Questions tagged [motivic-cohomology]
The motivic-cohomology tag has no usage guidance.
198 questions
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What is the relationship between motivic cohomology and the theory of motives?
I will begin by giving a rough sketch of my understanding of motives.
In many expositions about motives (for example, http://www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined ...
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What do higher Chow groups mean?
Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i,...
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Why is Milnor K-theory not ad hoc?
When Milnor introduced in "Algebraic K-Theory and Quadratic Forms" the Milnor K-groups he said that his definition is motivated by Matsumoto's presentation of algebraic $K_2(k)$ for a field $k$ but is ...
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What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences?
I come today to mathoverflow to showcase some genuine confusion about the motivic world. I want to ask some questions before actually starting to study the subject, to build some sense of direction.
I ...
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Beilinson conjectures
Continuing an amazingly interesting chain of answers about motivic cohomology, I thought I should learn about the Beilinson conjectures, referred there.
I have found some references, and they seem to ...
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Homotopy theory of schemes examples
Is it possible give an example of (or explain) how the Voevodsky et al.'s homotopy theory of schemes computes higher Chow groups?
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Motivic Cohomology vs. Chow for singular varieties?
I'm absolutely new to this stuff I'm asking about, so I hope this is not nonsense.
If X is a smooth scheme over a perfect field k, I can study its motivic cohomology in the sense of Voevodsky and ...
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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 $X$ ...
user87684
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What is the current knowledge of equivariant cohomology operations?
In Caruso's paper, "Operations in equivariant $Z/p$-cohomology," http://www.ams.org/mathscinet-getitem?mr=1684248, he shows that the integer-graded stable cohomology operations in $RO(\mathbb{Z}/p)$-...
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Applications of homotopy purity theorem of Morel-Voevodsky
One of the most important theorems in motivic homotopy theory is the homotopy purity theorem of Morel-Voevodsky which says that the motivic Thom space of the normal bundle $\mathcal N_{Z/X}$ of a ...
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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 $\...
- 16.9k
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Can we reconstruct positive weight invariants in algebraic topology using algebraic geometry?
I can't really say that I understand what a weight is, but the qualitative distinction between weight zero and positive weight has come up a couple times in MathOverflow questions:
The étale ...
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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 ...
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Relation between motivic homotopy category and the derived category of motives
What's the relation between the pointed motivic homotopy category $\mathcal{H}_*(k)$ and the derived category of motives $\mathbf{DM}^-_{eff}(k)$ besides the representability of motivic cohomology in ...
user19475
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Motivic cohomology with finite coefficients for singular varieties
Let $X$ be a smooth variety over a field $K$ whose characteristic does not divide a positive integer $m$. Then the motivic cohomology of $X$ with coefficients in $\mathbb Z/m(j)$ can be computed in ...
- 15.6k
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State of the art for Gersten's conjecture for K-theory?
Does anyone know (of a reference to) under what restrictions on the regular scheme $X$ it is known that we have an exact sequence
$$0 \to \mathcal{K}_n(X) \to \bigoplus_{x \in X^{(0)}} K_n(k(x)) \to \...
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Why $K(X) \longrightarrow G (X)$ is a Poincaré duality for K-theory?
It's well known that for Noetherian separated regular schemes the canonical map $$K(X) \longrightarrow G(X)$$ (Quillen uses $K'$ instead of $G$, though) is a weak equivalence.
This statement is ...
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Motivic cohomology and cohomology of Milnor K-theory sheaf
Let $X$ be a smooth variety over a field $k$. (Assume $k$ has characteristic 0 if it helps; in fact I'd be happy to assume that $k$ is a finite extension of either $\mathbf{Q}$ or $\mathbf{Q}_p$).
...
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Can there exist Chow motives/motivic cohomology for compact Kähler manifolds?
Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are there any ...
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Motivic cohomology vs. K-theory for singular varieties
As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for $\...
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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 ...
user92332
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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 $...
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Learning a little Motivic Cohomology
Simply because I find it interesting, I have spent some time studying motivic cohomology from the lectures by Mazza, Voevodsky and Weibel. However, I'm finding it hard to tell if the theory is ...
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Formalism of homotopy theory of schemes
I have some vague knowledge about the philosophy that schemes should be thought of as similar to topologic spaces, and we should divide everything by homotopy, and that the space should be actually ...
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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 ...
user95222
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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}...
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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 ...
- 17.4k
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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 ...
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Are there analogues of Beilinson's conjectures for motives with coefficients?
There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...
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Kunneth formula for motivic cohomology
I was wondering when the Kunneth formula holds for motivic cohomology:
$$
H^p(X,A(\alpha)) = \bigoplus_{i+j=p;\beta+\gamma = \alpha} H^j(X,A(\beta)) \otimes H^i(X,A(\gamma))
$$
where $H^p(X,A(\...
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How to think about infinite generatedness of motivic cohomology
In this question I previously asked how to think about the motivic complex $\mathbf{Z}(1)_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^*((-)\wedge S^{2n}...
user97068
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Motivic cohomology and pushforward maps
I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field.
According to Mazza--Voevodsky--Weibel "...
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When do the $\gamma$-filtration and codimension filtration of K-theory agree?
Let $X$ be a smooth quasiprojective algebraic variety over a field $k$. Then the $K$-groups $K_m(X)$ are defined, and there are two standard filtrations on them: the "codimension filtration" given by
...
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A question about the vanishing of motivic cohomology in negative Tate twist
Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.
Is it true that
$$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[...
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Stable motivic cohomology with finite coefficients?
In this question, which attracted no responces so far, I've asked whether it is possible to extend the Beilinson-Lichtenbaum etale descent rule for motivic cohomology to singular varieties, in ...
- 15.6k
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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 ...
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Correspondences in Topology
I had one or two little fights with correspondences in the context of algebraic geometry where an elementary correspondence $C:X\to Y$ of connected smooth $k$-Schemes seems to be defined as an ...
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the graded pieces of the gamma-filtration of Quillen K-theory and Chow groups of a regular scheme
Let $X$ be a regular scheme and consider Grothendieck's $\gamma$-filtration $F^nK(X)$ on $K(X)$. For the graded pieces, one has $Gr^0K(X) = CH^0(X)$ and $Gr^1K(X) = \mathrm{Pic}(X) = CH^1(X)$. Does ...
user19475
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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 ...
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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?
...
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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 ...
user113393
9
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Bass' conjecture implies the Parshin's conjecture
In the appendix of this paper. It is proved that Bass' conjecture for $K_n$ implies the rational Beilinson-Soulé conjecture for $K_n$. Then at the end the author claims that the same method can be ...
- 5,901
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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 ...
user95222
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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 ...
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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}}(\...
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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) $z_n(X,d) :=$...
user97068
8
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1
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When does the continuous Galois(=etale) cohomology of fields coincide with the naive one? Often true by the Bloch-Kato conjecture?
For a field $F$ I am interested in its $l$-adic (Galois=\'etale) cohomology; here $l$ is a prime distinct from the characteristic of $F$ (for simplicity one may assume that the latter is $0$).
For $...
- 16.9k
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Known norm varieties and the Bloch-Kato conjecture
The Bloch-Kato conjecture states that
$K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$.
A important part in the proof of the Bloch-Kato conjecture is to ...
- 1,479
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0
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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
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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 ...
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