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1
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1answer
127 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
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0answers
92 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 ...
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0answers
131 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
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1answer
424 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 ...
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1answer
203 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, ...
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0answers
97 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 ...
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0answers
89 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)?
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0answers
570 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
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3answers
281 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
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1answer
270 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 ...
11
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3answers
558 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
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0answers
326 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
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0answers
142 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
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1answer
853 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
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0answers
318 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
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1answer
179 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 ...
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0answers
102 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
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0answers
155 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
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1answer
394 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
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1answer
508 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
1answer
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
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0answers
210 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
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0answers
274 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
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1answer
414 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
1answer
236 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
2answers
421 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
1answer
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
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0answers
200 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
1answer
637 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
2answers
926 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
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0answers
450 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
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3answers
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
1answer
316 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 $\mathbb{Z}/n\mathbb{Z}$ ...
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0answers
218 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
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0answers
205 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
1answer
666 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 ...
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0answers
469 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). ...
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0answers
140 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
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1answer
277 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
1answer
401 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
1answer
483 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 ...
11
votes
1answer
693 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
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1answer
880 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
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2answers
797 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
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1answer
618 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
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3answers
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
1answer
414 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 ...
10
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0answers
886 views

Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
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0answers
753 views

What is the current knowledge of equivariant cohomology operations?

In Caruso's paper, "Operations in equivariant $Z/p$-cohomology," http://www.ams.org/mathscinet-getitem?mr=1684248, he shows that the integer-graded stable cohomology operations in ...
17
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1answer
561 views

Can we reconstruct positive weight invariants in algebraic topology using algebraic geometry?

I can't really say that I understand what a weight is, but the qualitative distinction between weight zero and positive weight has come up a couple times in MathOverflow questions: The étale ...