Questions tagged [motivic-cohomology]
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198 questions
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Cup product of $p$ first Galois Cohomologies of rationals, with coefficients in $\mu_{p}$
Let $p$ be an odd prime and $\mu_{p}$ be the group of $p^\text{th}$-roots of unity. Then, there exists a cup-product map which maps the product of $p$-copies of $H^{1}(\mathbb{Q}, \mu_{p})$ into $H^{p}...
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Isomorphism between motivic cohomology and algebraic cobordism
Let $MGL$ be the algebraic cobordism defined by Voevodsky, and $\Omega$ the algebraic cobordism constructed by Levine and Morel. For motivic cohomology $H^{p,q}$, we use Suslin-Voevodsky's definition. ...
- 121
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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'...
- 2,907
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Motives with compact support, Chow groups and proper pushforward maps
In Motivic cohomology of smooth geometrically cellular varieties (1999), Corollary 3.5, Bruno Kahn proves the following statement. Consider a cellular variety $X$ (i.e. it admits a filtration by ...
- 3,004
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Absolute Bloch-Kato Cohomology
The étale cohomology $R\Gamma_{\mathrm{ét}}(X;\mathbb{Z}_p(n))$ of a scheme $X/K$ can be computed by a Hochschild-Serre spectral sequence with terms of the form $H^i(K;H^j(X_{\overline{K}};\mathbb{Z}...
- 15.4k
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Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?
Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$.
Algebraically its $Spec$ is quite different from $k$. For example:
it has plenty non-trivial "line-...
- 24.9k
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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. ...
- 893
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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 $...
- 705
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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 ...
- 2,054
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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 ...
- 219
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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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150
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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 ...
- 5,901
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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 ...
- 5,901
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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 $\epsilon$ ...
- 5,901
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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 ...
- 5,901
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115
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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
$$ H^p(X,\mathcal{H}^q(\mathbb{Z}(n))) \Rightarrow H^{p+...
- 2,054
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158
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Map between Mordell-Weil group and Ext of (Mixed) Motives
We know that the motivic cohomology of an abelian variety $A$ over a number field $k$ computes the Mordell-Weil group up to torsion, and so if we were to grant the existence and nice behaviour of ...
- 4,418
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179
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Interpretation of Tate conjecture using motivic homotopy
For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps
$$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$
are surjective. To ...
- 679
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References for the construction of Beilinson's motivic Eisenstein classes
According to some authors, it is built in A.A.Beilinson "Higher regulator of modular curves" a class $\mathbf{Eis}_{\phi}$ in the motivic cohomology of the modular curve where $\phi$ is a ...
- 1,270
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Definition of Motivic cohomology via Ext
I have a little confusion about the definition of motivic cohomology assuming the existence of a category of (mixed) motives. I've seen it defined as either
$$\text{Ext}_{\mathcal{MM}_k}^i(1,M)$$
(for ...
- 4,418
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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.
- 918
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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 ...
- 5,901
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276
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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{...
- 5,901
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155
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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 ...
- 5,901
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119
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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 ...
- 2,054
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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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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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284
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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, ...
- 4,418
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207
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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 ...
- 893
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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 ...
- 2,871
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79
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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^{\...
- 309
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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
$$\begin{array}{ccc}\mathbb{G}_m\times\mathbb{G}_m&\longrightarrow&\mathbb{G}_m\times\mathbb{G}_m\\(...
- 918
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1
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154
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Grothendieck group and faithfully flat morpshim
For regular schemes $X$ and $Y$, and a faithfully flat morphism $f:Y \to X$, there is a flat pullback map of Grothendieck groups:
$$
f^*:K^0(X) \to K^0(Y).
$$
Is this map injective?
- 349
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Motivic cohomology commutes with field extension
$\DeclareMathOperator\Cor{Cor}$Let $X$ be a smooth scheme over $k$ and $k \subset F$ a field extension. Let $X_F$ be the field extension of $X$. Then there is a map
$$\varinjlim_{k\subset E \subset F} ...
- 1,168
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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 ...
- 918
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213
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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 ...
- 349
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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 ...
- 5,901
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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 ...
- 5,901
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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}...
- 8,167
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260
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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 ...
- 19
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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 ...
- 935
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108
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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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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 ...
- 5,901
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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 ...
- 5,901
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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 ...
- 5,901
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83
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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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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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459
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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. ...
- 2,054
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206
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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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588
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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 ...
- 5,901