Algebraic geometry II - schemes and sheaves

In Lecture 4, we covered the following topics:

-the stalks of the structure sheaf on Spec(R) (with proof this time);

-sections of the structure sheaf on principal open subsets D(f);

-ringed spaces, locally ringed spaces, and morphisms thereof;

-equivalence of categories between rings and spectra as locally ringed spaces (just statement);

-a ring homomorphism induces a continuous map between their spectra (in the reversed direction).