All Questions
17 questions with no upvoted or accepted answers
12
votes
0
answers
726
views
What is missing in the current constructions of pure and mixed motives?
Yo! Maybe this question is too broad, so maybe it should be community wiki? In summary, I want to known all the known comparisons between all the constructions of pure and mixed motives and what make ...
- 2,227
9
votes
0
answers
463
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Beilinson regulators and Bloch's mythological algebraic intermediate Jacobians
In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future:
Towards the end of page 270, he says, given a smooth projective variety ...
user95222
8
votes
0
answers
587
views
Values of cohomology theory on a point
$\DeclareMathOperator\Sm{Sm}$It is a well-known fact that in algebraic topology, generalized cohomology theories are determined by their values on the point. I was wondering whether anything similar ...
- 5,901
8
votes
0
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574
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Reference request: Motivic Cohomology and Cycle class maps
For a smooth projective variety $X$ over any field $K$, Voevodsky showed in his paper ``Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic" that the motivic ...
- 592
6
votes
0
answers
400
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Hodge Realisation of Mixed Tate Motives
For a field $k$ which satisfies Beilinson-Soule vanishing conjecture, the from Levine's paper,
https://www.uni-due.de/~bm0032/publ/TateMotives.pdf
There exists an abelian category of mixed Tate ...
- 2,971
6
votes
0
answers
351
views
Correspondences of schemes induced by a finite Galois extension
I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between schemes.
Recall that if $...
5
votes
0
answers
513
views
Poincaré duality for motivic cohomology
Is Poincaré duality for étale motivic cohomology known for projective regular (not necessarily smooth, just regular) schemes over Dedekind rings?
More precisely, two questions. Let $f: \mathcal{X}\to\...
user92332
5
votes
0
answers
801
views
Two questions on motivic homotopy theory
I am a beginner in motivic homotopy theory and motivic cohomology and have just begun reading books by Dundas et al. and Mazza-Voevodsky-Weibel. I have two basic questions:
Why is the question of $\...
- 805
4
votes
0
answers
205
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$\mathbf{A}^1$- contractibility
Suppose $U$ is an $\mathbf{A}^1$-contractible smooth scheme over a field $k$, that is, it is isomorphic to a point in the $\mathbf{A}^1$-homotopy category of smooth schemes over $k$.
Does motivic ...
user119470
4
votes
0
answers
244
views
Effectivity and Lower Shriek for Voevodsky Motives
I am in a situation where I need a result of the following form. Suppose $X$ is a smooth $k$-variety, $U$ is a dense open subvariety with complement $Z$ a smooth divisor. Let $\pi^X:X\rightarrow\text{...
- 143
3
votes
0
answers
206
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Generalization of conjectures involving Beilinson regulators
I had some questions about the Beilinson conjectures as mentioned in this page. I have to admit I do not know much about Deligne cohomology. The conjectures involve some form of comparison map between ...
- 5,901
3
votes
0
answers
638
views
Weak Lefschetz for torsion motivic cohomology (or torsion Chow groups)?
I am trying to prove the following Weak-Lefschetz-type statement for motivic cohomology: if $Z$ is a smooth hyperplane section of a smooth projective $X$ of dimension $d$ (say, over the field of ...
- 16.9k
2
votes
0
answers
326
views
Voevodsky's 'split standard triple' argument: an explanation; does it work with $Z/nZ$-coefficients?
For varieties over a perfect field (of some characteristic $p$ that could be 0) Voevodsky defines the notion of a 'standard triple' (see http://books.google.ru/books?id=TzUmk87bN9cC&pg=PA85&...
- 16.9k
1
vote
0
answers
213
views
Algebraic correspondence as morphisms in Betti cohomology
$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for ...
- 349
1
vote
0
answers
260
views
Non-examples of mixed Tate motives
I was trying to find examples of schemes (preferably smooth) over $\mathbb{C}$ which have motives that aren't mixed Tate. I wasn't able to come up with anything or find an argument that's been written ...
- 19
1
vote
0
answers
179
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Which "concrete" morphisms of varieties and motives induce bijections of their lower Chow groups?
This question is a continuation of Varieties with Chow groups supported in positive codimension: examples and properties?
What examples are known of morphisms of varieties and Chow motives (say, over ...
- 16.9k
0
votes
0
answers
288
views
What can one say about zero-cycle groups for products of Chow motives
What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes ...
- 16.9k