Foundations of Grothendieck Duality for Diagrams of Schemes

by Lipman, Joseph; Hashimoto, Mitsuyasu

• ISBN13: 9783540854197
• ISBN10: 3540854193
• Format: Paperback
• Copyright: 2009-03-06
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Joseph Lipman: Notes on Derived Functors and Grothendieck Duality
Abstract	p. 3
Introduction	p. 5
Derived and Triangulated Categories	p. 11
The Homotopy Category K	p. 12
The Derived Category D	p. 13
Mapping Cones	p. 15
Triangulated Categories (D-Categories)	p. 16
Triangle-Preserving Functors (D-Functors)	p. 25
D-Subcategories	p. 29
Localizing Subcategories of K; D-Equivalent Categories	p. 30
Examples	p. 33
Complexes with Homology in a Plump Subcategory	p. 35
Truncation Functors	p. 36
Bounded Functors; Way-Out Lemma	p. 38
Derived Functors	p. 43
Definition of Derived Functors	p. 43
Existence of Derived Functors	p. 45
Right-Derived Functors via Injective Resolutions	p. 52
Derived Homomorphism Functors	p. 56
Derived Tensor Product	p. 60
Adjoint Associativity	p. 65
Acyclic Objects; Finite-Dimensional Derived Functors	p. 71
Derived Direct and Inverse Image	p. 83
Preliminaries	p. 85
Adjointness of Derived Direct and Inverse Image	p. 89
D-Adjoint Functors	p. 97
Adjoint Functors between Monoidal Categories	p. 101
Adjoint Functors between Closed Categories	p. 110
Adjoint Monoidal D-Pseudofunctors	p. 118
More Formal Consequences: Projection, Base Change	p. 124
Direct Sums	p. 131
Concentrated Scheme-Maps	p. 132
Independent Squares; Kunneth Isomorphism	p. 144
Abstract Grothendieck Duality for Schemes	p. 159
Global Duality	p. 160
Sheafified Duality-Preliminary Form	p. 169
Pseudo-Coherence and Quasi-Properness	p. 171
Sheafified Duality, Base Change	p. 177
Proof of Duality and Base Change: Outline	p. 179
Steps in the Proof	p. 179
Quasi-Perfect Maps	p. 190
Two Fundamental Theorems	p. 203
Perfect Maps of Noetherian Schemes	p. 230
Appendix: Dualizing Complexes	p. 239
References	p. 253
Index	p. 257
Mitsuyasu Hashimoto: Equivariant Twisted Inverses
Introduction	p. 267
Commutativity of Diagrams Constructed from a Monoidal Pair of Pseudofunctors	p. 271
Sheaves on Ringed Sites	p. 287
Derived Categories and Derived Functors of Sheaves on Ringed Sites	p. 311
Sheaves over a Diagram of S-Schemes	p. 321
The Left and Right Inductions and the Direct and Inverse Images	p. 327
Operations on Sheaves Via the Structure Data	p. 331
Quasi-Coherent Sheaves Over a Diagram of Schemes	p. 345
Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes	p. 351
Simplicial Objects	p. 359
Descent Theory	p. 363
Local Noetherian Property	p. 371
Groupoid of Schemes	p. 375
Bokstedt-Neeman Resolutions and HyperExt Sheaves	p. 381
The Right Adjoint of the Derived Direct Image Functor	p. 385
Comparison of Local Ext Sheaves	p. 393
The Composition of Two Almost-Pseudofunctors	p. 395
The Right Adjoint of the Derived Direct Image Functor of a Morphism of Diagrams	p. 401
Commutativity of Twisted Inverse with Restrictions	p. 405
Open Immersion Base Change	p. 413
The Existence of Compactification and Composition Data for Diagrams of Schemes Over an Ordered Finite Category	p. 415
Flat Base Change	p. 419
Preservation of Quasi-Coherent Cohomology	p. 423
Compatibility with Derived Direct Images	p. 425
Compatibility with Derived Right Inductions	p. 427
Equivariant Grothendieck's Duality	p. 429
Morphisms of Finite Flat Dimension	p. 431
Cartesian Finite Morphisms	p. 435
Cartesian Regular Embeddings and Cartesian Smooth Morphisms	p. 439
Group Schemes Flat of Finite Type	p. 445
Compatibility with Derived G-Invariance	p. 449
Equivariant Dualizing Complexes and Canonical Modules	p. 451
A Generalization of Watanabe's Theorem	p. 457
Other Examples of Diagrams of Schemes	p. 463
Glossary	p. 467
References	p. 473
Index	p. 477
Table of Contents provided by Ingram. All Rights Reserved.