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Technische Universität München
Technische Universität München

Cohomology of sheaves on schemes

Time and venue

Lectures: Mo 16:15 - 17:45, Mi 10:15 - 11:45, in 02.06.020, Rechnerraum (5606.02.020)

Exercises: Di 12:15 - 13:45

Content

Serre was the first to consider cohomology groups of OX-modules on a variety X in the mid 1950's. The theory was then further developed by Grothendieck et al. for general schemes and also extended to a cohomology theory of topoi which became a fundamental tool in algebraic and arithmetic geometry. In this course we concentrate on the cohomology of OX-modules. If F is an OX-module and i is a non-negative natural number the ith cohomology group of F is an OX(X)-module Hi(X,F). These have some nice abstract properties (functorial in F, long exact sequence, etc.) but are in general very hard to describe. However, if for example X is projective over a field k and F is a coherent OX-module, then these cohomology groups are finite dimensional k-vector space which encode geometric properties of X. Hence one can use this to study geometric questions in the realm of linear algebra. Sometimes such cohomology groups help to decide whether two given varieties are not isomorphic (or not birational, i.e., don't have isomorphic function fields), they give a criterion when a variety is affine, or a given line bundle is ample, they can be used to find the global section of an OX-module, to classify varieties, etc.

In the lecture we will discuss (some of) the following topics:

  • some basic homological algebra
  • cohomology of sheaves on topological spaces 
  • Grothedieck's vanishing theorem 
  • Serre's cohomological affineness criterion
  • cohomology of the projective space
  • cohomological criterion for ampleness
  • finitness theorem for projective schemes
  • differential calculs 
  • smooth and étale morphisms
  • Serre duality
  • Grothendieck's formal function theorem
  • Zariski's connectedness theorem

Prerequisites

Basic knowledge of commutative algebra (as in Atiyah-Macdonald) and algebraic geometry, in particular the language of sheaves and schemes (as, e.g., in Hartshorne's book, II, 1-5).

Exam

There will be an oral exam on 19.08.2019 (Monday) during 10 - 14. Each oral exam will last about 30 minutes. The students will receive the time and place of their examination by email.

Exercises

  1. Exercise sheet (will be discussed on Tue, April 30)
  2. Exercise sheet (v2, will be discussed on Tue, May 7)
  3. Exercise sheet (will be discussed on Tue, May 14)
  4. Exercise sheet (will be discussed on Tue, May 21)
  5. Exercise sheet (will be discussed on Tue, May 28)
  6. Exercise sheet (will be discussed on Tue, June 4)
  7. Exercise sheet (will be discussed on Tue, June 18)
  8. Exercise sheet (v2, will be discussed on Tue, June 18)
  9. Exercise sheet (v2, will be discussed on Tue, July 2)
  10. Exercise sheet     (will be discussed on Tue, July 9)
  11. Exercise sheet     (will discussed on Tue, July 16)
  12. Exercise sheet     (will be discussed on Tue, July 23)

Lecture Notes

Here you find the handwritten lecture notes for the course (numbered by sections).

  1. Motivation
  2. (a little bit of) Homological Algebra
  3. Injectives
  4. Right Derived Functors
  5. Cohomology of Sheaves
  6. Grothendieck Vanishing on noetherian topological spaces
  7. Cech Cohomology
  8. Cohomology for quasi-coherent sheaves
  9. The cohomology of the projective space
  10. Ampleness Criteria and Finiteness Results
  11. Kähler Differentials
  12. Smooth Morphisms
  13. Ext-Groups
  14. Exterior Products
  15. Weak Duality
  16. Serre Duality for local complete intersections over a field

Literature

 

 

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