Notes on Pure Chow Motives

Notes on pure Chow motives.

I’ll be giving a talk soon on Pure Chow motives for the graduate algebra seminar at the university of Alberta. These are some notes I wrote up to fill in the details of what I’ll talk about.

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2 comments on “Notes on Pure Chow Motives

  1. alexyoucis says:

    Again, great post. The notes were great!

    A few remarks:

    1) This is pretty nit-picky, but you say that things like \ell-adic cohomology and de Rham cohomology are only defined in certain settings. It’s not that they’re only definable there, but that outside of that context they’re not Weil cohomology theories. Presumably this is what you meant.

    2) It’s all a matter of taste I suppose, but I think that computing the (etale) cohomology of \mathbb{P}^n is easier using the Mayer-Vietoris (spectral) sequence.

    3) You don’t just the Hodge-de Rham spectral sequence, you actually use the *degeneration* of the Hodge-de Rham spectral sequence. It is true in characteristic 0 (not characteristic p!) but the proof is highly non-trivial (you reduce to \mathbb{C} then use analysis, or use p-adic Hodge theory, or use Deligne-Illusie, but none of thoes are simple).

    4) I don’t know how interested in number theory you are, but I think that a lot of motivic hopes are obtained in the subject of p-adic Hodge theory in a pretty explicit way (e.g. comparisions of things like etale, crystalline, and de Rham cohomologies after base changing to a suitably large period ring).

    5) It’s also worth noting that (say we’re working over a number field) many of your realization functors are conjectured to be fully-faithful. This is (in the \ell-adic case to Galois representations), essentially, the content of the Tate conjecture (as you likely know).

    6) Typo “semminlgy” in first paragraph.

    7) An interesting thing to write an expository article on, if you’d ever be interested :), is how one can make the claim that abelian varieties (and their degenerations) are actually a rigorous category of 1-motives. I know that some notes exist, but I’d love to see someone suss out precisely the content of this. Of course it’s intuitive in the sense that all Weil cohomology theories factor through the Albanese variety, but it’d be interesting to make this all precisely.

    Thanks again!

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    • eoinmackall says:

      Ahh, now you’re starting to see some of my older posts…
      1), 2), 3) You’re absolutely correct. I’m less versed in a lot of this material and was trying to figure things out as best I could. I think there is another inaccuracy with (2.3) — I should have added a transversal assumption on the product.
      4) I think I’ve heard about some of this recently (maybe from some of your posts and cursory glances at some papers by Bhatt and Scholze) — I haven’t looked at p-adic Hodge theory in any detail though. I don’t know if this will change in the near future.
      5) Originally I planned on including a part on the motivic fundamental group (with some computations when possible!) — eventually I gave up on this plan though.
      6) Aww man 😦
      7) I’ve seen this somewhere as well — I was also under the impression that this was Grothendieck’s original motivation for defining his category of motives. I thought I knew of an article on this but I can’t seem to find it. I probably won’t be the one to make a new one though! I think if I turned back to writing (or even just thinking) about motives I’d have to reconsider my approach. Grothendieck has a really direct construction for a category but it takes some work (for me at least) to find examples. Most of the examples I know are consequences of an affine stratification and some functorial property that says these stratifications behave nicely over products (and then one can use Manin’s Yoneda functor to show the Chow motive is a sum of twisted tate motives depending on the stratification). To me, this sounds much more like a CW-complex type of construction and then an application of some homological machinery. If I were to start thinking about motives again I’d probably try learning more precisely Voevodsky’s construction (which contains Grothendieck’s) and some of its tools. There’s been a lot of progress relating Voevodsky’s motives to certain homotopy categories which probably provide a more intuitive approach to the subject. But there’s so much background necessary for this it’s completely daunting…
      Anyways, I think this p.o.v. is also probably not always as interesting to number theorists (although, there has also been a lot of work doing similar things for categories which are useful to number theorists, i.e. Ayoub’s construction of a motivic homotopy category for Rigid analytic spaces). But the proofs are so uninviting (large commutative diagrams from basic axioms for long stretches of pages — like hundreds of pages just checking diagrams) that I don’t know if you’d be interested in this subject.

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