Kato’s lecture notes are like a modernist novel about commutative algebra

Recorded and posted by U Chicago grad student Zev Chonoles.  What a strange and wonderful pleasure.

As we’ve seen, there is an analogy between Z and C[T]. In fact, the analogy between Z and Fp[T] is even stronger; for example the theory of zeta functions is very similar for Z and Fp[T]. We don’t know the true reason why they are so similar; perhaps they are children of the same parents. But we don’t know who their parents are; their parents are missing.


The class group is a bitter group and a sweet group. It is bitter because when it is non-trivial it
makes a mess. It is sweet because it makes things interesting.

There is a cake shop in Balmont, which is north of Chicago. The class group is the same as this
cake shop; it is a very nice cake shop.

I could go on but you should really just read these yourself.


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11 thoughts on “Kato’s lecture notes are like a modernist novel about commutative algebra

  1. These are fabulous Jordan, thank you for sharing!! Thank you to Zev, thank you to Kato!! Now everyone go learn the “Love song in the land of prime numbers” on p19, Lecture 8. After you learn it, this will mean you never forget it: “(q choose p) is how the girl q is reflected in the heart of the boy p, and (p choose q) is how the boy is reflected in the heart of the girl q. As you may some experience with, these are sometimes not related in our world. But in the world of prime numbers, they are related – this is very mysterious.” Now that’s a good teacher.

  2. byesac says:

    I like this, “If you look on the street, you never meet a commutative ring; that’s rather strange. They are rather shy I think. We need to ask them to come to this room. Rings, rings, please come! Rings, rings please come! *shuffles along the floor, playing the part of the ring* Finite fields, come! Rings of functions, come! *hops* I think they are here now.”

    This will be a nice supplement to my algebra class this semester.

  3. Zev Chonoles says:

    The other students and I have definitely been enjoying the course. There are even more little things about Professor Kato’s lectures that I’m not quick enough to type, or which simply can’t be captured in writing.

    The cake shop comment turned out to have a mundane explanation (?): after class one of my fellow grad students pointed out to me that in fact there is a cake shop called “Bittersweet” (http://www.bittersweetpastry.com/) in Belmont, so in all likelihood it was what Professor Kato was referring to.

  4. DS says:

    That’s just around the corner from our apartment. It is a nice cake shop.

  5. JSE says:

    Sure, but as nice as the ideal class group…??

  6. NDE says:

    Surely “(p choose q)” and “(q choose p)” were not combinatorial coefficients but Legendre symbols (with p and q separated by a horizontal line as if forming the fractions (p/q) and (q/p)); if it were really “choose” then one of these formulas must give zero (or both are 1 if p=q), which would make for a rather less romantic metaphor…

  7. Laura says:

    Thank you for the correction NDE!

  8. Joseph Nebus says:

    Thanks for a smile-generating observation. Mathematicians don’t seem to promote the beauty not just of the ideas but of the language we use. These are particularly good examples, certainly, but maybe the field would be less intimidating if we let people know there was melody in the writing too.

  9. Yohei Shiraishi says:

    In Japan, he is famous as a Great Storyteller; In his math books for Japanese, a lot of heartfelt episodes.are depicted. His lecture is also amazing :-))

    He looks like he is still quite well !! (so I’m happy.)

  10. ラッシュガード キッズ 長袖

  11. A. says:

    If you feel like updating the dead links on your blog, Zev Chonoles transcription of Kato’s lectures can now be found on Zev’s personal site: https://zevchonoles.org/math/course-notes/kato-commutative-algebra.pdf. The problem with personal sites is that people tend to abandon their own sites after a while, so here is a (hopefully) more stable link, on the Way Back Machine: https://web.archive.org/web/20181021152600/http://math.uchicago.edu/~chonoles/expository-notes/courses/2013/326/math326notes.pdf.

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