Lecture 4: Morphisms

We define morphisms between (embedded) affine varieties and see how these produce in a natural way a morphism of ringed spaces.

We see how there is an "arrow reversing" bijective correspondence between morphism of varieties and morphism of their rings of regular functions. 

Now that we have introduced morphism we can enlarge the definition of affine variety to a more abstract (not embedded) one.  In addition, we can see how the product of varieties is indeed a product. 

We conclude the lecture by showing that distinguished open sets in affine varieties are affine varieties, but this is not true for other open sets. (We did not have time to do Example 4.18 in class.)

Literature: [G] Chapter 4

Suggested exercises:  4.12–13, 4.19