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John Baez (May 11 2022 at 05:35):Is anyone here willing to talk about motives? I'll have a bunch of sort of basic questions about them.
John Baez (May 11 2022 at 05:37):Here's one, for example. I'd like to focus on pure Chow motives.
John Baez (May 11 2022 at 06:00):It seems the ingredients that go into setting up this category are the category of smooth projective varieties (which we then take spans of, called [[correspondences]]), the concept of [[algebraic cycle]], and an [[adequate equivalence relation]] on algebraic cycles.
John Baez (May 11 2022 at 06:04):But it seems we just need the concept of "algebraic 0-cycle" to actually get the category of motives. And it seemsthat the concept of "algebraic 0-cycle" and "rational equivalence of algebraic 0-cycles" can be defined using just categorical abstract nonsense starting from the category of smooth projective varieties. Is that right?
John Baez (May 11 2022 at 06:14):The general goal of my question here is to try to strip down the necessary input data for constructing something resembling a category of pure motives to the bare minimum.
Matteo Capucci (he/him) (May 11 2022 at 08:00)::+1: following this
Tim Hosgood (May 11 2022 at 08:55):i’d like to talk about motives (so i’m leaving this comment to remind me to write something properly when i get to my laptop)
John Baez (May 11 2022 at 20:23):I think some of what I said was a bit confused... I hope the intent was somewhat clear nonetheless.
John Baez (May 11 2022 at 20:27):Here's something I'm mildly confused about, though maybe I'm working out my confusion by asking this question. Wikipedia defines pure Chow motives starting with "degree k correspondences" meaning spans of smooth projective variety where the apex of the span is equipped with a codimension-k Chow cycle.
John Baez (May 11 2022 at 20:28):But then, when they get going, they only use degree zero correspondences.
John Baez (May 11 2022 at 20:29):Am I right that this means we can just forget about equipping our correspondence with a codimension-k Chow cycle and get the same category of pure Chow motives in the end?
John Baez (May 11 2022 at 21:00):I think that must be true.
John Baez (May 12 2022 at 07:18):Wow, I was really confused. Please ignore all the mistakes I just made. :nauseated:
John Baez (May 16 2022 at 00:06):I asked a more interesting question about motives on MathOverflow, and Will Sawin gave a very nice answer. Later I found part of that answer here too:
(This seems to be a draft of a book. I like it.)
The first part of the question was whether you can restate the Riemann hypothesis part of the Weil conjectures directly in terms of a formula for the number of points of a variety. The second part was whether the summands in my guessed formula correspond to motives. The answers were "yes" and "yes".
Matteo Capucci (he/him) (May 16 2022 at 10:45):Are you in that part of your career when you try to prove RH, John? :)
John Baez (May 16 2022 at 23:24):Yes, and buying a red sports car.
John Baez (May 16 2022 at 23:28):Actually I'm sublimating my desire to prove the Riemann Hypothesis by trying instead to learn enough algebraic geometry to understand how Grothendieck and Deligne proved Weil's watered-down version, the "Riemann hypothesis for curves over finite fields".
John Baez (May 16 2022 at 23:29):Manin and Connes have ideas about how to generalize the proof of the watered-down version and prove the real version, using a mystical entity called the "field with one element". So I'm trying to learn that stuff too.
John Baez (May 16 2022 at 23:30):But I will avoid trying to actually prove anything. I find it really fun to just learn math and explain it.
Matteo Capucci (he/him) (May 17 2022 at 12:08):That's great, I'm very curious about these things as well (especially Connes, Consani & co new ideas) but it's a bit too advanced for me to easily grasp them, so I'd be very glad if you start churning out expositions of them
Matteo Capucci (he/him) (May 17 2022 at 12:10):I remember you writing about RH for finite fields (and not) on the nCafè once, I remember because it was one of the nicest things I've read about the argument
Matteo Capucci (he/him) (May 17 2022 at 12:11):I can't find it right now though
John Baez (May 17 2022 at 16:14):There are 3 posts starting here:
and the other two are directly after that on the n-Cafe (look on top of the page).
John Baez (May 17 2022 at 16:15):What I'm doing now is preparing a talk where I cover that stuff and go further, leading up to Grothendieck's definition of motive and maybe a bit about how he got stuck on the "Standard Conjectures".
John Baez (May 17 2022 at 16:16):I'd like to expand this talk into some continuation posts on the n-Cafe... but first I have to finish writing the talk!