Questions tagged [motivic-cohomology]
The motivic-cohomology tag has no usage guidance.
198 questions
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Grayson filtration and weight filtration
I had a question that might be well-known but I'm not sure where to find it. Grayson defined a filtration on the algebraic $K$-theory of affine regular rings via commuting automorphisms which you can ...
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Vanishing of Motivic Cohomology
In these notes, page $10$, bullet $(5)$, it is stated that if $X$ is a scheme of finite type over a field $k$, then the motivic cohomology $\mathrm{H}^{p,q}(X,R)$ of $X$ over $k$, where $R$ is a ring, ...
user62675
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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. ...
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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 $H^i(K;H^j(X_{\overline{K}};\mathbb{Z}...
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Are higher Chow groups and motivic cohomology isomorphic for smooth schemes over a Dedekind domain?
Voevodsky famously proved that his motivic cohomology defined by presheaves with transfers was isomorphic to Bloch's higher Chow groups for smooth schemes over a field. There have long been ...
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$\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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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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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.
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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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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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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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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 ...
- 5,901
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Lefschetz type theorems/conjectures for algebraic $K$-theory
Lefschetz hyperplane theorem, compares the homology/cohomology of a projective variety with a hyperplane section of it and claims they are isomorphic in certain ranges. There are Lefschetz type ...
- 5,901
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Few questions about the algebraic cycles and the conjectures of Beilinson and Tate
I have three slightly related questions about algebraic cycles which I am just going to list them. I'd really appreciate any answers:
1) Is there any example of a smooth projective variety $X$ over a ...
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Chow group of a pair
In a paper by S. Landsburg the (higher) Chow groups of a pair $(X,Y)$ are defined when $Y$ is a smooth closed subvariety of a smooth variety $X$ as follows.
We consider the sub-complex $z^{*}(X;.)_{Y}...
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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 ...
user19475
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Voevodsky's 'split standard triple' argument: an explanation; does it work with $Z/nZ$-coefficients?
For varieties over a perfect field (of some characteristic $p$ that could be 0) Voevodsky defines the notion of a 'standard triple' (see http://books.google.ru/books?id=TzUmk87bN9cC&pg=PA85&...
- 16.9k
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a counterexample of Serre vs. motivic cohomology
There is a counterexample of Serre showing that there is no Weil cohomology theory with coefficients in $\mathbf{Q}, \mathbf{Q}_p, \mathbf{R}$ over $\mathbf{F}_{p^2}$ (a supersingular elliptic curve). ...
user19475
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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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Is there a projection formula for motivic étale cohomology?
Let $f: X \to Y$ be a finite morphism of varieties with $Y$ smooth. Is there a projection formula for $f$ and $H^{i}_{et}(-,\mathbf{Z}(1)) = H^i(-,\mathbf{G}_m)$?
Background: I want to show that for ...
user19475
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Pseudo-Abelian Completion in the constrution of Motifs (by Y. Manin)
I reading Yuri Manin's famous paper on "CORRESPONDENCES, MOTIFS AND MONOIDAL TRANSFORMATIONS" and struggle with his definition for so called pseudo-abelian completion given on page 453 by a reason I ...
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Definition of Hurewicz map relating $SH(k)$ with $DM_\_^{eff}(k)$
In "Motivic Homotopy Theory" on page 153 it is stated that there exists a canonical Hurewicz map relating the motivic stable homotopy category with the category $\operatorname{DM}_\_^{eff}(k)$. ...
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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 ...
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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 $\epsilon$ ...
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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
$$ H^p(X,\mathcal{H}^q(\mathbb{Z}(n))) \Rightarrow H^{p+...
- 2,044
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Interpretation of Tate conjecture using motivic homotopy
For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps
$$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X_{\bar{k}},\mathbb{Q}_l(i))^{G_k}$$
are surjective. To ...
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Proof of Geisser-Levine
I am trying to understand the proof of the Geisser-Levine theorem (Thm 8.4 here ) which claims that for a smooth variety $X$ over a perfect field of characteristic $p$ we have an isomorphism
$$H^s(X, ...
- 4,418
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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^{\...
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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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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 ...
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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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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 ...
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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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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 ...
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Contractibility of a $K_0^{\oplus}$ presheaf
Let's assume $X$ is a smooth projective variety over a field. Let $\Delta^{\bullet}$ be the cosimplicial scheme over the same field, where at level $n$ is just the $n$-th algebraic simplex. We can ...
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What can be said about the Chow rings of classifying spaces of semi-direct products of groups?
For instance, what can we say about the Chow ring of the classifying space of a semi-direct product $CH^*(B(G\ltimes H))$, in terms of the Chow rings of $CH^*(BG)$, $CH^*(BH)$, and the singular ...
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Example of a non-strongly $A^1$ invariant sheaf of groups
A Nisnevich sheaf of groups is called strongly $A^1$-invariant if its classifying space $BG$ is $A^1$-invariant. I would like an example that is $A^1$ invariant but not strongly $A^1$-invariant for an ...
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Norm quadrics and their motives
Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective ...
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Which "concrete" morphisms of varieties and motives induce bijections of their lower Chow groups?
This question is a continuation of Varieties with Chow groups supported in positive codimension: examples and properties?
What examples are known of morphisms of varieties and Chow motives (say, over ...
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Differences and relationships between Motivic cohomology and Universal cohomology theory? [duplicate]
Differences and relationships between Motivic cohomology (Beilinson, Lichtenbaum and Voevodsky) and Universal cohomology theory (Grothendieck)?
user56696
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Homotopy theory of schemes
I have seen the notion of Homotopy come up in several contexts in schemes. For example, the book "Lectures on Motivic Cohomology" by Mazza, Weibel and Voevodsky uses this language to some extent. I.e. ...
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What can one say about a smooth variety whose lower cohomology is trivial?
Let $X$ be a smooth (quasi-affine) complex variety; suppose that its cohomology (say, with integral coefficients) is trivial in degrees $0 < i\le s$ (for some $s>0$). What can one say about such ...
- 16.9k
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On motivic cohomology with compact support
For a smooth projective variety $X$ and its closed non-smooth subvariety $Z$ I would like to say that a cone of the morphism between the motivic cohomology of $Z$ and those of $X$ is the motivic ...
- 16.9k
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tensor product of motivic complexes $\mathbf{Z}(n)$
Is the morphism $\mathbf{Z}(n) \otimes^L \mathbf{Z}(m) \to \mathbf{Z}(n+m)$ from the Beilinson-Lichtenbaum conjectures a quasi-isomorphism?
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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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What can one say about zero-cycle groups for products of Chow motives
What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes ...
- 16.9k
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Vanishing of motivic cohomology with finite coefficients in negative degrees
I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not.
STATEMENT:
Let $X$ be a smooth and projective scheme over a finite field $\...
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