This final chapter starts with an overview, relating the material of the book to more advanced subjects. There is a brief discussion of extra topics that I would like to have included, but that would have stretched the material beyond what can be covered in a 30 hour undergraduate lecture course. In 9.4–9.5 I treat some counterexamples, for example to finiteness of normalisation, that are traditionally regarded as “difficult”. Apart from the evangelical aspiration of making these examples accessible to a wider public, my purpose here is to illustrate some ways in which Noetherian rings can be more general than geometric rings k[V], while retaining a strong geometric flavour.

I also discuss in 9.7–9.9 the current sociological status of algebra in math. With research experience restricted to algebraic geometry, I do not claim any particular authority to write on algebra, but it is clear that there is room in the subject for views other than those of professional algebraists.

Where we've come from

Chapters 1–4 and 6 discuss fairly harmless general ideas in algebra such as generators of a module and Noetherian rings. The material is standard, but I have mixed in some more substantial things like the picture of Spec ℤ[X, Y] in Chapter 1 and the discussion of resolution of singularities of curves in Chapter 4, partly in the hope of keeping the reader awake.

Chapters 5, 7 and 8 are three substantial individual topics, and the undergraduate exam always contains questions on at least two of these three chapters.