Foundations of Grothendieck Duality for Diagrams of Schemes

• Main Authors: Lipman, Joseph. (Author), Hashimoto, Mitsuyasu. (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg, 2009.
• Series: Lecture Notes in Mathematics, 1960
• Subjects: Mathematics.; Geometry, algebraic.; Algebra.; Algebraic Geometry.; Category Theory, Homological Algebra.
• Online Access: https://ezaccess.library.uitm.edu.my/login?url=http://dx.doi.org/10.1007/978-3-540-85420-3

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:

• Part I
• 1. Derived and Triangulated Categories
• 2. Derived Functors
• 3. Derived Direct and Inverse Image
• 4. Abstract Grothendieck Duality for schemes
• Part II
• 1. Commutativity of diagrams constructed from a monoidal pair of pseudofunctors
• 2. Sheaves on ringed sites
• 3. Derived categories and derived functors of sheaves on ringed sites
• 4. Sheaves over a diagram of S-schemes
• 5. The left and right inductions and the direct and inverse images
• 6. Operations on sheaves via the structure data
• 7. Quasi-coherent sheaves over a diagram of schemes
• 8. Derived functors of functors on sheaves of modules over diagrams of schemes
• 9. Simplicial objects
• 10. Descent theory
• 11. Local noetherian property
• 12. Groupoid of schemes
• 13. Boekstedt-Neeman resolutions and hyperExt sheaves
• 14. The right adjoint of the derived direct image functor
• 15. Comparison of local Ext sheaves
• 16. The Composition of two almost-pseudofunctors
• 17. The right adjoint of the derived direct image functor of a morphism of diagrams
• 18. Commutativity of twisted inverse with restrictions
• 19. Open immersion base change
• 20. The existence of compactification and composition data for diagrams of schemes over an ordered finite category
• 21. Flat base change
• 22. Preservation of Quasi-coherent cohomology
• 23. Compatibility with derived direct images
• 24. Compatibility with derived right inductions
• 25. Equivariant Grothendieck's duality
• 26. Morphisms of finite flat dimension
• 27. Cartesian finite morphisms
• 28. Cartesian regular embeddings and cartesian smooth morphisms
• 29. Group schemes flat of finite type
• 30. Compatibility with derived G-invariance
• 31. Equivariant dualizing complexes and canonical modules
• 32. A generalization of Watanabe's theorem
• 33. Other examples of diagrams of schemes.