--- title: Cohomology description: Cohomology is a mathematical tool that associates algebraic invariants with geometric spaces to measure global topological obstructions. Learners will understand sheaf cohomology, Čech cohomology, and derived functors, enabling them to solve global existence problems in algebraic geometry. category: mathematics subcategory: algebraic-geometry difficulty: beginner, intermediate, advanced url: /subject/cohomology --- # Cohomology Cohomology is a mathematical tool that associates algebraic invariants with geometric spaces to measure global topological obstructions. Learners will understand sheaf cohomology, Čech cohomology, and derived functors, enabling them to solve global existence problems in algebraic geometry. ## Available Resources 2 Books • 6 Websites ## Websites ### 1. Hatcher's Algebraic Topology (free PDF) **Author:** Allen Hatcher Allen Hatcher's standard graduate textbook, freely downloadable from Cornell, covering fundamental groups, homology, cohomology and homotopy theory, with CW complexes introduced in Chapter 0 and used throughout. Readers learn to build spaces from cells and compute their algebraic invariants. **Difficulty:** Intermediate | **Price:** Free **Link:** https://pi.math.cornell.edu/~hatcher/AT/ATpage.html **Tags:** algebraic-topology, cw-complexes, homology, homotopy-theory, fundamental-group ### 2. nLab Cohomology nLab reference entry defining cohomology in the general framework of homotopy theory and higher category theory, as maps into classifying objects. Relates singular, sheaf, group and generalized cohomology, letting readers see how the classical theories fit one abstract definition. **Difficulty:** Beginner | **Price:** Free **Link:** https://ncatlab.org/nlab/show/cohomology **Tags:** cohomology, homotopy-theory, higher-category-theory, sheaf-cohomology ### 3. Milne's Etale Cohomology Notes J.S. Milne's free lecture notes on étale cohomology, covering étale morphisms, sites and sheaves, cohomology of curves, compact supports, and the Lefschetz trace formula. Readers work toward proofs of the rationality and functional equation in the Weil conjectures. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.jmilne.org/math/CourseNotes/LEC.pdf **Tags:** etale-cohomology, algebraic-geometry, sheaf-cohomology, weil-conjectures, lecture-notes ### 4. Stacks Project **Author:** The Stacks Project Authors A collaborative, open-source reference on algebraic geometry run from Columbia University, with thousands of pages building from commutative algebra to schemes and algebraic stacks. Every result has a stable tag, so readers can look up precise statements and complete proofs. **Difficulty:** Advanced | **Price:** Free **Link:** https://stacks.math.columbia.edu/ **Tags:** algebraic-geometry, schemes, commutative-algebra, algebraic-stacks ### 5. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ### 6. nLab Collaborative wiki for mathematics, physics and philosophy written from a category-theoretic viewpoint, with cross-linked entries on categories, functors, adjunctions, topos theory, higher category theory and homotopy theory. Readers can look up precise definitions, examples and references for advanced structural mathematics. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://ncatlab.org **Tags:** category-theory, higher-category-theory, topos-theory, homotopy-theory, reference-wiki ## Books ### 1. Elements of Algebraic Topology **Author:** James R. Munkres Textbook on homology theory built from simplicial complexes, then singular homology, cohomology, duality on manifolds, and the Künneth and universal coefficient theorems. Readers learn to compute homology groups concretely and follow the homological algebra underlying them. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0201045869?tag=edmonddante07-20 **Tags:** algebraic-topology, homology, cohomology, simplicial-complexes, homological-algebra ### 2. An Introduction to Homological Algebra **Author:** Charles A. Weibel Graduate textbook covering chain complexes, derived functors, Tor and Ext, spectral sequences, group and Lie algebra cohomology, simplicial methods, and derived categories. Readers learn to compute cohomology with resolutions and spectral sequences, the tools underlying algebraic topology and geometry. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0521559871?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, algebraic-geometry --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/cohomology