Foundations of Grothendieck Duality for Diagrams of Schemes / by Joseph Lipman, Mitsuyasu Hashimoto.

Format:

Book

Author/Creator:

Lipman, Joseph, author.

Hashimoto, Mitsuyasu, 1962- author.

Contributor:

SpringerLink (Online service)

Series:

Lecture Notes in Mathematics, 0075-8434 ; 1960.

Lecture Notes in Mathematics, 0075-8434 ; 1960

Language:

English

Subjects (All):

Geometry, Algebraic.

Algebra.

Algebraic Geometry.

Category Theory, Homological Algebra.

Local Subjects:

Algebraic Geometry.

Category Theory, Homological Algebra.

Physical Description:

1 online resource (X, 478 pages).

Contained In:

Springer eBooks

Place of Publication:

Berlin, Heidelberg : Springer Berlin Heidelberg, 2009.

System Details:

text file PDF

Summary:

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.

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.

Other Format:

Printed edition:

ISBN:

9783540854203

Access Restriction:

Restricted for use by site license.