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Chow motive, a quick definition
Well, there is some overlap with pure motive. I guess there should be a single page titled pure Chow motive...
Ah, thanks.
I have added quick cross-references. Why “pure”? That should be explained in the entry!
(And why does no expert ever write a decent account of the various definitions in motivic cohomology?)
In my opinion, we really need an expert to rewrite basically everything on the nLab about motives!
Why “pure”?
In my (very minimal) understanding, the category of pure motives has smooth projective varieties as its objects, while the category of mixed motives is supposed to be constructed from all smooth varieties. It is supposed to be an abelian tensor category which contains the pure motives as the full category of semisimple objects. So far there is no realisation of such a category, but there are proposals by Voevodsky and Levine of triangulated categories that behave as its derived category is expected to.
In my opinion, we really need an expert to rewrite basically everything on the nLab about motives!
Yes, I think that’s true. But what I find surprising is that without being an expert having some inside knowledge, the literature alone makes it hard to just extract the definitions. There are many fields in which I am not expert, but for which I can easily state the fundamental definitions, simply by looking them up in the literature. For motivic cohomology that is much less so.
I remember my enthusiasm when I was finally told where the actual definition of derived motives is stated, at least essentially precisely (namely in the introduction to Cisinki-Déglise). This I had tried to record at motve in the section Idea of the precise abstract definition of derived motives.
Maybe with enough joint effort we can eventually collect enough information and eventually feel confident to rewite the motivic entries from scratch and coherently.
In my (very minimal) understanding, the category of pure motives has smooth projective varieties as its objects, while the category of mixed motives is supposed to be constructed from all smooth varieties. It is supposed to be an abelian tensor category which contains the pure motives as the full category of semisimple objects. So far there is no realisation of such a category, but there are proposals by Voevodsky and Levine of triangulated categories that behave as its derived category is expected to.
Thanks!
I have used that to create a stub for mixed motive and and I have given pure motive one more lead-in sentence in order to reflect this.
But in view of this it seems to actually make sense to have pure motive and Chow motive be distinct entries, doesn’t it? For I suppose we can (and already do, it seems) consider other types of categories not built via Chow motives and still distinguish for them between pure and mixed motives. No?
FYI, the wikipedia entry on motives is not that bad. One should also point out the related distinction pure versus mixed Tannakian categories which is something of the sort of single fiber functor vs. the fiber functor for every such and such field extension, what boils down to working with a gerbe.
Kontsevich’s motives which extend the motive idea to categorical framework (categories enriched in spectra) has lots of simplicifications with respect to the usual theory. For example, no need for starting technical tools like Chow moving lemma and the framework of spectral categories is itself manifestly close to the topological picture. It is also a bit more general, as it includes the noncommutative examples like Landau-Ginzburg models. Thanks to Cisinski and Tabuada the comparison with the usual motives is written out in rigorous detail. Finally, MK mentioned few times in his talks that he believed that there is a version which could be stated in terms of nuclear spaces or alike, thus bringing the whole picture of algebraic motives to operator algebraic framework, but this is just a conjecture so far.
Maybe with enough joint effort we can eventually collect enough information and eventually feel confident to rewite the motivic entries from scratch and coherently.
I am trying to learn this stuff now so I'll try to contribute what I can.
But in view of this it seems to actually make sense to have pure motive and Chow motive be distinct entries, doesn’t it? For I suppose we can (and already do, it seems) consider other types of categories not built via Chow motives and still distinguish for them between pure and mixed motives. No?
Okay, well you can take any adequate equivalence relation, and what you get probably can be called a category of pure motives. And pure Chow motives would then be the specific case of rational equivalence.
Okay, well you can take any adequate equivalence relation, and what you get probably can be called a category of pure motives.
This difference is not essential from the theoretic point of view.
It seems to be very essential in the theory at least right now, since everybody is talking about all those different flavors of motives like “Chow motives”, “Voevodsky motives”, “derived motives”, “Nori motives” and whatnot. As far as I can see for all of them it makes sense in principle to distinguish between the pure and the mixed version.
Maybe one day in 50 years when the dust here has settled and everybody knows what is supposed to be meant by “motive” this may be different. But right now this does not seem to be the case.
Okay, well you can take any adequate equivalence relation, and what you get probably can be called a category of pure motives.
This difference is not essential from the theoretic point of view.
What I mean is that only the category obtained using rational equivalence should be called "pure Chow motives". So Urs is right to keep the pages separate, I guess.
Though to be honest, I don't understand in which respect the difference is not essential. After all the categories Mot_rat and Mot_num are certainly not the same category in general.
Adel: This is the discussion which we had before. Some people call Chow rings for any adequate equivalence relation, more classical default case is for the rational equivalence. The same is for the Chow motives.
Urs: I am not discussing the difference between derived and nonderived, mixed and pure, zero characteristics and positive etc. which ARE essential to discuss separately. But I see no reason to have separate entry for Chow motives for the classical case of rational equivalence, vs. the other Weil cohomology theories, like for numerical equivalence. It is just one parameter different (as far as the definitions are concerned).
Maybe one day in 50 years when the dust here has settled and everybody knows what is supposed to be meant by “motive” this may be different.
Not quite, the differences I stated above will stay different (like positive vs zero characteristics, mixed vs pure etc.).
I think what makes most sense is to have a page pure motives which mentions pure Chow motives as the case \sim = rat.
A separate page to mention a parameter in one line ?? I disagree. Many people will get there by coincidence and better they see the full uniform treatment with the line inserted at some place. One line difference does not qualify for the separate page, we often have more than one notion in the same page if common treatment is beneficial. Of course, if you have a detailed material of properties which are specific for motives for rational equivalence then one needs pure Chow motive separate from pure motive in general.
Sorry, what I meant was to merge the current pages Chow motive and pure motive into a page called "pure motive", mentioning pure Chow motives as the special case.
Right, that is what I think is sensible at this point !
Please don’t do away with the page Chow motive as is. I want a page to which people can go and see right away that a Chow motive is a certain equivalence classes of linear combinations of spans, without much distraction around it.
But if you work on the entry pure motive to make it better, that would be great.
Urs, if you want that the merged page starts with the material now at Chow motive and the other remarks below in scrolling down this is OK, but I see no reason that the two would not be covered in the same page with redirects.
Why don’t you first do the major revision of the pure motive entry and then when done we can still think about killing the entry on Chow motives or not.
I edited pure motive.
Thanks!
I have looked through it:
I gave the definition of each of the several categories its own three-hashed subsection (where you had boldface keywords).
in the direct sum formua for the homs in $Corr$ I replaced $\oplus_\alpha$ with $\oplus_i$, for I suppose that was a typo, but check;
in the section on “Category of effective pure motives” I fixed what I think was a typo by setting $p \in Corr(h(X), h(X))$ instead of $Corr(h(X), h(Y))$. But then I am still unsure about your description of the morphisms. Could you check if that’s what you mean to say? Maybe I am mixed up.
still in the section “Category of effective pure motives” I added a statement of the functor $h : SmProj(k) \to Mor^{eff}_\sim(k,A)$ and that its images are called “the motives of a variety”, so that this term has been introduced before it is used in the next section
Ah, and I gave several of the “$Corr$” a subscript “$Corr_\sim$”, for I thought that was missing.
Maybe you should go through the whole thing and check the notation again for typos – possibly new typos now introduced by me. ;-)
I have now adapted the notation and terminology at Chow motive to that at pure motive. But please check.
Thanks. Everything looks good to me!
Okay, thanks.
One remaining question: where it says
and morphisms from $(h(X), p)$ to $(h(Y), q)$ are compositions $q \circ \alpha \circ p$ with $\alpha \in Corr_{\sim}(h(X), h(Y))$
should it not read
and morphisms from $(h(X), p)$ to $(h(Y), q)$ are elements $\alpha \in Corr_{\sim}(h(X), h(Y))$ such that $p \circ \alpha = \alpha = \alpha \circ q$
i.e. the standard formula for the morphisms in the Karoubi envelope? Or do you mean something else? (Or maybe I am missing something.)
These are actually the same. If $\beta = q \circ \alpha \circ p$ for some $\alpha$, then $q \circ \beta = \beta = \beta \circ p$ is clear, and so is the converse (taking $\alpha = \beta$).
Okay, fair enough. But how about we write more explicitly
and morphisms from $(h(X), p)$ to $(h(Y), q)$ are morphisms $h(X) \to h(Y)$ of the form $q \circ \alpha \circ p$ with $\alpha \in Corr_{\sim}(h(X), h(Y))$
Yes, that’s a good idea.
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