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.
Contents:
- Preface
- Introduction
- Homology and Cohomology
- de Rham Cohomology
- Singular Homology and Cohomology
- Simplicial Homology and Cohomology
- Homology and Cohomology of CW Complexes
- Poincaré Duality
- Presheaves and Sheaves; Basics
- Čech Cohomology with Values in a Presheaf
- Presheaves and Sheaves; A Deeper Look
- Derived Functors, δ-Functors, and ∂-Functors
- Universal Coefficient Theorems
- Cohomology of Sheaves
- Alexander and Alexander–Lefschetz Duality
- Spectral Sequences
- Bibliography
- Index
Readership: Senior undergraduates of maths major who are familiar with some basic notions of linear algebra and abstract algebra, in particular the notion of a module. Also good for graduate students of abstract algebra courses.
Key Features:
- Provides a historical overview of the development of homology/cohomology, while at the same time gently introducing the reader to pertinent fundamental concepts such as exact sequences, chain complexes, presheaves/sheaves, stalk spaces, universal functors, and sheaf cohomology
- Five step illustrated presentation in Chapter 7 of the Poincare duality theorem. This presentation explicitly works out the omitted details of the "canonical" Milnor and Stasheff presentation
