# 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

given.

### 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

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