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.

shouldbe taught in a first schemes course, but it's something that I'd love to see exposited more fully. Jim Borger gave an outline of a program to jump straight into algebraic spaces, skipping schemes entirely. Maybe you could figure out a way to do it? sbseminar.wordpress.com/2009/08/06/… $\endgroup$1more comment