The fifth post is the October 23 version here. This fortnight’s reading is 6.5-7.5. By the end of it, you will be able to work with morphisms of schemes. In case you have seen these ideas before, I’ve included two more advanced sections, 7.6 and 7.7 (starred or double-starred), as well as the first section of chapter 8.

I will continue to fall behind on responding to comments. I’ve chosen to post notes and move forward, but you’ll notice that I’m gradually catching up. (I expect to fully catch up in mid-December, and then not fall behind for any appreciable time thereafter.)

For learners.

Exercises: As usual, what you should do depends on who you are and where you (mentally) are. Likely you should do a subset of these, and then a selection of other exercises. But the following exercises include some key concepts: 6.5.B (meaning of associated prime), 6.5.D (commutative algebra to properties of associated points), 6.5.F (“being 0” is stalk-local), 7.2.A (morphisms of ringed spaces glue), 7.2.C (maps of stalks induced by maps of ringed spaces), 7.2.D (maps of rings induce maps of ringed spaces), 7.3.B (the previous construction is a map of locally ringed spaces), 7.3.F (maps to Spec A “are” maps of global sections in the other direction), 7.4.A (maps of graded rings give maps of Proj’s), 7.5.C (dominant maps are maps of function fields in the opposite direction), 7.5.E (categories of “varieties” and “finitely generated field extensions” are “basically the same”). The following exercises let you get your hands dirty: 6.5.A, 6.5.K, 7.2.E, 7.3.E, 7.4.C, 7.5.A, 7.5.H, 7.5.J (and 7.5.K and 7.5.L are fun). If you reading 7.6, try 7.6.B and 7.6.C

Some questions for you: Can you get insight out of section 6.5 on associated points? Are the problems do-able? Are there any hints I should have included to make them easier? (I want these problems to be as easy as possible, yet still be problems. I realize that many of them are not easy.) By the end, you should be perplexed by the pictures, but hopefully intrigued and not put off. These pictures are often secret knowledge, never mentioned in writing, because they are so hard to convey outside of a conversation at a blackboard. If you decided try to understand them only in the case of reduced schemes (which is reasonable, and which is even recommended in the text), did that turn out okay for you?

Does the hint make Exercise 7.5.D do-able? I can’t “unremember” what I now know to judge how reasonable this problem is.

For experts.

I had earlier hoped to duck associated points, and “star” section 6.5. But I realized that I later needed it, both for modules and for rings. Perhaps it should be starred on a first reading. I’ve tried to make the mathematical development as painless as possible, excluding all but that necessary, either for future use, or for the development of intuition. (I’ve changed the exposition from course notes in earlier years. Primary ideals appear only as an aside!) I’ve also taken the position that what matters most about associated points are their properties, and their definition should be in service of their properties, and not vice versa. (I do this a lot, e.g. for cohomology.) Associated points make sense for locally Noetherian and integral schemes, which makes for awkward exposition: we want both. I’m unhappy that rational functions are so hard to explain on locally Noetherian schemes, and don’t want to throw the reader. I introduce the total fraction ring Q(A), but have noticed that I never use it! (No, I don’t do Cartier divisors.)

I don’t define radiciel morphisms. If you find this a horrific omission, please say so, and make a case that they should be here (even in a double-starred exercise).

Any pedagogical improvements to the section on representable functors would be appreciated. As one of the topics where the main new content is a point of view (rather than explicitly mathematical content), it is an interesting challenge to explain this well. And yes, I know the initial definition of the Grassmannian is the wrong one (just as my initial definition of projective space is the wrong one). But I think the Grassmannian is unusual in that it tends not to be defined in any introductory source (what are the counterexamples?), yet more advanced sources assume that you know what it is. Somewhere you have to see it for the first time, and I like to introduce it as early as possible. I further think that the right first definition to see isn’t in terms of the quot functor, any more than projective space shouldn’t be first seen in terms of its functorial definition. Everyone “knows” what projective space is, and our job is to connect the “right” definition to that initial idea.

For everyone.

There are so many properties of morphisms of schemes that it is easy and natural to become lost. It still seems best (as someone explaining them) to introduce them in a huge torrent. Anything that can help to make them individually memorable is a good thing — ideas are appreciated!