[Download] "Derived Functors and Sheaf Cohomology" by Ugo Bruzzo & Beatriz Graña Otero * eBook PDF Kindle ePub Free
eBook details
- Title: Derived Functors and Sheaf Cohomology
- Author : Ugo Bruzzo & Beatriz Graña Otero
- Release Date : January 10, 2020
- Genre: Mathematics,Books,Science & Nature,
- Pages : * pages
- Size : 17037 KB
Description
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.
Contents: Basic Notions:Category TheoryElements of Homological AlgebraAdditional ExercisesDerived Functors:Injective ObjectsRight Derived FunctorsFree ResolutionsLong Exact Sequence of a Derived FunctorAcyclic Resolutionsδ-FunctorsLeft Derived FunctorsAdditional ExercisesIntroduction to Sheaves:Presheaves and SheavesÉtalé SpaceSome ConstructionsDirect and Inverse ImagesAdditional ExercisesCohomology of Sheaves:Čech CohomologySheaf CohomologyComparing Čech and Sheaf CohomologyFine Sheaves and de Rham CohomologySheaf Cohomology of SchemesAdditional ExercisesSpectral Sequences:Filtered ComplexesThe Spectral Sequence of a Filtered ComplexThe Spectral Sequences Associated with a Double ComplexHypercohomologyThe Spectral Sequence of Hyperderived FunctorsSome ApplicationsAdditional ExercisesEpilogue: Derived CategoriesAppendices:Pushouts and PullbacksSnake LemmaBaer's CriterionAdditional Exercises
Readership: Graduate students and researchers interested in derived functors and sheaf cohomology.Derived Functors;Sheaves;Spectral Sequences0Key Features:Provides a general introduction to the theory of derived functors, and also establish in detail its application to sheaves that is a unique featureThe comparison between Čech and sheaf cohomology is developed in greater detailSpectral sequences are introduced in a very pedagogical wayMoreover, all available textbooks on these topics are quite dated