Foundations of Grothendieck Duality for Diagrams of Schemes

• Main Authors: Lipman, Joseph (Author), Hashimoto, Mitsuyasu (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg, 2009.
• Series: Lecture Notes in Mathematics, 1960
• Subjects: Mathematics.; Algebraic geometry.; Category theory (Mathematics).; Homological algebra.; Algebraic Geometry.; Category Theory, Homological Algebra.

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 f...

Table of Contents:

• Joseph Lipman: Notes on Derived Functors and Grothendieck Duality
• Derived and Triangulated Categories
• Derived Functors
• Derived Direct and Inverse Image
• Abstract Grothendieck Duality for Schemes
• Mitsuyasu Hashimoto: Equivariant Twisted Inverses
• Commutativity of Diagrams Constructed from a Monoidal Pair of Pseudofunctors
• Sheaves on Ringed Sites
• Derived Categories and Derived Functors of Sheaves on Ringed Sites
• Sheaves over a Diagram of S-Schemes
• The Left and Right Inductions and the Direct and Inverse Images
• Operations on Sheaves Via the Structure Data
• Quasi-Coherent Sheaves Over a Diagram of Schemes
• Derived Functors of Functors on Sheaves of Modules Over Diagrams of Schemes
• Simplicial Objects
• Descent Theory
• Local Noetherian Property
• Groupoid of Schemes
• Bökstedt—Neeman Resolutions and HyperExt Sheaves
• The Right Adjoint of the Derived Direct Image Functor
• Comparison of Local Ext Sheaves
• The Composition of Two Almost-Pseudofunctors
• The Right Adjoint of the Derived Direct Image Functor of a Morphism of Diagrams
• Commutativity of Twisted Inverse with Restrictions
• Open Immersion Base Change
• The Existence of Compactification and Composition Data for Diagrams of Schemes Over an Ordered Finite Category
• Flat Base Change
• Preservation of Quasi-Coherent Cohomology
• Compatibility with Derived Direct Images
• Compatibility with Derived Right Inductions
• Equivariant Grothendieck's Duality
• Morphisms of Finite Flat Dimension
• Cartesian Finite Morphisms
• Cartesian Regular Embeddings and Cartesian Smooth Morphisms
• Group Schemes Flat of Finite Type
• Compatibility with Derived G-Invariance
• Equivariant Dualizing Complexes and Canonical Modules
• A Generalization of Watanabe's Theorem
• Other Examples of Diagrams of Schemes.