Algebraic geometry II - schemes and sheaves

In lecture 7, we covered the following topics:

-geometric interpretation of the primary decomposition, with examples;

-closed subsets of Spec can be uniquely written as union of irreducible closed subsets.

-Krull's Hauptidealsatz (height theorem), its geometric interpretation, with examples;

-dimension of a polynomial ring over a Noetherian ring and of a local Noetherian ring (via system of parameters);

- review of dimension theory;

-going up, going down, Noether normalization;

-height estimates beyond the case of Krull's theorem (left as Exercise);

-review of the classical projective space as a group quotient and motivation for Proj (as space of lines in the affine space).