What is Inferred Algebraic Geometry

Lecture: Algebraic Geometry 2

The lecture is the continuation of the lecture on algebraic geometry in the summer semester. Appropriate prior knowledge is required.

Time and place: Di, 8-10, SR B; Thu, 8-10, SR B.

Exercises: Thu, 14-16, SR F.

Literature: The same recommendations apply as for algebraic geometry 1. I also mention the cohomology theory

  • R. Godement, Topologie Algébrique et théorie des faisceaux, Hermann 1960.
  • A. Grothendieck, Sur quelques points d'algèbre homologique, Tohoku Math. J. 9 (1957), 119-221.
  • C. Weibel, An introduction to homological algebra, Cambridge University Press.

A nice-to-read introduction to the concept of spectral sequence is given in

  • T. Chow, You could have invented spectral sequences, Notices of the AMS, January 2006


Content of the lecture:

9. Projective morphisms
9.1 The Proj construction
9.2 Quasi-coherent sheaves on Proj S.
9.3 Very ample and ample bundles of straight lines

10. Affine and proper morphisms
10.1 Affine, finite, whole morphisms
10.2 Actual morphisms
10.3 Evaluation Criteria

11th dimension, flat morphisms
11.1 dimension
11.2 Flat morphisms

12. Normal schemes
12.1 Definition, simple properties
12.2 normalization
12.3 Rational mappings
12.4 Tsariski's main theorem
12.5 Algebraic Curves

13. Cohomology of sheaves
13.1 Derived functors
13.2 Sheaf cohomology
13.3 Cech cohomology
13.4 Spectral Sequences
13.5 Higher direct images

14. Cohomology of affine schemes, of projective space
14.1 Cohomology of quasi-coherent sheaves on affine schemes
14.2 Applications
14.3 Cohomology of Projective Space
14.4 Finiteness Theorems

15. Serre duality
1. Duality for the projective space
2. The dualizing sheaf, Serre duality
3. Sheaves of differential forms, smooth morphisms
4. The Riemann-Roch theorem

16. Theorem about formal functions, cohomology and base change
1. Theorem about formal functions
2. Cohomology and change of base