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7
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0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
2answers
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
3answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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 ...