Derived Functors And Sheaf Cohomology
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Derived Functors And Sheaf Cohomology
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Author : Ugo Bruzzo
language : en
Publisher: World Scientific
Release Date : 2020-03-10
Derived Functors And Sheaf Cohomology written by Ugo Bruzzo and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-10 with Mathematics categories.
The aim of the book is to present a precise and comprehensive introduction to the basic theory of derived functors, with an emphasis on sheaf cohomology and spectral sequences. It keeps the treatment as simple as possible, aiming at the same time to provide a number of examples, mainly from sheaf theory, and also from algebra.The first part of the book provides the foundational material: Chapter 1 deals with category theory and homological algebra. Chapter 2 is devoted to the development of the theory of derived functors, based on the notion of injective object. In particular, the universal properties of derived functors are stressed, with a view to make the proofs in the following chapters as simple and natural as possible. Chapter 3 provides a rather thorough introduction to sheaves, in a general topological setting. Chapter 4 introduces sheaf cohomology as a derived functor, and, after also defining Čech cohomology, develops a careful comparison between the two cohomologies which is a detailed analysis not easily available in the literature. This comparison is made using general, universal properties of derived functors. This chapter also establishes the relations with the de Rham and Dolbeault cohomologies. Chapter 5 offers a friendly approach to the rather intricate theory of spectral sequences by means of the theory of derived triangles, which is precise and relatively easy to grasp. It also includes several examples of specific spectral sequences. Readers will find exercises throughout the text, with additional exercises included at the end of each chapter.
Derived Functors And Sheaf Cohomology
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Author : Ugo Bruzzo
language : en
Publisher:
Release Date : 2020
Derived Functors And Sheaf Cohomology written by Ugo Bruzzo and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with Electronic books categories.
"The aim of the book is to present a precise and comprehensive introduction to the basic theory of derived functors, with an emphasis on sheaf cohomology and spectral sequences. It keeps the treatment as simple as possible, aiming at the same time to provide a number of examples, mainly from sheaf theory, and also from algebra. The first part of the book provides the foundational material: Chapter 1 deals with category theory and homological algebra. Chapter 2 is devoted to the development of the theory of derived functors, based on the notion of injective object. In particular, the universal properties of derived functors is stressed, with a view to make the proofs in the following chapters as simple and natural as possible. Chapter 3 provides a rather thorough introduction to sheaves, in a general topological setting. Chapter 4 introduces sheaf cohomology as a derived functor, and, after also defining Čech cohomology, develops a careful comparison between the two cohomologies which is a detailed analysis not easily available in the literature. This comparison is made using general, universal properties of derived functors. This chapter also establishes the relations with the de Rham and Dolbeault cohomologies. Chapter 5 offers a friendly approach to the rather intricate theory of spectral sequences by means of the theory of derived triangles, which is precise and relatively easy to grasp. It also includes several examples of specific spectral sequences. Readers will find exercises throughout the text, with additional exercises included at the end of each chapter"--
The Heart Of Cohomology
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Author : Goro Kato
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-11-08
The Heart Of Cohomology written by Goro Kato and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-08 with Mathematics categories.
If you have not heard about cohomology, The Heart of Cohomology may be suited for you. The book gives Fundamental notions in cohomology for examples, functors, representable functors, Yoneda embedding, derived functors, spectral sequences, derived categories are explained in elementary fashion. Applications to sheaf cohomology. In addition, the book examines cohomological aspects of D-modules and of the computation of zeta functions of the Weierstrass family.
Equivariant Sheaves And Functors
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Author : Joseph Bernstein
language : en
Publisher: Springer
Release Date : 2006-11-15
Equivariant Sheaves And Functors written by Joseph Bernstein and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-15 with Mathematics categories.
The equivariant derived category of sheaves is introduced. All usual functors on sheaves are extended to the equivariant situation. Some applications to the equivariant intersection cohomology are given. The theory may be useful to specialists in representation theory, algebraic geometry or topology.
Homology Cohomology And Sheaf Cohomology For Algebraic Topology Algebraic Geometry And Differential Geometry
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Author : Jean H Gallier
language : en
Publisher: World Scientific
Release Date : 2022-01-19
Homology Cohomology And Sheaf Cohomology For Algebraic Topology Algebraic Geometry And Differential Geometry written by Jean H Gallier and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-01-19 with Mathematics categories.
For more than thirty years the senior author has been trying to learn algebraic geometry. In the process he discovered that many of the classic textbooks in algebraic geometry require substantial knowledge of cohomology, homological algebra, and sheaf theory. In an attempt to demystify these abstract concepts and facilitate understanding for a new generation of mathematicians, he along with co-author wrote this book for an audience who is familiar with basic concepts of linear and abstract algebra, but who never has had any exposure to the algebraic geometry or homological algebra. As such this book consists of two parts. The first part gives a crash-course on the homological and cohomological aspects of algebraic topology, with a bias in favor of cohomology. The second part is devoted to presheaves, sheaves, Cech cohomology, derived functors, sheaf cohomology, and spectral sequences. All important concepts are intuitively motivated and the associated proofs of the quintessential theorems are presented in detail rarely found in the standard texts.
Homology Cohomology And Sheaf Cohomology For Algebraic Topology Algebraic Geometry And Differential Geometry
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Author : Jean H. Gallier
language : en
Publisher:
Release Date : 2022
Homology Cohomology And Sheaf Cohomology For Algebraic Topology Algebraic Geometry And Differential Geometry written by Jean H. Gallier and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022 with Algebraic topology categories.
"For more than thirty years the senior author has been trying to learn algebraic geometry. In the process he discovered that many of the classic textbooks in algebraic geometry require substantial knowledge of cohomology, homological algebra, and sheaf theory. In an attempt to demystify these abstract concepts and facilitate understanding for a new generation of mathematicians, he along with co-author wrote this book for an audience who is familiar with basic concepts of linear and abstract algebra, but who never has had any exposure to the algebraic geometry or homological algebra. As such this book consists of two parts. The first part gives a crash-course on the homological and cohomological aspects of algebraic topology, with a bias in favor of cohomology. The second part is devoted to presheaves, sheaves, Cech cohomology, derived functors, sheaf cohomology, and spectral sequences. All important concepts are intuitively motivated and the associated proofs of the quintessential theorems are presented in detail rarely found in the standard texts"--
Completion Ech And Local Homology And Cohomology
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Author : Peter Schenzel
language : en
Publisher: Springer
Release Date : 2018-09-15
Completion Ech And Local Homology And Cohomology written by Peter Schenzel and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-15 with Mathematics categories.
The aim of the present monograph is a thorough study of the adic-completion, its left derived functors and their relations to the local cohomology functors, as well as several completeness criteria, related questions and various dualities formulas. A basic construction is the Čech complex with respect to a system of elements and its free resolution. The study of its homology and cohomology will play a crucial role in order to understand left derived functors of completion and right derived functors of torsion. This is useful for the extension and refinement of results known for modules to unbounded complexes in the more general setting of not necessarily Noetherian rings. The book is divided into three parts. The first one is devoted to modules, where the adic-completion functor is presented in full details with generalizations of some previous completeness criteria for modules. Part II is devoted to the study of complexes. Part III is mainly concerned with duality, starting with those between completion and torsion and leading to new aspects of various dualizing complexes. The Appendix covers various additional and complementary aspects of the previous investigations and also provides examples showing the necessity of the assumptions. The book is directed to readers interested in recent progress in Homological and Commutative Algebra. Necessary prerequisites include some knowledge of Commutative Algebra and a familiarity with basic Homological Algebra. The book could be used as base for seminars with graduate students interested in Homological Algebra with a view towards recent research.
Methods Of Homological Algebra
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Author : Sergei I. Gelfand
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Methods Of Homological Algebra written by Sergei I. Gelfand and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.
Homological algebra first arose as a language for describing topological prospects of geometrical objects. As with every successful language it quickly expanded its coverage and semantics, and its contemporary applications are many and diverse. This modern approach to homological algebra, by two leading writers in the field, is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory (sheaf cohomology, spectral sequences, etc.) are described. In most cases complete proofs are given. Basic concepts and results of homotopical algebra are also presented. The book addresses people who want to learn about a modern approach to homological algebra and to use it in their work.
Homological Algebra
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Author : S.I. Gelfand
language : en
Publisher: Springer Science & Business Media
Release Date : 1994-03-29
Homological Algebra written by S.I. Gelfand and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994-03-29 with Mathematics categories.
This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.
Homotopical Algebra
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Author : Daniel G. Quillen
language : en
Publisher: Springer
Release Date : 2006-11-14
Homotopical Algebra written by Daniel G. Quillen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.