Questions tagged [motivic-cohomology]

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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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Who proved the motivic 6-functor formalism?

In the recent beautiful talk "Motives and ring stacks" Peter Scholze states the theorem saying that there exists an initial 6-functor formalism on $\mathit{Sch}_\mathbb{Z}$ such that when $...
Ola Sande's user avatar
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Are higher Chow groups and motivic cohomology isomorphic for smooth schemes over a Dedekind domain?

Voevodsky famously proved that his motivic cohomology defined by presheaves with transfers was isomorphic to Bloch's higher Chow groups for smooth schemes over a field. There have long been ...
xir's user avatar
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Universal properties for Bloch's higher Chow groups

I work in the category of varieties over some field of characteristic zero. Assume that for any variety I can define the group $\widetilde{CH}^r(X,n)$ which behave like classical Bloch's higher Chow ...
Galois group's user avatar
3 votes
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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*} \...
The Thin Whistler's user avatar
5 votes
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140 views

Analytical Dold-Thom

Let's $X$ be a projective smooth variety over a field that has an embedding into $\mathbb{C}$. Let's denote the infinite symmetric power of $X$ by $\text{Sym}^{\infty}(X)$. Denote the algebraic ...
user127776's user avatar
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Motivic complex on arithmetic schemes

If we believe the finite generation of motivic cohomology for regular arithmetic schemes like $X$ then we can see that (using Quillen-Lichtenbaum)for infinitely many primes $l$ we have an isomorphism ...
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Integral Beilinson-Lichtenbaum truncation issue

In this book page 10 the section about Beilinson–Lichtenbaum Conjecture, it mentions that Bloch-Kato implies that $\mathbb{Z}(n) \cong \tau ^{\leq n+1} R\epsilon_*\mathbb{Z}(n)_{ét}$ where $\epsilon$ ...
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2 votes
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$\mathbb{A}^1$-invariance and cdh descent

It is known that cdh-sheafification of algebraic $K$-theory coincides with homotopy $K$-theory. Although I haven't gone through the details of the proof, I was wondering whether there is a general set ...
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Action of correspondences on motivic cohomology sheaves

Writing $\mathcal{H}^a(\mathbb{Z}(b))$ for the Zariski sheaf of motivic cohomology groups, there is a hypercohomology/descent spectral sequence $$ H^p(X,\mathcal{H}^q(\mathbb{Z}(n))) \Rightarrow H^{p+...
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Map between Mordell-Weil group and Ext of (Mixed) Motives

We know that the motivic cohomology of an abelian variety $A$ over a number field $k$ computes the Mordell-Weil group up to torsion, and so if we were to grant the existence and nice behaviour of ...
curious math guy's user avatar
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Interpretation of Tate conjecture using motivic homotopy

For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps $$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$ are surjective. To ...
TCiur's user avatar
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6 votes
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References for the construction of Beilinson's motivic Eisenstein classes

According to some authors, it is built in A.A.Beilinson "Higher regulator of modular curves" a class $\mathbf{Eis}_{\phi}$ in the motivic cohomology of the modular curve where $\phi$ is a ...
Marsault Chabat's user avatar
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Definition of Motivic cohomology via Ext

I have a little confusion about the definition of motivic cohomology assuming the existence of a category of (mixed) motives. I've seen it defined as either $$\text{Ext}_{\mathcal{MM}_k}^i(1,M)$$ (for ...
curious math guy's user avatar
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102 views

Is $\mathbb{Z}_{\mathrm{tr}}(X)$ a cdh sheaf?

Suppose $X\in \mathrm{Sm}/k$. Is the sheaf with transfers $\mathbb{Z}_{\mathrm{tr}}(X)$ a cdh sheaf? Its sections are finite correspondences.
Nanjun Yang's user avatar
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Purity of truncated Zariski sheaves of roots of unity

By Quillen-Lichtenbaum theorem the weight $i$ mod $l$ motivic complex is quasi-isomorphic to $\tau^{\leq i}R\alpha_{*}\mu_l^{\otimes i}$ where $\alpha$ is the forgetful functor sending etale sheaves ...
user127776's user avatar
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Hodge conjecture for generic points

I was reading the following paper: "Beilinson’s Hodge Conjecture For Smooth Varieties". They study the cycle class map $cl_{m,r}: H^{2r-m}_{\mathcal{M}}(U, \mathbb{Q}(r))\rightarrow \text{...
user127776's user avatar
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2 votes
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Constructions of motivic complex that is only supported on positive degrees

It is expected by Beilinson-Soule vanishing conjecture that negative motivic cohomology groups are zero so the motivic complexes are supported on nonnegative degrees. My question is about ...
user127776's user avatar
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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 ...
xir's user avatar
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3 votes
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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. ...
gdb's user avatar
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3 votes
1 answer
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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, ...
Marsault Chabat's user avatar
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Proof of Geisser-Levine

I am trying to understand the proof of the Geisser-Levine theorem (Thm 8.4 here ) which claims that for a smooth variety $X$ over a perfect field of characteristic $p$ we have an isomorphism $$H^s(X, ...
curious math guy's user avatar
3 votes
0 answers
189 views

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 ...
Alexey Do's user avatar
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1 answer
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Functoriality conjectures on the slice filtration

Voevodsky wrote on his paper "Open Problems in the Motivic Stable Homotopy Theory, I" that Three other groups of conjectures in motivic homotopy theory, not included in to this paper, seem ...
Tintin's user avatar
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Localization with or without transfers

Let $Sh_{Nis}^{tr}$ be the category of Nisnevich sheaves with transfers of abelian groups over a perfect field. Let $u\colon Sh_{Nis}^{tr}\to Sh_{Nis}$ be the functor “forget transfers” and let $h_0^{\...
user197402's user avatar
4 votes
2 answers
365 views

On the swapping map of $\mathbb{G}_m$

On Voevodsky's paper 'Cancellation theorem', Lemma 4.8, he stated in the proof that the map $$\begin{array}{ccc}\mathbb{G}_m\times\mathbb{G}_m&\longrightarrow&\mathbb{G}_m\times\mathbb{G}_m\\(...
Nanjun Yang's user avatar
1 vote
1 answer
144 views

Grothendieck group and faithfully flat morpshim

For regular schemes $X$ and $Y$, and a faithfully flat morphism $f:Y \to X$, there is a flat pullback map of Grothendieck groups: $$ f^*:K^0(X) \to K^0(Y). $$ Is this map injective?
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Motivic cohomology commutes with field extension

$\DeclareMathOperator\Cor{Cor}$Let $X$ be a smooth scheme over $k$ and $k \subset F$ a field extension. Let $X_F$ be the field extension of $X$. Then there is a map $$\varinjlim_{k\subset E \subset F} ...
XT Chen's user avatar
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5 votes
1 answer
297 views

Motivic cohomology with $\mathbb{Z}/2$ coefficients in positive characteristic

In G. M. L. Powell's note 'Steenrod operations in motivic cohomology', he stated that if $\mathrm{char}(k)=0$, $$H^{*,*}(k,\mathbb{Z}/2)=K_*^M(k)/2[\tau]$$ where $\tau\in H^{0,1}$ is the unique ...
Nanjun Yang's user avatar
1 vote
0 answers
201 views

Algebraic correspondence as morphisms in Betti cohomology

$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for ...
OOOOOO's user avatar
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Hodge's conjecture as a quasi-isomorphism between two complexes of sheaves

A version of Hodge's conjecture due to Beilinson, expects that the Betti cycles class map $H_{\mathcal{M}}^i(X,\mathbb{Q}(j))\rightarrow hom_{MHS}(\mathbb{Q}(0),H^{i}(X,\mathbb{Q}(j) ))$ is surjective ...
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4 votes
1 answer
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Support of torsion in the Borel–Moore homology

Given a complex quasi-projective variety $X$, let $\alpha$ be an element of the Borel–Moore homology $H_i^\text{BM}(X)$ such that it can be killed by a prime $p$. Under what conditions one can say ...
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12 votes
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Can Quillen-Lichtenbaum recover Borel's computation?

Borel famously used analysis on symmetric spaces to compute the rationalised algebraic $K$-theory groups of rings of integers $\mathcal{O}_F$ in number fields, e.g. $K_i(\mathbb{Z}) \otimes \mathbb{Q}...
skupers's user avatar
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1 vote
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Non-examples of mixed Tate motives

I was trying to find examples of schemes (preferably smooth) over $\mathbb{C}$ which have motives that aren't mixed Tate. I wasn't able to come up with anything or find an argument that's been written ...
Arpith's user avatar
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2 votes
1 answer
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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 ...
Xing Gu's user avatar
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3 votes
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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 ...
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4 votes
1 answer
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Etale $K$ theory coincides with algebraic one in high enough degrees

I have seen the claim that Beilinson Lichtenbaum implies that higher algebraic $K$ groups coincides with etale ones integrally in high enough degrees. Is this statement accurate? What conditions are ...
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2 votes
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When mod $l$ algebraic $K$-groups inject into the mod $l$ etale algebraic $K$-group?

I was wondering whether in general it is known that for an invertible prime $l$, the mod-$l$ algebraic $K$-group of a regular Noetherian scheme $X$ injects into the mod-$l$ etale $K$-groups? I just ...
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Continuity of motivic cohomology under direct limit

Given the motivic complexes $\mathbb{Z}(n)$ on the big Zariski site of finite type smooth $k$-schemes denoted by $FinSm_k$, we pullback it to the smooth $k$-schemes i.e. $Sm_k$. For example for a ...
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3 votes
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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 $\...
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4 votes
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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 ...
user127776's user avatar
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4 votes
1 answer
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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. ...
xir's user avatar
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3 votes
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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 ...
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8 votes
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Values of cohomology theory on a point

$\DeclareMathOperator\Sm{Sm}$It is a well-known fact that in algebraic topology, generalized cohomology theories are determined by their values on the point. I was wondering whether anything similar ...
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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^...
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2 votes
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Motivic complexes associated to adequate equivalence relations

Motivic cohomology sheaves $\mathbb{Z}(n)$ are homotopy invariant sheaves with transfers (under finite maps) and they satisfy excision long exact sequence when everything is smooth. The motivic ...
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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. ...
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Motivic cohomology of Weil restriction

hopefully this isn't too obvious or well-known, but I couldn't find it by searching. The motivic cohomology of $\mathbb{G}_m$ and its powers over any base with known motivic cohomology can be computed ...
xir's user avatar
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Is this etale motivic or motivic cohomology?

I am trying to reconcile my understanding of motivic cohomology (based on the Lecture Notes by Mazza-Voevodsky-Weibel) with the homotopic point of view. I am currently struggling to answer this ...
J.P. Gimori's user avatar
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0 answers
132 views

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 ...
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