Algebraic geometry II - schemes and sheaves

In lecture 13, we covered the following topics:

-definition of discrete valuation with examples;

-(locally) finite morphisms;

-a finitely generated field extension of an algebraically closed field k corresponds to the function field of a k-variety;

-interpretation of DVR's (under suitable assumptions) as germs of curves;

-separatedness and its valuative criterion;

-properness and its valuative criterion;

-definition of varieties in the realm of schemes;

-going-up theorem.