Lecture 5: Varieties

In this lecture we define abstract (pre)varieties: ringed spaces that are locally affine. We can construct examples by gluing (affine) varieties. Two examples: the affine line with doubled origin and the projective line.

Open and closed subsets of varieties are varieties. The product of two varieties is a variety and equals the usual Cartesian product as sets but not as topological spaces.

Finally, we introduced the notion of separated varieties (diagonal is closed). Terminology: prevariety = not necessarily separated variety; variety = separated variety.

Literature [G] Chapter 5

Suggested exercises:  5.7–9, 5.21, 5.22 (first do 2.24), 5.23, 5.24