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LECTURE 22, THURSDAY MAY 13, 2004

Our goal is to sketch a generalization of algebraic geometry from curves with coordinate rings to more general objects whose coordinate rings are arbitrary rings; the coordinate rings K[C] of curves C in the affine plane over K all contain the field K, so rings like Z or Z/mZ cannot occur. Our motivation is drawn from the fact that the rings Z(p) have a lot of properties in common with the local rings OP (C). We started out with a general ring R and defined the spectrum of R, Spec R, as the set of prime ideals in R. These prime ideals will be our “points”, and R itself will play the role of the coordinate ring. In fact, ring homomorphisms R −→ S induce maps Spec S −→ Spec R, and this corresponds to the fact that ring homomorphisms K[D] −→ K[cC] induce polynomial maps C −→ D. Next we made Spec R into a topological space: its closed sets are, by definitions, sets of the form