I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a number of reasons (most of all the students, who
were smart, hard-working, and from a variety of fields). I've
particularly enjoyed talking with experts (some in nearby fields, many
active on mathoverflow) about what one should (or must!) do in a first
schemes course. I've been pleasantly surprised to find that those who
have actually thought about teaching such a course (and hence who know
how little can be covered) tend to agree on what is important, even if
they are in very different parts of the subject. I want to raise this
question here as well:
What topics/examples/ideas etc. really really should be learned in a
year-long first serious course in schemes?
Here are some constraints. Certainly most excellent first courses
ignore some or all of these constraints, but I include them to focus
the answers. The first course in question should be
purely algebraic. (The reason for this constraint: to avoid a
debate on which is the royal road to algebraic geometry — this is
intended to be just one way in. But if the community thinks that a
first course should be broader, this will be reflected in the voting.)
The course should be intended for people in all parts of algebraic
geometry. It should attract smart people in nearby areas. It should
not get people as quickly as possible into your particular area of
research. Preferences: It can (and, I believe, must) be hard. As
much as possible, essential things must be proved, with no handwaving
(e.g. "with a little more work, one can show that…", or using
exercises which are unreasonably hard). Intuition should be given
when possible.
Why I'm asking: I will likely edit the notes further, and hope to post
them in chunks over the 2010-11 academic year to provoke further debate. Some hastily-written thoughts are
here,
if you are curious.
As usual for big-list questions: one topic per answer please. There
is little point giving obvious answers (e.g. "definition of a
scheme"), so I'm particularly interested in things you think others
might forget or disagree with, or things often omitted, or things you
wish someone had told you when you were younger. Or propose dropping
traditional topics, or a nontraditional ordering of traditional topics. Responses
addressing prerequisites such as "it shouldn't cover any commutative
algebra, as participants should take a serious course in that subject
as a prerequisite" are welcome too. As the most interesting
responses might challenge (or defend) conventional wisdom, please give
some argument or evidence in favor of your opinion.
Update later in 2010: I am posting the notes, after suitable editing, and trying to take into account the advice below, here. I hope to reach (near) the end some time in summer 2011. Update July 2011: I have indeed reached near the end some time in summer 2011.
Best Answer
One of the wholly unnecessary reasons that schemes are regarded with such fear by so many mathematicians in other fields is that three, largely orthogonal, generalizations are made simultaneously.
Considering a "variety" to be Spec or Proj of a domain finitely generated over an algebraically closed field, the generalizations are basically
Allowing nilpotents in the ring
Gluing affine schemes together
Working over a base ring that isn't an algebraically closed field (or even a field at all).
For many years I got by with only #1. More recently I've been interested in #1 + #3. Presumably someday I'll care about #2, but not yet. Anyway I think it's crazy to give the impression that the three are a package deal that one must buy all of simultaneously, rather than in much easier installments.
I think it could be useful to explain which subfield of mathematics, or which important example, motivates which of #1,#2,#3 is really a necessary generalization.