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Grothendieck Duality and Base Change

Author: Brian Conrad

Publisher: Springer

Published: 2003-07-01

Total Pages: 302

ISBN-13: 354040015X

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Grothendieck's duality theory for coherent cohomology is a fundamental tool in algebraic geometry and number theory, in areas ranging from the moduli of curves to the arithmetic theory of modular forms. Presented is a systematic overview of the entire theory, including many basic definitions and a detailed study of duality on curves, dualizing sheaves, and Grothendieck's residue symbol. Along the way proofs are given of some widely used foundational results which are not proven in existing treatments of the subject, such as the general base change compatibility of the trace map for proper Cohen-Macaulay morphisms (e.g., semistable curves). This should be of interest to mathematicians who have some familiarity with Grothendieck's work and wish to understand the details of this theory.

Book Synopsis Grothendieck Duality and Base Change by : Brian Conrad

Download or read book Grothendieck Duality and Base Change written by Brian Conrad and published by Springer. This book was released on 2003-07-01 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt: Grothendieck's duality theory for coherent cohomology is a fundamental tool in algebraic geometry and number theory, in areas ranging from the moduli of curves to the arithmetic theory of modular forms. Presented is a systematic overview of the entire theory, including many basic definitions and a detailed study of duality on curves, dualizing sheaves, and Grothendieck's residue symbol. Along the way proofs are given of some widely used foundational results which are not proven in existing treatments of the subject, such as the general base change compatibility of the trace map for proper Cohen-Macaulay morphisms (e.g., semistable curves). This should be of interest to mathematicians who have some familiarity with Grothendieck's work and wish to understand the details of this theory.

Foundations of Grothendieck Duality for Diagrams of Schemes

Author: Joseph Lipman

Publisher: Springer

Published: 2009-03-07

Total Pages: 471

ISBN-13: 3540854207

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Part One of this book covers the abstract foundations of Grothendieck duality theory for schemes in part with noetherian hypotheses and with some refinements for maps of finite tor-dimension. Part Two extends the theory to the context of diagrams of schemes.

Book Synopsis Foundations of Grothendieck Duality for Diagrams of Schemes by : Joseph Lipman

Download or read book Foundations of Grothendieck Duality for Diagrams of Schemes written by Joseph Lipman and published by Springer. This book was released on 2009-03-07 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: Part One of this book covers the abstract foundations of Grothendieck duality theory for schemes in part with noetherian hypotheses and with some refinements for maps of finite tor-dimension. Part Two extends the theory to the context of diagrams of schemes.

Introduction to Grothendieck Duality Theory

Author: Allen Altman

Publisher: Springer

Published: 2006-11-15

Total Pages: 188

ISBN-13: 3540363092

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Book Synopsis Introduction to Grothendieck Duality Theory by : Allen Altman

Download or read book Introduction to Grothendieck Duality Theory written by Allen Altman and published by Springer. This book was released on 2006-11-15 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes

Author: Leovigildo Alonso Tarrío

Publisher: American Mathematical Soc.

Published: 1999

Total Pages: 138

ISBN-13: 0821819429

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This volume contains three papers on the foundations of Grothendieck duality on Noetherian formal schemes and on not-necessarily-Noetherian ordinary schemes. The first paper presents a self-contained treatment for formal schemes which synthesizes several duality-related topics, such as local duality, formal duality, residue theorems, dualizing complexes, etc. Included is an exposition of properties of torsion sheaves and of limits of coherent sheaves. A second paper extends Greenlees-May duality to complexes on formal schemes. This theorem has important applications to Grothendieck duality. The third paper outlines methods for eliminating the Noetherian hypotheses. A basic role is played by Kiehl's theorem affirming conservation of pseudo-coherence of complexes under proper pseudo-coherent maps. This work gives a detailed introduction to the subject of Grothendieck Duality. The approach is unique in its presentation of a complex series of special cases that build up to the main results.

Book Synopsis Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes by : Leovigildo Alonso Tarrío

Download or read book Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes written by Leovigildo Alonso Tarrío and published by American Mathematical Soc.. This book was released on 1999 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains three papers on the foundations of Grothendieck duality on Noetherian formal schemes and on not-necessarily-Noetherian ordinary schemes. The first paper presents a self-contained treatment for formal schemes which synthesizes several duality-related topics, such as local duality, formal duality, residue theorems, dualizing complexes, etc. Included is an exposition of properties of torsion sheaves and of limits of coherent sheaves. A second paper extends Greenlees-May duality to complexes on formal schemes. This theorem has important applications to Grothendieck duality. The third paper outlines methods for eliminating the Noetherian hypotheses. A basic role is played by Kiehl's theorem affirming conservation of pseudo-coherence of complexes under proper pseudo-coherent maps. This work gives a detailed introduction to the subject of Grothendieck Duality. The approach is unique in its presentation of a complex series of special cases that build up to the main results.

Grothendieck Duality for Flat Morphisms

Author: Muhammad Hafiz Khusyairi

Publisher:

Published: 2017

Total Pages: 0

ISBN-13:

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Traditionally, the twisted inverse image functor of Grothendieck duality (upper-shriek) is defined by means of compactification on a class of morphisms between noetherian schemes. Recently, Iyengar, Lipman, and Neeman introduced a formula for this pseudo-functor which works for flat, separated, essentially of finite-type morphisms between noetherian schemes. This formula raised some important questions. Not only it is not well understood why the formula is free of compactification but the process of how this formula satisfies all the standard properties of (upper-shriek) is also unclear. Another important question is whether this new formula can be expanded outside the class of flat, separated, essentially of finite-type morphisms between noetherian schemes.In this thesis, we talk about the motivations behind the two twisted inverse image pseudo-functors of Grothendieck duality (upper-times) and (upper-shriek). We also recall the sufficient conditions and properties of these pseudo-functors. These properties are presented as the existence of some morphisms and compatibility diagrams satisfied by these morphisms. Then we discuss the surprising compactification-free formula of the functor on the subclass of flat morphisms. A simplified proof that this formula is isomorphic to (upper-shriek) is also given.This recently discovered formula satisfies the properties of (upper-shriek) defined classically. As in the classical definition, the properties of this formula will also be presented via the existence of some morphisms and some compatibility diagrams. Extracting the essential information from the proofs, especially regarding the flat base change morphism, we discuss how understanding these proofs may enable us to generalize this Grothendieck Duality formula for flat morphisms to non-noetherian schemes.

Book Synopsis Grothendieck Duality for Flat Morphisms by : Muhammad Hafiz Khusyairi

Download or read book Grothendieck Duality for Flat Morphisms written by Muhammad Hafiz Khusyairi and published by . This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Traditionally, the twisted inverse image functor of Grothendieck duality (upper-shriek) is defined by means of compactification on a class of morphisms between noetherian schemes. Recently, Iyengar, Lipman, and Neeman introduced a formula for this pseudo-functor which works for flat, separated, essentially of finite-type morphisms between noetherian schemes. This formula raised some important questions. Not only it is not well understood why the formula is free of compactification but the process of how this formula satisfies all the standard properties of (upper-shriek) is also unclear. Another important question is whether this new formula can be expanded outside the class of flat, separated, essentially of finite-type morphisms between noetherian schemes.In this thesis, we talk about the motivations behind the two twisted inverse image pseudo-functors of Grothendieck duality (upper-times) and (upper-shriek). We also recall the sufficient conditions and properties of these pseudo-functors. These properties are presented as the existence of some morphisms and compatibility diagrams satisfied by these morphisms. Then we discuss the surprising compactification-free formula of the functor on the subclass of flat morphisms. A simplified proof that this formula is isomorphic to (upper-shriek) is also given.This recently discovered formula satisfies the properties of (upper-shriek) defined classically. As in the classical definition, the properties of this formula will also be presented via the existence of some morphisms and some compatibility diagrams. Extracting the essential information from the proofs, especially regarding the flat base change morphism, we discuss how understanding these proofs may enable us to generalize this Grothendieck Duality formula for flat morphisms to non-noetherian schemes.

Foundations of Grothendieck Duality for Diagrams of Schemes

Author: Joseph Lipman

Publisher: Springer Science & Business Media

Published: 2009-02-05

Total Pages: 471

ISBN-13: 3540854193

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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.

Book Synopsis Foundations of Grothendieck Duality for Diagrams of Schemes by : Joseph Lipman

Download or read book Foundations of Grothendieck Duality for Diagrams of Schemes written by Joseph Lipman and published by Springer Science & Business Media. This book was released on 2009-02-05 with total page 471 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Arithmetic Duality Theorems

Author: J. S. Milne

Publisher:

Published: 1986

Total Pages: 440

ISBN-13:

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Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, ,tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Book Synopsis Arithmetic Duality Theorems by : J. S. Milne

Download or read book Arithmetic Duality Theorems written by J. S. Milne and published by . This book was released on 1986 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, ,tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Introduction to Grothendieck Duality Theory

Author: Allen Altman

Publisher: Springer

Published: 2014-01-15

Total Pages: 192

ISBN-13: 9783662165522

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Book Synopsis Introduction to Grothendieck Duality Theory by : Allen Altman

Download or read book Introduction to Grothendieck Duality Theory written by Allen Altman and published by Springer. This book was released on 2014-01-15 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Étale Cohomology

Author: James S. Milne

Publisher: Princeton University Press

Published: 2016-10-11

Total Pages: 338

ISBN-13: 1400883989

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One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Book Synopsis Étale Cohomology by : James S. Milne

Download or read book Étale Cohomology written by James S. Milne and published by Princeton University Press. This book was released on 2016-10-11 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series. Originally published in 1980. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Variance and Duality for Cousin Complexes on Formal Schemes

Author: Joseph Lipman

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 290

ISBN-13: 0821837052

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Robert Hartshorne's book, Residues and Duality (1966, Springer-Verlag), introduced the notion of residual complexes and developed a duality theory (Grothendieck duality) on the category of maps of noetherian schemes. The three articles in this volume constitute a reworking of the main parts of the corresponding chapters in Hartshorne's 1966 book in greater generality using a somewhat different approach. In particular, throughout this volume, the authors work with arbitrary (quasi-coherent, torsion) Cousin complexes on formal schemes, not only with residual complexes on ordinary schemes. Additionally, their motivation is to help readers gain a better understanding of the relation between local properties of residues and global properties of the dualizing pseudofunctor. The book is suitable for graduate students and researchers working in algebraic geometry.

Book Synopsis Variance and Duality for Cousin Complexes on Formal Schemes by : Joseph Lipman

Download or read book Variance and Duality for Cousin Complexes on Formal Schemes written by Joseph Lipman and published by American Mathematical Soc.. This book was released on 2005 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: Robert Hartshorne's book, Residues and Duality (1966, Springer-Verlag), introduced the notion of residual complexes and developed a duality theory (Grothendieck duality) on the category of maps of noetherian schemes. The three articles in this volume constitute a reworking of the main parts of the corresponding chapters in Hartshorne's 1966 book in greater generality using a somewhat different approach. In particular, throughout this volume, the authors work with arbitrary (quasi-coherent, torsion) Cousin complexes on formal schemes, not only with residual complexes on ordinary schemes. Additionally, their motivation is to help readers gain a better understanding of the relation between local properties of residues and global properties of the dualizing pseudofunctor. The book is suitable for graduate students and researchers working in algebraic geometry.