Sheaves, Manifolds and Cohomology 21

Info. The course is organized by me and Andrea Gagna. The lectures will be delivered at the Maths Department of Charles University in room K5, every Friday from 14 to 15.45 o'clock. We start the 8th of October.

If you send me or Andrea an email about this course make sure to include the word SMC21 in the subject. This makes our life easier.

Lectures may be decorated by additional material. Click on the for the exercise sheet. Check out for a very synthetic summary of the lecture.


Lecture Material Reference
1. Background in Categories. Lei, Ch. 1 and 5.
2. Adjunctions, Yoneda. Lei, Ch. 2 and 4.
3. Sheaves, towards étalé spaces. Wed, parts of Ch. 3.
4. Sheaves, sheafification. Wed, parts of Ch. 3.
5. Geometric morphisms, Manifolds as ringed spaces. Wed, the rest of Ch. 3 + main ideas from Ch. 4.
6. Bundles and 𝒪-modules. Wed, 8.2-3-4.
7. Cohomology of Sheaves. Wed, §10.1.
8. Cohomology of Sheaves.
9. Cohomology of Sheaves.
10. Cohomology of Sheaves.
11. Cohomology of Constant Sheaves.
12. Cohomology of Constant Sheaves.

Description. The aim of the course is to introduce the student to the concept of sheaf over a topological space. We choose manifolds as our object of study, but most of the theory would stay in place for any ringed space, and indeed this point of view makes the framework very flexible and applicable to both differential topology and algebraic geometry. The main content of the course is the cohomology of sheaves, which generalizes singular, simplicial, De Rham cohomology and offers a broad framework for homological algebra.

Audience. The course is open to master and Ph.D. students. The audience is expected to have some familiarity with the following topics and notions: manifolds and classical examples of manifolds, categories and basic notions of category theory, rings and modules over a rings.

Syllabus. Sheaves over a topological space, Manifolds as ringed spaces, Bundles and 𝒪-modules, Cohomology of Sheaves, Cohomology of Constant Sheaves. These cover approximately the chapters 3-4-8-10-11 of the book by Wedhorn in the Bibliography.

Exam rules. 2 exercise sheets during the course + oral examination. The exercise sheets () will be released on the 1st and 7th lecture. Since the release the student has 7 weeks to hand the solutions. The exercise sheets will be scored from F (fail) to A (very good) and each of them will be worth 15% of the final grade. They should be understood as a necessary condition to stand the oral examination, which is worth most of the final score (70%).

Bibliography.