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created a stub for motivic cohomology in reply to a question here
Zoran (or anyone), can you say something about how that related to motives?
I daringly added to motive an "Idea" section and a rough idea of the Definition.
still don't know what the relation to motivic cohomology is, though...
These lecture notes by Voevodsky et al. are useful. They would be even more useful if the authors stopped just for a second to tell the reader what the heck it actually is they are heading for, but okay.
Prop 14.16 tells us how motivic cohomology knows about certain hom-sets in the category of motives.
I have included a brief remark at As hom-sets of motives.
Okay, I don't really know about this and don't mean to be proposing any specific picture. I just tried to summarize what I saw in these notes. I liked them, because this was the first time that I actually saw an explicit definition of what a motive is. But I understand when you say that this definition is not necessarily what other people are looking for.
Zoran, I would like to see what you would write for an Idea section at motive. If Urs's article is too orthogonal, go the bottom, add ***
to make a horizontal rule, and then put your article underneath. We can fit them together afterwards.
Zoran, what you put at Voevodsky motive, does that refer to the definition in Mazza-Voevodsky-Weibel that I quote at motive? Or is that yet another definition?
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it is important that he found the correct category of "finite correspondences"
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<p>And that's the one the entry points to, or isn't it?</p>
<p>I am not sure what you want to have changed. Should we add a sentence saying "This is one proposed definition, other people are thinking about other definitions." ?</p>
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How do "pure motives" relate to the definition in the Voevodsky-Weibel lectures?
Do you think these lectures give a wrong impression of the subject?
I have now added a section Motivic cohomology -- Idea where I try to say clearly that there was first Grothendieck's hypothetical ideas and then later Voevodsky's proposal for a concrete realization.
I renamed the "Definition"-section into "Voevodsky's definition", similarly at motive itself (to which i haven't yet added a similar expanded "Idea" section).
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I do not understand at all the statement of "abelian cohomology" with the "coefficients in motivic complexes"
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<p>I really did nothing but open the Voevodsky lectures and copied the statements from there. He defines a complex of sheaves called the motivic complex and defines motivic cohomology to be sheaf cohomology with coefficients in that.</p>
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I added few clarifications at motive
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<p>Thanks.</p>
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in the previous version of motivic cohomology I said that one forms the stabilization of . That's not quite precise: the point is -- as explained in the references linked to there -- that one stabilizes not with respect to the "categorical" or "simplicial" sphere, but with respect to the "geometric" sphere, the Tate sphere . I have added remarks about this now, but just ever so briefly. Needs to be expanded on later.
This happens to be closely related to the discussion we are having over G-equivariant stable homotopy theory, elsewhere. There, too, it seems the point is that the spectra are defined not with respect to the categorical/simplicial spheres, but with respect to the "geometric" spheres.
I am being told that for the G-equivariant case a very insightful discussion of this aspect is in two article Andrew Blumberg. Probably the first two ones listed here. I am going to look at these now.
Thanks. Misplaced quotation mark: here
But I have added the reference to the entry, meanwhile.
added a section grading and bigrading to motivic cohomology where I try to say precisely what the Tate sphere is and precisely what the bigrading is.
I think I know what i am doing (following the cited references) but some expert should eventually check that I got the conventions right and everything.
Concerning my query box at cohomology in the section on bigrading, where I propose that one should look at as that gives something that is generated from categorical and geometric speheres:
maybe that's precisely what captures the passage to the Tate sphere , i.e. the smasch product of both spheres. Both spheres at once.
Peter Arndt made two useful comments at motivic cohomology. One of them I reworked a bit to fit into the text. See the history for details.
Is Peter also in nforum or only in nlab ? Say hello to this enthusiastic young fellow :)
Peter Arndt further added a few paragraphs to the beginning of the section on Chow groups
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