All Questions
7 questions
71
votes
1
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8k
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What is the relationship between motivic cohomology and the theory of motives?
I will begin by giving a rough sketch of my understanding of motives.
In many expositions about motives (for example, http://www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined ...
- 5,929
8
votes
1
answer
989
views
How to think about $\mathbf{Z}(n)_{\mathcal{M}}$
One definition of motivic cohomology for smooth schemes $X$ over a field, is via Friedlander-Suslin complexes.
A refresher (you may skip to the question at the bottom)
One defines
(1) $z_n(X,d) :=$...
user97068
17
votes
1
answer
1k
views
Why presheaves with transfer?
Presheaves with transfer are a main technical ingredient in Voevodsky's construction of his category of mixed motives. From the perspective of motivic homotopy theory, the only difference between SH ...
- 171
12
votes
3
answers
2k
views
Motivic vs Deligne cohomology
Where can I find the construction of the cycle class map from motivic cohomology to Deligne cohomology of smooth projective varieties over the complex numbers?
It should be a construction by Bloch ...
user95222
11
votes
1
answer
967
views
How to think about infinite generatedness of motivic cohomology
In this question I previously asked how to think about the motivic complex $\mathbf{Z}(1)_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^*((-)\wedge S^{2n}...
user97068
5
votes
1
answer
468
views
Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?
Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$.
Algebraically its $Spec$ is quite different from $k$. For example:
it has plenty non-trivial "line-...
- 24.9k
2
votes
1
answer
589
views
Are finite correspondances flat?
In Voevodsky&Mazza&Weibel's book on motivic cohomology they define "an elementary correspondance from X (Smooth connected scheme over $k$) to Y (separated scheme over $k$) as an irreducible ...
- 2,871