图书简介
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 Č ech 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.
Key Features:
○Provides a general introduction to the theory of derived functors, and also establish in detail its application to sheaves that is a unique feature
○The comparison between Čech and sheaf cohomology is developed in greater detail
○Spectral sequences are introduced in a very pedagogical way
○Moreover, all available textbooks on these topics are quite dated
Basic Notions: Category Theory, Homological Algebra; Derived Functors: Injective Objects, Right Derived Functors, Free Resolutions, the Long Exact Sequence of a Derived Functor, Acyclic Resolutions, δ-Functors, Left Derived Functors, Additional Exercises; Sheaves: Presheaves and Sheaves, Étalé Space, Direct and Inverse Images, Additional Exercises; Cohomology of Sheaves: Čech Cohomology, Sheaf Cohomology, Comparing Sheaf and Čech Cohomology, Fine Sheaves and de Rham Cohomology, Sheaf Cohomology of Schemes, Additional Exercises; Spectral Sequences: Filtered Complexes, the Spectral Sequence of a Filtered Complex, the Spectral Sequences Associated with a Double Complex, Hypercohomology, the Spectral of Hyperderived Functors, Some Applications, Additional Exercises; Appendix: Pushouts and Pullbacks, Snake Lemma, Baer’s Criterion, Additional Exercises;
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