Foundations of Grothendieck duality for diagrams of schemes

Joseph Lipman, Mitsuyasu Hashimoto (auth.)3540854193, 9783540854197, 3540854207, 9783540854203

The first part written by Joseph Lipman, accessible to mid-level graduate students, 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, written independently by Mitsuyasu Hashimoto, 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 :
Front Matter….Pages i-x
Front Matter….Pages 1-3
Introduction….Pages 5-10
Derived and Triangulated Categories….Pages 11-42
Derived Functors….Pages 43-81
Derived Direct and Inverse Image….Pages 83-158
Abstract Grothendieck Duality for Schemes….Pages 159-252
Back Matter….Pages 253-259
Front Matter….Pages 253-257
Introduction….Pages 259-262
Commutativity of Diagrams Constructed from a Monoidal Pair of Pseudofunctors….Pages 263-278
Sheaves on Ringed Sites….Pages 279-302
Derived Categories and Derived Functors of Sheaves on Ringed Sites….Pages 303-312
Sheaves over a Diagram of S -Schemes….Pages 313-317
The Left and Right Inductions and the Direct and Inverse Images….Pages 319-321
Operations on Sheaves Via the Structure Data….Pages 323-336
Quasi-Coherent Sheaves Over a Diagram of Schemes….Pages 337-342
Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes….Pages 343-349
Simplicial Objects….Pages 351-353
Descent Theory….Pages 355-361
Local Noetherian Property….Pages 363-365
Groupoid of Schemes….Pages 367-371
Bökstedt—Neeman Resolutions and HyperExt Sheaves….Pages 373-376
The Right Adjoint of the Derived Direct Image Functor….Pages 377-383
Back Matter….Pages 467-478
Front Matter….Pages 253-257
Comparison of Local Ext Sheaves….Pages 385-386
The Composition of Two Almost-Pseudofunctors….Pages 387-391
The Right Adjoint of the Derived Direct Image Functor of a Morphism of Diagrams….Pages 393-395
Commutativity of Twisted Inverse with Restrictions….Pages 397-403
Open Immersion Base Change….Pages 405-406
The Existence of Compactification and Composition Data for Diagrams of Schemes Over an Ordered Finite Category….Pages 407-409
Flat Base Change….Pages 411-413
Preservation of Quasi-Coherent Cohomology….Pages 415-416
Compatibility with Derived Direct Images….Pages 417-417
Compatibility with Derived Right Inductions….Pages 419-420
Equivariant Grothendieck’s Duality….Pages 421-422
Morphisms of Finite Flat Dimension….Pages 423-426
Cartesian Finite Morphisms….Pages 427-430
Cartesian Regular Embeddings and Cartesian Smooth Morphisms….Pages 431-436
Group Schemes Flat of Finite Type….Pages 437-439
Compatibility with Derived G -Invariance….Pages 441-442
Equivariant Dualizing Complexes and Canonical Modules….Pages 443-448
A Generalization of Watanabe’s Theorem….Pages 449-453
Other Examples of Diagrams of Schemes….Pages 455-457
Back Matter….Pages 467-478
Back Matter….Pages 459-478