All Questions
Tagged with motivic-cohomology reference-request
19 questions
3
votes
2
answers
342
views
Reference Request: Beilinson-Bloch conjecture in terms of Beilinson regulator isomorphism
I'm looking for a reference that provides a concise statement of the Beilinson-Bloch conjecture, specifically formulated in terms of an isomorphism under the Beilinson regulator map.
More precisely, I'...
- 2,907
4
votes
0
answers
178
views
Every stable homotopical functor factors through $\mathbf{SH}$
In this nlab page, it says that the fact that every stable homotopical functor factors through $\mathbf{SH}$ (the motivic stable homotopy category of Morel-Voevodsky) is proven in Ayoub's thesis. ...
- 883
1
vote
0
answers
115
views
Action of correspondences on motivic cohomology sheaves
Writing $\mathcal{H}^a(\mathbb{Z}(b))$ for the Zariski sheaf of motivic cohomology groups, there is a hypercohomology/descent spectral sequence
$$ H^p(X,\mathcal{H}^q(\mathbb{Z}(n))) \Rightarrow H^{p+...
- 2,044
7
votes
1
answer
327
views
References for the construction of Beilinson's motivic Eisenstein classes
According to some authors, it is built in A.A.Beilinson "Higher regulator of modular curves" a class $\mathbf{Eis}_{\phi}$ in the motivic cohomology of the modular curve where $\phi$ is a ...
- 1,270
3
votes
1
answer
197
views
Motivic cohomology as $\mathit{Hom}$ in the category of geometric motives, with coefficient in a Chow motive
The main references for this question are
1 : V.Voevodsky's paper Triangulated categories of motives over a field
2 : the book "Lecture notes in motivic cohomology" written by Carlo Mazza, ...
- 1,270
4
votes
1
answer
458
views
Motivic cohomology of rigid analytic spaces
There is a satisfactory theory of B1-homotopy theory for rigid analytic spaces defined by Ayoub in the style of Voevodsky, and I'm aware of some work about the corresponding theory of motives, e.g. ...
- 2,044
7
votes
0
answers
278
views
Adequate equivalence relations and algebraic $K$-theory
I have a somewhat vague question. We know that Adams operation gives a filtration on $K_i(X)\otimes \mathbb{Q}$ for the scheme $X$ such that the weight $j$ elements are isomorphic to higher Bloch Chow ...
- 5,901
9
votes
1
answer
713
views
Bass' conjecture implies the Parshin's conjecture
In the appendix of this paper. It is proved that Bass' conjecture for $K_n$ implies the rational Beilinson-Soulé conjecture for $K_n$. Then at the end the author claims that the same method can be ...
- 5,901
6
votes
1
answer
510
views
A question about the (motivic) integral cohomology of the Eilenberg-MacLane spectrum
Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum. Let $n\geq 0$ be any integer.
Is it known the structure of the group $[H\mathbb{Z},\Sigma^{n}H\mathbb{Z}]$?
Is there any reference in this ...
- 761
7
votes
1
answer
709
views
A question on Voevodsky´s categories
I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions:
1.- ...
- 761
18
votes
1
answer
1k
views
Which motivic cohomology groups of complex numbers are non-torsion?
I would like to know which motivic cohomology groups of complex numbers are non-zero and ("better") non-torsion, i.e., for which $(i,j)$ the $i$th cohomology of the complex ${\mathbb{Q}}(j)$ over $\...
- 16.9k
3
votes
1
answer
338
views
Reference request for the relation of Ext groups and bar construction
I need a reference for the description of Ext groups in mixed categories (i.e. abelian categories with a weight filtration and semisimple graded quotients) by using the bar complex, as mentioned in "...
- 4,484
1
vote
0
answers
179
views
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
2
votes
1
answer
749
views
Vanishing of Motivic Cohomology
In these notes, page $10$, bullet $(5)$, it is stated that if $X$ is a scheme of finite type over a field $k$, then the motivic cohomology $\mathrm{H}^{p,q}(X,R)$ of $X$ over $k$, where $R$ is a ring, ...
user62675
4
votes
0
answers
217
views
The 'most general' papers on rational Borel-Moore motivic homology and K'-theory?
There are two ways to define Borel-Moore motivic homology (of schemes) with rational coefficients: one should either consider certain complexes of algebraic cycles, or the $\gamma$-filtrations of ...
- 16.9k
13
votes
1
answer
2k
views
Learning a little Motivic Cohomology
Simply because I find it interesting, I have spent some time studying motivic cohomology from the lectures by Mazza, Voevodsky and Weibel. However, I'm finding it hard to tell if the theory is ...
- 3,555
14
votes
1
answer
746
views
Can there exist Chow motives/motivic cohomology for compact Kähler manifolds?
Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are there any ...
- 16.9k
14
votes
1
answer
1k
views
Motivic cohomology vs. K-theory for singular varieties
As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for $\...
- 16.9k
4
votes
0
answers
255
views
On (the cohomology of) Hensel pairs
I would like to study the cohomology of the Henselization $H_X(Z)$ of a closed subvariety $Z$ of a variety $X$.
I would like the following facts to be true (and to make sense!:)).
a.) The motivic ...
- 16.9k