Math 216

Please take a look at Ravi’s blog about his Math 216 graduate course at Stanford university. Students and others have been chiming in leading to a total of 265 comments in 6 months. A group of people (mainly graduate students?) are working through the material as it gets updated on Ravi’s blog, and these people provide most of the which are helpful and constructive comments on the blog. Moreover, even though Ravi is not actually teaching his course this year, the blog gives one a sense of activity much like for a real course. Of course, since I am teaching my algebraic geometry course this year based on Ravi’s lecture notes, I may be more inclined to say so than others.

You can download the latest version of Ravi’s notes here. Let me give you a bit of my own preliminary impression of these notes; you can read Ravi’s philosophy behind them on his blog and in the introduction to the notes.

As everybody who has taught an algebraic geometry course knows it is virtually impossible to feel satisfied with the end result. In my experience it actually works well when younger people teach it because they have a fresh take on it, want to get to some particular material that is important to them and they are less likely to get stuck in details. I personally never teach algebraic geometry the same way twice, and I usually end up covering a fair amount of material despite feeling like I did not at the end of it.

One of the pleasing aspects of teaching the material out of Ravi’s notes is that I do not have to organize the material as much as I usually do. Mostly I am happy with the order in which things get done, although I moved the material on quasi-coherent O_X modules and on morphisms of schemes earlier in my lectures. Also, in hindsight, I should probably have skipped chapter 2 (category theory) and jumped straight to the chapter on sheaves. A key feature of Ravi’s notes is that more than 75% of the proofs of lemmas, propositions, and theorems are left as exercises. As lecture notes often Ravi explains why things are true, with lots of examples, rather than providing a formal proof. Results from previous exercises are used throughout the text, not always with explicit references (especially in the exercises themselves of course). When lecturing it sometimes made me wonder to what extend I’ve really built up the theory from scratch (which is the stated goal of the course). Of course here you can rely on outside references and ask that students read those, ask that the students do lots of exercise, and so on. One of the standing assumptions underlying the setup is that students will work hard on their own to understand the material. Moreover, I think no matter how you teach algebraic geometry you cannot build it up completely from scratch in your lectures, i.e., the students are always going to have to do a lot themselves, and maybe by building it into the course material they are more likely to do it?

Is it a good idea to have many different algebraic geometry texts? Tentatively, I would say more is better. I have personally found Ravi’s notes useful in the following way: if you can find what you’re looking for in Ravi’s notes (e.g. by googling) then you’ll quickly find pointers unencumbered by details or generalities.

Overall I am very happy with my course and the notes so far. One of my questions is how much commutative algebra I will cover teaching the course in this way (traditionally at Columbia we teach a first semester of commutative algebra and then a second semester on schemes — in one semester focused entirely on commutative algebra you can cover quite a bit). I’ll report on this in another post about Ravi’s notes at the end of the next semester, so stay tuned.

6 thoughts on “Math 216

  1. I like Hartshorne’s book. It can be difficult. But after working through Chapters 2 and 3, students have a sense of accomplishment as well as “bragging rights”. Students develop confidence in the foundations; they don’t feel that Hartshorne has “cheated” and claimed a result without proof (except for the many commutative algebra results labeled with an “A”). And most importantly, Hartshorne’s book is a standard reference. If a student knows where results are in Hartshorne, they can use that knowledge forever in citing results in their own articles.

    Of course it is great to have other references for more perspective (and for topics which Hartshorne doesn’t discuss, like coproducts). And also, of course, eventually students need to move on to reading parts of EGA. But in the first instance, I recommend beginning algebraic geometry graduate students to read Hartshorne’s book.

    • “Eventually” in my case meant about twenty years…
      I agree with everything that Jason says.
      But I also second Brian’s recommendation to look at several books.
      Hartshorne is a beautiful distillation of EGA, gives complete proofs, is virtually error-free, and includes many good exercises.
      But one needs a source of concrete examples: Harris, Griffiths-Harris, Shafarevich are good for this.
      Also, some topics fall between the cracks in Hartshorne: normality, for example, and smoothness (the key point, as I see it, is the relationship between formal smoothness, having smooth fibers, and being ├ętale-locally modeled on affine space).

  2. A good analogy is with learning class field theory, for which it seems best to really look at multiple sources (Lang, Cassels-Frohlich, Serre’s “Local Fields”, ….) that provide diverse points of view. I recommend to students that they keep an eye on both Hartshorne and Qing Liu’s book as they learn the material: each has merits that the others lack, and both have a wealth of instructive exercises. Ravi’s notes are great for giving an intuition and “big picture” and sense of technique which is missing elsewhere, but I imagine that his decision to put so much of the details of certain proofs in the exercises may be frustrating to some students. On the other hand, some topics in Hartshorne are quite opaque, such as blow-ups and the universal property of projective space (let alone how to think about the role of valuative criteria) and are nicely done in Ravi’s notes. In the future I will recommend to students that they look at all three references.

  3. As a user of algebraic geometry, i.e., someone who wishes to be able to understand the basic statements and language, and to write down very elementary arguments or to explain questions clearly to more expert colleagues, I agree that there is an invaluable feeling of confidence that comes from having actually gone through something like Hartshorne Ch. 2+3 in detail. In fact, I regret not having gone through a similar phase with Liu’s book, because some of the arithmetic aspects can be missed from Hartshorne. (E.g. I remember being very puzzled and requiring assistance to understand which statements in Katz-Mazur explained how to compute how many cusps of various modular curves were defined over Z/pZ, because being finite ├ętale over something like Spec(Z) was not a property I had an intuitive feeling about at the time…)

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