The motivic-cohomology tag has no usage guidance.

**7**

votes

**0**answers

204 views

### Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?

It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence.
This statement is ...

**8**

votes

**1**answer

236 views

### Motivic cohomology is universal with respect to what (co)homology theories?

I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem?
...

**9**

votes

**2**answers

268 views

### A question about the vanishing of motivic cohomology in negative Tate twist

Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.
Is it true that
$$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[...

**4**

votes

**0**answers

255 views

### A question on the slice filtration and the slice of the motive of the projective space

In the following $k$ is an algebraically closed field of characteristic $0$.
Consider the category $SH(k)$ (the Morel-Voevodsky stable motivic homotopy category).
By the work of Voevodsky (see for ...

**3**

votes

**1**answer

230 views

### Thom class in motivic cohomology

Let $E$ be a vector bundle over a smooth scheme $X$. The Thom space of $E$ is $Th(E)=E/E-i(X)$ where $i\colon X \longrightarrow E$ is the zero section. This space is
$\mathbb{A}^{1}$ isomorphic to $\...

**0**

votes

**0**answers

62 views

### $\mathbb{A}^{1}$-weak equivalences $K_{n,R} \longrightarrow K(2n,n,R)$?

Theorem 2.1 in "REDUCED POWER OPERATIONS IN MOTIVIC COHOMOLOGY" written by V.Voevodsky. He said that "there are $\mathbb{A}^{1}$-weak equivalences $K_{n,R} \longrightarrow K(2n,n,R)$ which are ...

**6**

votes

**2**answers

240 views

### Galois descent in motivic cohomology

Let $X_N$ denote the Fermat curve defined over $\mathbb{Q}$ by the equation $x^N+y^N-z^N=0$ and let $X_{N,\mathbb{Q}(\mu_N)}$ be the base change. Let $G$ be the Galois group of $\mathbb{Q}(\mu_N)/\...

**7**

votes

**0**answers

285 views

### 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 ...

**10**

votes

**0**answers

174 views

### Injectivity of regulator maps

Let $X$ be a scheme which is smooth and quasi-projective over $\operatorname{Spec} \mathbf{Z}[1/N]$, and let $\ell$ be a prime dividing $N$ (hence invertible on $X$). Then then there is a regulator ...

**4**

votes

**1**answer

383 views

### A quite puzzling question on Deligne cohomology sheaves and cycle maps

Intro. I would be deeply grateful if someone could please clarify the following to me.
The question. (the main point is (4))
Let $X$ be a smooth projective variety over $\mathbf{C}$, and $\mathbf{Z}(...

**5**

votes

**1**answer

236 views

### Blowup formula for motivic cohomology

If $X$ is a smooth projective variety over a field, $Z\subset X$ a smooth closed subvariety of codimension $d$, $X'\to X$ the blowup of $X$ along $Z$, there's the blowup formula
$$H^j(X'_{et},\mathbf{...

**7**

votes

**1**answer

463 views

### Two motivic complexes, compared

Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986).
Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) :...

**4**

votes

**0**answers

190 views

### $\mathbf{A}^1$- 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 ...

**11**

votes

**1**answer

779 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}...

**6**

votes

**1**answer

500 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) $z_n(X,d) :=$...

**7**

votes

**1**answer

544 views

### Torsion in Deligne cohomology

Let $X$ be a smooth projective complex analytic space, $i,p\ge 0$ integers, $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex of $X$, $H^i_{\mathcal{D}}(X,\mathbf{Z}(p))$ its hypercohomology.
What ...

**5**

votes

**1**answer

254 views

### Around algebraic equivalence of cycles

Let $k$ be a finitely generated field, $X$ a smooth projective $k$-variety, $\ell$ a prime number, $\ell\in k^{\times}$, $r\ge 0$ an integer.
The Tate conjecture asserts surjectivity of the cycle ...

**9**

votes

**0**answers

323 views

### 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 ...

**12**

votes

**0**answers

171 views

### Homology of Gersten complex for singular schemes

It is one of the important facts in K-theory/motivic cohomology that the Gersten-type complexes (for Quillen K-theory, Milnor K-theory or more generally Rost's cycle modules) are exact for smooth ...

**11**

votes

**2**answers

881 views

### Is Deligne cohomology the motivic cohomology of analytic spaces?

Let $X$ be a smooth projective complex analytic space.
We can cook up a complex analytic version of Bloch's cycle complex by declaring
$z^n(X^{\rm an}, m)$
is the free abelian group on all ...

**12**

votes

**3**answers

922 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 ...

**4**

votes

**0**answers

294 views

### Homotopical enhancements of cycle class maps

Fix a smooth projective variety $X$ over the complex numbers.
We write $H^n(X,\mathbf{Z}(d)) = \text{CH}^d(X, 2d-n)$ for Bloch's higher Chow groups.
Notation
For a field $k$, recall $\Delta^n_{k} :=...

**5**

votes

**0**answers

270 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\...

**6**

votes

**1**answer

505 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.-...

**4**

votes

**0**answers

231 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{...

**6**

votes

**1**answer

594 views

### Higher Chow groups revisited

Let $X$ be an algebraic variety over a field $k$.
Bloch defines the "algebraic singular complex" using the algebraic simplices
$$\Delta^n = \text{Spec}(k[x_0,\dots,x_n]/(x_0+x_1+\dots+x_n=1) \subset ...

**21**

votes

**1**answer

775 views

### Spectral sequences in $K$-theory

There is an algebraic analogue of the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological $K$-theory of a topological space.
For a field $k$, let $X$ be smooth variety $X$ ...

**6**

votes

**0**answers

213 views

### 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 ...

**4**

votes

**1**answer

171 views

### Nori's Mixed Motives and Realisation Functors

The conjecture 51 in Levine's Mixed Motives in Handbook of K-theory
http://www.math.illinois.edu/K-theory/handbook/1-429-522.pdf
states that the functor induced by $hs:\text{ECM} \rightarrow \text{...

**4**

votes

**1**answer

144 views

### Explicit description of Verdier quotient of effective motives

Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory ...

**2**

votes

**0**answers

225 views

### representability of étale and motivic cohomology by schemes

One has $\mathrm{H}^1_{\mathrm{et}}(X,\mathbf{G}_m) = \mathrm{H}^2(X,\mathbf{Z}(1)) = \mathrm{Pic}(X)$ (étale and motivic cohomology).
Is étale or motivic cohomology in other dimensions also ...

**16**

votes

**1**answer

428 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 $\...

**2**

votes

**1**answer

263 views

### Dualizability and motivic cohomology

Suppose $k$ is an algebraically closed field of characteristic $p$. Let $A=\mathbb{Z}/\ell\mathbb{Z}$, $\ell$ a prime coprime to $p$. Denote by $MA$ the motivic Eilenberg-Maclane spectrum over $k$. Is ...

**7**

votes

**0**answers

382 views

### Arakelov Motivic Cohomology and Hodge Theory

Lately I have been studying these two papers (first and second) that introduce a new cohomology theory called Arakelov motivic cohomology. While most of the applications presented in the papers are ...

**6**

votes

**1**answer

213 views

### Obtaining the Hilbert symbol from cup product on motivic cohomology

Let F be a number field. Does the Hilbert symbols at the various places of F arise from the cup product on the motivic cohomology groups of Spec(F)? And if so, is it possible to interpret Moore's ...

**7**

votes

**0**answers

361 views

### Galois descent for etale motivic cohomology

I am interested in the map from etale motivic cohomology of a smooth and projective variety over a field $K$ to the Galois invariants of etale
motivic cohomology over the algebraic closure $\bar K$:
$...

**11**

votes

**1**answer

664 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 ...

**12**

votes

**0**answers

536 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 ...

**4**

votes

**1**answer

315 views

### Proof of Theorem 6.8 in the paper “Singular homology of abstract algebraic varieties”

Background:
Theorem 6.8 in Suslin and Voevodsky's article "Singular homology of abstract algebraic varieties" states that there is an isomorphism between effective relative zero cycles $z_0^c(Z)^{eff}(...

**7**

votes

**0**answers

170 views

### Comparison of cdh and h cohomology

I expect that if $X$ is a finite type $k$-scheme, $k$ a field of characterisic $p$, then for primes $\ell\neq p$, $H^*_{cdh}(X;\mathbb{Z}/\ell\mathbb{Z})\cong H^*_{h}(X;\mathbb{Z}/\ell\mathbb{Z})$. Is ...

**3**

votes

**0**answers

155 views

### Vanishing of Zariski motivic sheaf

Let X be a smooth variety over a field. Is it true in general that the sheaf $\mathcal{H}^{r}(\mathbb{Z}(s))=0$ if $r>s \geq 0$? Here $\mathcal{H}^{r}(\mathbb{Z}(s))=0$ is the sheaf associated to ...

**8**

votes

**0**answers

220 views

### Injectivity of map in Beilinson's conjectures

In Beilinson's conjectures on special values of L-functions, he uses the image of the motivic cohomology of a a regular proper model in the motivic cohomology of the generic fiber to state the ...

**5**

votes

**0**answers

145 views

### Fulton's pullback vs. Pullback via Gersten complexes vs. Pullback coming from motivic homotopy SH(k)

Let $f: X \rightarrow Y$ be a morphism of smooth projective $k$-schemes (let's assume $f$ flat or even smooth). There is pullback in Fulton's style $f^*_{Ful}: CH^p(Y) \rightarrow CH^p(X)$ given by $[...

**4**

votes

**0**answers

162 views

### On non-vanishing of Milnor K-groups for infinite fields

It is well-known that for $n \geq 2$ and a finite field $k$, the Milnor $K$-group $K_n ^M (k)$ vanishes. I don't know who proved this first, but if curious, you may look at somewhere in Srinivas's ...

**6**

votes

**1**answer

359 views

### Intuition behind the definition of finite correspondences

Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over ...

**9**

votes

**0**answers

525 views

### Motivic cohomology of a point

I was wondering how much is known about the integral motivic cohomology groups of $\mathrm{Spec}\, k$, $H^{n,p}_{\mathrm{mot}}(\mathrm{Spec}\, k,\mathbb{Z})$. One knows that $H^{n,n}_{\mathrm{mot}}(\...

**4**

votes

**0**answers

194 views

### Motivic Interpretation of Rationally Trivial Cycles

The Chow groups are defined by taking groups of cycles modulo rationally trivial cycles. One then has a cycle class map to etale cohomology (over the base field), and for a number field, one expects ...

**8**

votes

**1**answer

332 views

### What is the etale sheafification of the (unramified) Milnor-Witt $K$-theory

I would like a reference/argument for the truth/falsity of the following statement:
The etale sheafification of the unramified Milnor-Witt K-theory (Nisnevich) sheaves are the (etale sheafification ...

**9**

votes

**1**answer

460 views

### Motivic cohomology and pushforward maps

I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field.
According to Mazza--Voevodsky--Weibel "...

**3**

votes

**1**answer

295 views

### Do there exist nontrivial motivic cohomology operations preserving weights?

Suppose that for each field $F$ a linear map $X(F): H_M^{p,q}(F, \mathbb{Q}) \longrightarrow H_M^{p,q}(F,\mathbb{Q})$ is given, such that $X$ commutes with inclusions of fields and transfers for ...