The motivic-cohomology tag has no wiki summary.

**3**

votes

**1**answer

246 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 ...

**6**

votes

**0**answers

119 views

### Exactness of pure functors

I can't prove this lemma in "Notes on motivic cohomology, Beilinson, Macpherson, Schechtman":
Lemma. A pure functor is exact.
Definitions: A mixed category $\mathcal{M}$ is a ...

**8**

votes

**1**answer

206 views

### When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?

For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For ...

**2**

votes

**1**answer

158 views

### About embedding pure motives into the triangulated category of mixed motives and some further questions about motivic cohomology

I have tried reading some texts about motives, mainly motivic cohomology (Bloch's " Lectures on Algebraic Cycles", Voevodsky's "Motivic Cohomology" etc). However some things confused me. I don't know ...

**0**

votes

**0**answers

106 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 ...

**1**

vote

**0**answers

135 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 ...

**1**

vote

**1**answer

451 views

### Standard conjectures on positive characteristic

In this MO answer of M. Bondarko, he says:
"the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..."
and in Remarks on Grothendieck's ...

**2**

votes

**1**answer

231 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, ...

**4**

votes

**0**answers

107 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 ...

**1**

vote

**0**answers

91 views

### Differences and relationships between Motivic cohomology and Universal cohomology theory? [duplicate]

Differences and relationships between Motivic cohomology (Beilinson, Lichtenbaum and Voevodsky) and Universal cohomology theory (Grothendieck)?

**10**

votes

**0**answers

721 views

### Has there been any progress on the standard conjectures on algebraic cycles?

What's the current state of these conjectures?
Who is working on them?
In "Standard conjectures on algebraic cycles" Grothendieck says:
"They would form the basis of the so-called "theory of ...

**5**

votes

**3**answers

296 views

### Chow motive of a fibration

If we have a fibration of smooth projective complex varieties $F\to E\to B$, which is locally trivial in the analytic topology, and the global monodromy is trivial. Then is it true that the Chow ...

**8**

votes

**1**answer

283 views

### Known norm varieties and the Bloch-Kato conjecture

The Bloch-Kato conjecture states that
$K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$.
A important part in the proof of the Bloch-Kato conjecture is to ...

**12**

votes

**3**answers

584 views

### Applications of homotopy purity theorem of Morel-Voevodsky

One of the most important theorems in motivic homotopy theory is the homotopy purity theorem of Morel-Voevodsky which says that the motivic Thom space of the normal bundle $\mathcal N_{Z/X}$ of a ...

**3**

votes

**0**answers

344 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 ...

**3**

votes

**0**answers

149 views

### Regulator maps for ordinary varieties

Let $K$ be a finite extension of $\mathbf{Q}_p$ and $\mathcal{X}$ a smooth proper scheme over the ring of integers $\mathcal{O}_K$. For $i, j$ integers with $i \ne 2j$, there's a regulator map
$$ ...

**14**

votes

**1**answer

878 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 ...

**1**

vote

**0**answers

337 views

### Homotopy theory of schemes

I have seen the notion of Homotopy come up in several contexts in schemes. For example, the book "Lectures on Motivic Cohomology" by Mazza, Weibel and Voevodsky uses this language to some extent. I.e. ...

**3**

votes

**1**answer

187 views

### the graded pieces of the gamma-filtration of Quillen K-theory and Chow groups of a regular scheme

Let $X$ be a regular scheme and consider Grothendieck's $\gamma$-filtration $F^nK(X)$ on $K(X)$. For the graded pieces, one has $Gr^0K(X) = CH^0(X)$ and $Gr^1K(X) = \mathrm{Pic}(X) = CH^1(X)$. Does ...

**0**

votes

**0**answers

104 views

### Reference for Cech cohomology for Nisnevich topology

I need the theorems that prove that Nisnevich cohomology can be computed by the Cech complex similar to what happens in the étale topology.
I know this has to be true since (Morel-Voevodsky: ...

**3**

votes

**0**answers

159 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 ...

**4**

votes

**1**answer

400 views

### Voevodsky's proof in any characteristic (for motivic and Chow)

Regarding the published version of "Motivic cohomology groups are isomorphic to higher Chow groups in any characterstic" (IMRN) available at:
...

**8**

votes

**1**answer

540 views

### Motivic cohomology and cohomology of Milnor K-theory sheaf

Let $X$ be a smooth variety over a field $k$. (Assume $k$ has characteristic 0 if it helps; in fact I'd be happy to assume that $k$ is a finite extension of either $\mathbf{Q}$ or $\mathbf{Q}_p$).
...

**1**

vote

**1**answer

178 views

### Definition of Hurewicz map relating $SH(k)$ with $DM_\_^{eff}(k)$

In "Motivic Homotopy Theory" on page 153 it is stated that there exists a canonical Hurewicz map relating the motivic stable homotopy category with the category $\operatorname{DM}_\_^{eff}(k)$. ...

**0**

votes

**0**answers

213 views

### Vanishing of motivic cohomology with finite coefficients in negative degrees

I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not.
STATEMENT:
Let $X$ be a smooth and projective scheme over a finite field ...

**2**

votes

**0**answers

276 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 ...

**11**

votes

**1**answer

429 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 ...

**2**

votes

**1**answer

245 views

### Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?

For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):
$E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...

**5**

votes

**2**answers

458 views

### When do the $\gamma$-filtration and codimension filtration of K-theory agree?

Let $X$ be a smooth quasiprojective algebraic variety over a field $k$. Then the $K$-groups $K_m(X)$ are defined, and there are two standard filtrations on them: the "codimension filtration" given by
...

**39**

votes

**1**answer

2k views

### 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, www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined to be ...

**1**

vote

**0**answers

201 views

### What can one say about a smooth variety whose lower cohomology is trivial?

Let $X$ be a smooth (quasi-affine) complex variety; suppose that its cohomology (say, with integral coefficients) is trivial in degrees $0 < i\le s$ (for some $s>0$). What can one say about such ...

**11**

votes

**1**answer

651 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 ...

**2**

votes

**2**answers

965 views

### Milnor-Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture

Voevodsky reformulated the Milnor-Bloch-Kato conjecture as a change-of-topology morphism from the Zariski to the étale topology of a field with torsion coefficients being an isomorphism. The ...

**3**

votes

**0**answers

462 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 ...

**9**

votes

**3**answers

1k views

### State of the art for Gersten's conjecture for K-theory?

Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence
$$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to ...

**3**

votes

**1**answer

323 views

### An example of an affine variety with non-zero Chow groups

Are there any examples known of an affine variety $A$ over an algebraically closed field such some Chow group (say, of codimension at least $2$) of $A$ with coefficients in $\mathbb{Z}/n\mathbb{Z}$ ...

**1**

vote

**0**answers

219 views

### On motivic cohomology with compact support

For a smooth projective variety $X$ and its closed non-smooth subvariety $Z$ I would like to say that a cone of the morphism between the motivic cohomology of $Z$ and those of $X$ is the motivic ...

**3**

votes

**0**answers

208 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 ...

**3**

votes

**1**answer

679 views

### Higer Chow groups and singular cohomology theory

Let $X$ be a regular scheme over a field $k$ and $Δ^m$ be the algebraic $m$-simplex $\mathrm{Spec} k[t_0,...,t_m]/(1-\sum_jt_j)$. The group $z^i(X,m)$ is the free abelian group generated by all closed ...

**2**

votes

**0**answers

475 views

### a counterexample of Serre vs. motivic cohomology

There is a counterexample of Serre showing that there is no Weil cohomology theory with coefficients in $\mathbf{Q}, \mathbf{Q}_p, \mathbf{R}$ over $\mathbf{F}_{p^2}$ (a supersingular elliptic curve). ...

**1**

vote

**0**answers

141 views

### tensor product of motivic complexes $\mathbf{Z}(n)$

Is the morphism $\mathbf{Z}(n) \otimes^L \mathbf{Z}(m) \to \mathbf{Z}(n+m)$ from the Beilinson-Lichtenbaum conjectures a quasi-isomorphism?

**2**

votes

**1**answer

281 views

### Is there a universal coefficient theorem for motivic cohomology?

Is there some kind of universal coefficient theorem for motivic cohomology?
In particular, suppose we have a ring morphism $R\to S$, then I would like to know when
$$ H^{\star\star}(-,S)\simeq ...

**1**

vote

**1**answer

402 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 ...

**10**

votes

**1**answer

501 views

### Stable motivic cohomology with finite coefficients?

In this question, which attracted no responces so far, I've asked whether it is possible to extend the Beilinson-Lichtenbaum etale descent rule for motivic cohomology to singular varieties, in ...

**12**

votes

**1**answer

712 views

### Motivic cohomology with finite coefficients for singular varieties

Let $X$ be a smooth variety over a field $K$ whose characteristic does not divide a positive integer $m$. Then the motivic cohomology of $X$ with coefficients in $\mathbb Z/m(j)$ can be computed in ...

**15**

votes

**1**answer

908 views

### Motivic Cohomology vs. Chow for singular varieties?

I'm absolutely new to this stuff I'm asking about, so I hope this is not nonsense.
If X is a smooth scheme over a perfect field k, I can study its motivic cohomology in the sense of Voevodsky and ...

**5**

votes

**2**answers

808 views

### Relation between motivic cohomology and Quillen K-theory

What's the relation between Voevodsky's motivic cohomology and Quillen K-theory of a scheme?

**2**

votes

**1**answer

619 views

### The motivic cohomology of projective space

What is the motivic cohomology $H^{p,q}(\mathbf{P}^n,\mathbf{Z})$ of projective space? By the projective bundle formula, one has
$H^{p,q}(\mathbf{P}^n,\mathbf{Z})$ =
...

**14**

votes

**3**answers

1k views

### Relation between motivic homotopy category and the derived category of motives

What's the relation between the pointed motivic homotopy category $\mathcal{H}_*(k)$ and the derived category of motives $\mathbf{DM}^-_{eff}(k)$ besides the representability of motivic cohomology in ...

**1**

vote

**1**answer

424 views

### Is there a projection formula for motivic étale cohomology?

Let $f: X \to Y$ be a finite morphism of varieties with $Y$ smooth. Is there a projection formula for $f$ and $H^{i}_{et}(-,\mathbf{Z}(1)) = H^i(-,\mathbf{G}_m)$?
Background: I want to show that for ...