Foundations of Grothendieck Duality for Diagrams of Schemes

Häftad, Engelska, 2009

Del 1960 i serien Lecture Notes in Mathematics

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Beskrivning

This volume contains two related, though independently written, mo- graphs. In Notes on Derived Functors and Grothendieck Duality the ?rst three chapters treat the basics of derived categories and functors, and of the rich formalism, over ringed spaces, of the derived functors, for unbounded com- ? plexes,ofthesheaffunctors?,Hom,f andf wheref isaringed-spacemap. ? Included are some enhancements, for concentrated (i.e., quasi-compact and quasi-separated) schemes, of classical results such as the projection and K.. unneth isomorphisms. The fourth chapter presents the abstract foun- tions of Grothendieck Duality-existence and tor-independent base change for the right adjoint of the derived functor Rf when f is a quasi-proper ? map of concentrated schemes, the twisted inverse image pseudofunctor for separated ?nite-type maps of noetherian schemes, re?nements for maps of ?nite tor-dimension, and a brief discussion of dualizing complexes. In Equivariant Twisted Inverses the theory is extended to the context of diagrams of schemes, and in particular, to schemes with a group-scheme action. An equivariant version of the twisted inverse-image pseudofunctor is de?ned, and equivariant versions of some of its important properties are proved, including Grothendieck duality for proper morphisms, and ?at base change. Also, equivariant dualizing complexes are dealt with. As an appli- tion,ageneralizedversionofWatanabe'stheoremontheGorensteinproperty of rings of invariants is proved. More detailed overviews are given in the respective Introductions.

Produktinformation

Utgivningsdatum:2009-02-05
Mått:155 x 235 x 27 mm
Vikt:744 g
Format:Häftad
Språk:Engelska
Serie:Lecture Notes in Mathematics
Antal sidor:478
Upplaga:2009
Förlag:Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
ISBN:9783540854197

Utforska kategorier

Matematik inom Naturvetenskap och teknik

Recensioner i media

From the reviews: "The appearance of a well-planned, detailed and up-to-date exposition of a topic in abstract algebraic geometry is good news, and the book by J. Lipman and M. Hashimoto definitely has all the above qualities. ... get the book in its current state now than to wait for years until the authors produce a more unified presentation. To conclude, the book by Joseph Lipman and Mitsuyasu Hashimoto is an important contribution to an important task of explaining the main ideas of abstract algebraic geometry ... ." (George Shabat, Bulletin of the London Mathematical Society, March, 2010)

Innehållsförteckning

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.