem on the Gorenstein property of invariant subrings.
Part I Joseph Lipman: Notes on Derived Functors and Grothendieck DualityAbstract 3Introduction 51 Derived and Triangulated Categories 111.1 The Homotopy Category K 121.2 The Derived Category D 131.3 Mapping Cones 151.4 Triangulated Categories (D-Categories) 161.5 Triangle-Preserving Functors (D-Functors) 251.6 D-Subcategories 291.7 Localizing Subcategories of K; D-Equivalent Categories 301.8 Examples 331.9 Complexes with Homology in a Plump Subcategory 351.10 Truncation Functors 361.11 Bounded Functors; Way-Out Lemma 382 Derived Functors 432.1 Definition of Derived Functors 432.2 Existence of Derived Functors 452.3 Right-Derived Functors via Injective Resolutions 522.4 Derived Homomorphism Functors 562.5 Derived Tensor Product 602.6 Adjoint Associativity 652.7 Acyclic Objects; Finite-Dimensional Derived Functors 713 Derived Direct and Inverse Image 833.1 Preliminaries 853.2 Adjointness of Derived Direct and Inverse Image 893.3 D-Adjoint Functors 973.4 Adjoint Functors between Monoidal Categories 1013.5 Adjoint Functors between Closed Categories 1103.6 Adjoint Monoidal D-Pseudofunctors 1183.7 More Formal Consequences: Projection, Base Change 1243.8 Direct Sums 1313.9 Concentrated Scheme-Maps 1323.10 Independent Squares; Kunneth Isomorphism 1444 Abstract Grothendieck Duality for Schemes 1594.1 Global Duality 1604.2 Sheafified Duality-Preliminary Form 1694.3 Pseudo-Coherence and Quasi-Properness 1714.4 Sheafified Duality, Base Change 1774.5 Proof of Duality and Base Change: Outline 1794.6 Steps in the Proof 1794.7 Quasi-Perfect Maps 1904.8 Two FundamentalTheorems 2034.9 Perfect Maps of Noetherian Schemes 2304.10 Appendix: Dualizing Complexes 239References 253Index 257Part II Mitsuyasu Hashimoto: Equivariant Twisted InversesIntroduction 2671 Commutativity of Diagrams Constructed from a Monoidal Pair of Pseudofunctors 2712 Sheaves on Ringed Sites 2873 Derived Categories and Derived Functors of Sheaves on Ringed Sites 3114 Sheaves over a Diagram of S-Schemes 3215 The Left and Right Inductions and the Direct and Inverse Images 3276 Operations on Sheaves Via the Structure Data 3317 Quasi-Coherent Sheaves Over a Diagram of Schemes 3458 Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes 3519 Simplicial Objects 35910 Descent Theory 36311 Local Noetherian Property 37112 Groupoid of Schemes 37513 Bokstedt-Neeman Resolutions and HyperExt Sheaves 38114 The Right Adjoint of the Derived Direct Image Functor 38515 Comparison of Local Ext Sheaves 39316 The Composition of Two Almost-Pseudofunctors 39517 The Right Adjoint of the Derived Direct Image Functor of a Morphism of Diagrams 40118 Commutativity of Twisted Inverse with Restrictions 40519 Open Immersion Base Change 41320 The Existence of Compactification and Composition Data for Diagrams of Schemes Over an Ordered Finite Category 41521 Flat Base Change 41922 Preservation of Quasi-Coherent Cohomology 42323 Compatibility with Derived Direct Images 42524 Compatibility with Derived Right Inductions 42725 Equivariant Grothendieck's Duality 42926 Morphisms of Finite Flat Dimension 43127 Cartesian Finite Morphisms 43528 Cartesian Regular Embeddings and Cartesian Smooth Morphisms 43929 Group Schemes Flat of Finite Type 44530 Compatibility with Derived G-Invariance 44931 Equivariant Dualizing Complexes and Canonical Modules 45132 A Generalization of Watanabe's Theorem 45733 Other Examples of Diagrams of Schemes 463Glossary 467References 473Index 477
