Questions tagged [motivic-cohomology]
The motivic-cohomology tag has no usage guidance.
198 questions
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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 $\...
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Suspension Theorem in $\mathbb{A}^1$-homotopy
In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have
$$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$
So I'm wondering if this has an analogue in ...
- 4,418
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A question regarding the Suslin's proof on Grayson motivic cohomology
This question is regarding the proof strategy presented in the paper, "On The Grayson Spectral Sequence", which its overview is explained in page 1 and 2. It seems a very general approach is ...
- 5,901
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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. ...
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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}(...
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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. ...
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When is the degree $(2,2)$ motivic cohomology generated by products of units?
The motivic coniveau spectral sequence tells us that for a scheme $X/k$, its cohomology $H^2(X,\mathbb{Z}(2))$ is the kernel of the tame symbol $K_2^M(k(X))\to \oplus_{Y} K_1^M(k(Y))$ where $Y$ runs ...
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Borel-Moore variant of the Lichtenbaum conjecture
A conjecture of Lichtenbaum expects that for a smooth proper variety $X$ over a finite field, the etale motivic cohomology groups $H^i(X_{et}, \mathbb{Z}(n))$ are finite for $i\neq 2n, 2n+2$, finitely ...
- 5,901
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Filtrations of motivic spectral sequences
I had a general question about motivic spectral sequences. In order to derive them we first begin with a filtration of the algebraic $K$-theory spectra. Something like this $\cdots \rightarrow W^2(X)\...
- 5,901
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On the Beilinson's conjecture regarding the proper flat integral models
I had a question which seems to be straightforward but I wasn't able to figure it out. In
page 13 of this paper a conjecture of beilison is mentioned that if $\mathcal{X}_{\mathbb{Z}}$ is a proper ...
- 5,901
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Brauer groups over local fields
Let X be a smooth projective variety over a local field of characteristic $(0,p)$. The Brauer group of X is a torsion group whose $l$-part is of cofinite type of some corank.
Is it know that the $l$-...
- 1,553
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$\mathbf{A}^1$- 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 ...
user119470
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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} :=...
user97068
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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{...
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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 ...
- 15.4k
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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
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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
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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 ...
user19475
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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
$H^{p,q}(\mathbf{P}^n,\mathbf{Z})$ =
$\oplus_{i=0}^n\mathrm{Hom}_\...
user19475
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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'...
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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 ...
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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 ...
- 2,227
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Higher 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 ...
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A class in the motivic cohomology group $H^{0,1}(\operatorname{Spec}k;\mathbb{Z}/p)$
In the following paper
N. Yagita, Examples for the Mod p Motivic Cohomology of Classifying Spaces,
on the first page, below display (1.1), it says "It is known that there is an element $\tau\in H^...
- 935
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Arc space & formal loops in motivic integration
One of the most essential ingredients in the theory of motivic integration are the space of arcs of a given $k$-variety
$X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points ...
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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
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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 H^{\...
- 33
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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
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Beilinson-Lichtenbaum conjecture for algebraic extensions of $\mathbb{Z}/m$
Let $X$ be smooth over some field $k$ and $m\in\mathbb{Z}$ so that $m$ maps to a unit in $k^{\times}$. By Beilinson-Lichtenbaum one has an isomorphism of cohomology groups
\begin{equation*}
\...
- 1,805
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Symmetrical monoidal $2$-category of cohomological correspondences
My question is whether a symmetric monoidal $2$-category of ``cohomological correspondences'' has been been rigorously constructed anywhere in the literature.
Let me be more precise about what I mean.
...
- 2,923
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Compute the nearby cycles functor for the category of mixed motives
I am reading the survey of J. Ayoub, The motivic nearby cycles and the conservation conjecture (see here), in which he introduced the original version motivic nearby cycles (another note by Illusie is ...
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Periodicity of algebraic $K$-theory in high enough degrees with finite coefficients
Given this it seems that higher algebraic $K$-theory and the etale one coincide in high enough degrees. The etale $K$-theory with finite coefficients is also Bott inverted $K$-theory, so it should be ...
- 5,901
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Do rationally contractible presheaves have rationally contractible injective resolution
Given a presheaf $\mathcal{F}: Sm/k\rightarrow Ab$ we define a new presheaf $C\mathcal{F}= \varinjlim\limits_{X\times \{0,1\}\subset U \subset X\times \mathbb{A}^1}\mathcal{F}(U)$. The presheaf $\...
- 5,901
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Generalization of conjectures involving Beilinson regulators
I had some questions about the Beilinson conjectures as mentioned in this page. I have to admit I do not know much about Deligne cohomology. The conjectures involve some form of comparison map between ...
- 5,901
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Relations between the morphic cohomology and Hodge theory
The main question can be summarized in the following form:
For a smooth projective complex variety $X$, is the cohomology $H^{2p}(X, \tau^{\leq p}\Omega_{alg}^{\bullet})$ supposed to surject onto $(H^...
- 5,901
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Are cubical higher Chow groups of field $CH^{n-1}(F,n)$ generated by linear cycles?
In the paper "The linearization of higher Chow cycles of dimension one" W. Gerdes proved that Higher Chow homology group $CH^{n-1}(F,n)=H^{n}(z^{n-1}(F,*))$ are generated by linear cycles.
...
user21167
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Comparing $K$-cohomology groups and weight filtration on the $K$-groups
The second page of the Quillen-Brown-Gersten is in the following form:
$$E_2^{p,q}=H^{p}(X, \mathcal{K}_{-q})\Rightarrow K_{-q-p}(X)$$
Here $\mathcal{K}_n$ is sheafification of the $U\mapsto K_n(U)$ ...
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Constructible motivic sheaves
Motivic complexes are a complex of Zariski sheaves that their Zariski hypercohomology gives us the motivic cohomology groups. There are various constructions of these complexes. As far as I know they ...
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Loop spaces of motivic Eilenberg-Mac Lane spaces
Consider the unstable $\mathbb{A}^1$-homotopy category (say over $\mathbb{C}$). By the loop space $\Omega X$ of an object $X$, we mean the homotopy fiber of $pt\to X$.
For an abelian group A and the ...
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Descent and Chow groups
One of the features of the $\mathbb{A}^1$-homotopy theory is the existence of the motivic Eilenberg-MacLane space $K(\mathbb{Z}(n),2n)$ such that for $k$-schemes $X$, we have
$$[X,K(\mathbb{Z}(n),2n)]\...
- 4,418
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Vanishing of a Higher Brauer group of a field
Let $k$ be a field. I am interested in the notion of the higher Brauer group defined as follows: For X a smooth scheme over $k$, $Br^r(X):=H^{2r+1}_{et}(X, \mathbb{Z}(r))$, an etale motivic cohomology ...
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Proof that $Sing^IX$ is $I$-invariant for an interval object in a site by "simplicial decomposition"
I just saw a proof showing that $Sing^{A^1}X$ is $A^1$ invariant where they use an algebraic topological analogue of ``simplcial decomposition'' defined as follow:
The argument works by showing that ...
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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 ...
user97464
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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
$$ H^i_{...
- 37k
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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 ...
- 16.9k
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Vanishing in etale motivic cohomology
As far as vanishing is concerned, the usual motivic cohomology has the following two properties (for a smooth scheme $X$ over a field):
$H^{p,q}(X, \mathbb Z) = 0$, if $p > q + dim(X)$; and
$H^{p,...
- 23
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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 ...
- 2,871
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The multiplicativity of the (complex) geometric realization of motivic cohomology
Consider the (complex) geometric realization of the motivic cohomology theory on simplicial presheaves over complex smooth schemes, which is a functorial homomorphism of $R$-modules, where $R$ is the ...
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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 ...
- 21
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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 ...
- 945