Questions tagged [motivic-cohomology]
The motivic-cohomology tag has no usage guidance.
198 questions
8
votes
0
answers
669
views
Arakelov Motivic Cohomology and Hodge Theory
Lately I have been studying these two papers (first and second) that introduce a new cohomology theory called Arakelov motivic cohomology. While most of the applications presented in the papers are ...
- 341
7
votes
1
answer
919
views
Intuition behind the definition of finite correspondences
Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over ...
- 596
7
votes
1
answer
710
views
A question on Voevodsky´s categories
I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions:
1.- ...
- 761
7
votes
3
answers
718
views
Chow motive of a fibration
If we have a fibration of smooth projective complex varieties $F\to E\to B$, which is locally trivial in the analytic topology, and the global monodromy is trivial. Then is it true that the Chow ...
- 311
7
votes
1
answer
689
views
Two motivic complexes, compared
Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986).
Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) :...
user120812
7
votes
1
answer
327
views
References for the construction of Beilinson's motivic Eisenstein classes
According to some authors, it is built in A.A.Beilinson "Higher regulator of modular curves" a class $\mathbf{Eis}_{\phi}$ in the motivic cohomology of the modular curve where $\phi$ is a ...
- 1,270
7
votes
1
answer
551
views
Motivation for Suslin’s Rigidity Conjecture
Suslin Rigidity conjecture states that motivic cohomology
$$
H_{\mathcal{M}}^1(\operatorname{Spec}(F),\mathbb{Q}(n))
$$
of the field $F$ coincides with motivic cohomology for the subfield of ...
- 4,316
7
votes
0
answers
279
views
Adequate equivalence relations and algebraic $K$-theory
I have a somewhat vague question. We know that Adams operation gives a filtration on $K_i(X)\otimes \mathbb{Q}$ for the scheme $X$ such that the weight $j$ elements are isomorphic to higher Bloch Chow ...
- 5,901
7
votes
0
answers
230
views
Motivic cohomology of $n$-sphere
All motivic cohomology groups are taken with $\mu_2$ coefficient and $k$ has characteristic different from $2$.
Consider the affine variety $X$ with coordinate ring $k[x_1,\ldots,x_n]/(x_1^2+\ldots + ...
- 191
7
votes
0
answers
480
views
Galois descent for etale motivic cohomology
I am interested in the map from etale motivic cohomology of a smooth and projective variety over a field $K$ to the Galois invariants of etale
motivic cohomology over the algebraic closure $\bar K$:
$...
- 1,553
7
votes
0
answers
209
views
Comparison of cdh and h cohomology
I expect that if $X$ is a finite type $k$-scheme, $k$ a field of characterisic $p$, then for primes $\ell\neq p$, $H^*_{cdh}(X;\mathbb{Z}/\ell\mathbb{Z})\cong H^*_{h}(X;\mathbb{Z}/\ell\mathbb{Z})$. Is ...
- 71
6
votes
2
answers
882
views
Idempotent completions in K-theory
I have a reference request on following comment I found in
nLab article on Karoubian categories & envelopes. It states:
The Karoubian envelope is also used in the construction of the
category of ...
- 6,028
6
votes
1
answer
510
views
A question about the (motivic) integral cohomology of the Eilenberg-MacLane spectrum
Let $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum. Let $n\geq 0$ be any integer.
Is it known the structure of the group $[H\mathbb{Z},\Sigma^{n}H\mathbb{Z}]$?
Is there any reference in this ...
- 761
6
votes
2
answers
540
views
Galois descent in motivic cohomology
Let $X_N$ denote the Fermat curve defined over $\mathbb{Q}$ by the equation $x^N+y^N-z^N=0$ and let $X_{N,\mathbb{Q}(\mu_N)}$ be the base change. Let $G$ be the Galois group of $\mathbb{Q}(\mu_N)/\...
- 63
6
votes
1
answer
1k
views
Higher Chow groups revisited
Let $X$ be an algebraic variety over a field $k$.
Bloch defines the "algebraic singular complex" using the algebraic simplices
$$\Delta^n = \text{Spec}(k[x_0,\dots,x_n]/(x_0+x_1+\dots+x_n=1) \subset ...
user97068
6
votes
1
answer
621
views
Representable cohomology theories in motivic homotopy theory
I am reading Mazza's, Voevodsky's and Weibel's book Lecture Notes on Motivic Cohomology and have grown curious about the following question:
Which cohomology theories on $Sm/k$ are representable, i.e. ...
- 133
6
votes
1
answer
655
views
An example of an affine variety with non-zero Chow groups
Are there any examples known of an affine variety $A$ over an algebraically closed field such some Chow group (say, of codimension at least $2$) of $A$ with coefficients in $\mathbb{Z}/n\mathbb{Z}$ ($...
- 16.9k
6
votes
1
answer
411
views
Obtaining the Hilbert symbol from cup product on motivic cohomology
Let F be a number field. Does the Hilbert symbols at the various places of F arise from the cup product on the motivic cohomology groups of Spec(F)? And if so, is it possible to interpret Moore's ...
- 61
6
votes
0
answers
400
views
Hodge Realisation of Mixed Tate Motives
For a field $k$ which satisfies Beilinson-Soule vanishing conjecture, the from Levine's paper,
https://www.uni-due.de/~bm0032/publ/TateMotives.pdf
There exists an abelian category of mixed Tate ...
- 2,971
6
votes
0
answers
277
views
On non-vanishing of Milnor K-groups for infinite fields
It is well-known that for $n \geq 2$ and a finite field $k$, the Milnor $K$-group $K_n ^M (k)$ vanishes. I don't know who proved this first, but if curious, you may look at somewhere in Srinivas's ...
- 937
6
votes
0
answers
165
views
Exactness of pure functors
I can't prove this lemma in "Notes on motivic cohomology, Beilinson, Macpherson, Schechtman":
Lemma. A pure functor is exact.
Definitions: A mixed category $\mathcal{M}$ is a $\mathbb{Q}$-abelian ...
- 4,484
6
votes
0
answers
351
views
Correspondences of schemes induced by a finite Galois extension
I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between schemes.
Recall that if $...
5
votes
2
answers
1k
views
Relation between motivic cohomology and Quillen K-theory
What's the relation between Voevodsky's motivic cohomology and Quillen K-theory of a scheme?
user19475
5
votes
1
answer
468
views
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
5
votes
1
answer
739
views
Surjective étale morphisms étale locally split
The actual question is slightly more general than that in the title:
Let $p: U\to Y$ be a surjective étale morphism and $Y\to X$ be a finite morphism of schemes. Is there an étale cover $V\to X$ (...
- 1,906
5
votes
1
answer
488
views
Is $B\mathbb{G}_m$ strongly $A^1$-invariant?
I have just seen the definition of strongly ${A}_1$ invariance:
A sheaf of group $G$ is strongly $A_1$ invariance , if both $H^0(-;G)$ and $H^1(-;G)$ is $A_1$-invariant.
I haven't got too much ...
- 631
5
votes
1
answer
316
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 ...
- 918
5
votes
1
answer
482
views
Around algebraic equivalence of cycles
Let $k$ be a finitely generated field, $X$ a smooth projective $k$-variety, $\ell$ a prime number, $\ell\in k^{\times}$, $r\ge 0$ an integer.
The Tate conjecture asserts surjectivity of the cycle ...
user92332
5
votes
1
answer
169
views
Is transfinite composition of weak equivalences of simplicial presheaves a weak equivalence?
In a left Bousfield localization of the projective model structure on the category of simplicial presheaves, what is the condition that transfinite composition preserves weak equivalences? How about ...
- 631
5
votes
1
answer
324
views
Nori's Mixed Motives and Realisation Functors
The conjecture 51 in Levine's Mixed Motives in Handbook of K-theory
http://www.math.illinois.edu/K-theory/handbook/1-429-522.pdf
states that the functor induced by $hs:\text{ECM} \rightarrow \text{...
- 2,971
5
votes
1
answer
552
views
A quite puzzling question on Deligne cohomology sheaves and cycle maps
Intro. I would be deeply grateful if someone could please clarify the following to me.
The question. (the main point is (4))
Let $X$ be a smooth projective variety over $\mathbf{C}$, and $\mathbf{Z}(...
user119470
5
votes
1
answer
426
views
Blowup formula for motivic cohomology
If $X$ is a smooth projective variety over a field, $Z\subset X$ a smooth closed subvariety of codimension $d$, $X'\to X$ the blowup of $X$ along $Z$, there's the blowup formula
$$H^j(X'_{et},\mathbf{...
user95222
5
votes
0
answers
157
views
Motives with compact support, Chow groups and proper pushforward maps
In Motivic cohomology of smooth geometrically cellular varieties (1999), Corollary 3.5, Bruno Kahn proves the following statement. Consider a cellular variety $X$ (i.e. it admits a filtration by ...
- 3,004
5
votes
0
answers
150
views
Analytical Dold-Thom
Let's $X$ be a projective smooth variety over a field that has an embedding into $\mathbb{C}$. Let's denote the infinite symmetric power of $X$ by $\text{Sym}^{\infty}(X)$. Denote the algebraic ...
- 5,901
5
votes
0
answers
234
views
Definition of Motivic cohomology via Ext
I have a little confusion about the definition of motivic cohomology assuming the existence of a category of (mixed) motives. I've seen it defined as either
$$\text{Ext}_{\mathcal{MM}_k}^i(1,M)$$
(for ...
- 4,418
5
votes
0
answers
496
views
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 ...
- 121
5
votes
0
answers
167
views
Mod $l$ algebraic $K$-theory of product of an algebra with a complete algebra
By Gabber's rigidity the mod-$l$ $K$-theory of $k[[t]]$ and $k$ are isomorphic for a field $k$. Is there anything that we can say about the mod $l$ $K$-theory of $A\otimes_kk[[t]]$? Note that this is ...
- 5,901
5
votes
0
answers
279
views
Stalk of motivic homotopy sheaves
In contrast to "classical" homotopy theory, in the motivic homotopy theory, we don't have homotopy group but rather homotopy sheaves in the Nisnevich topology, which is associated to the ...
- 4,418
5
votes
0
answers
181
views
What is the fibrant replacement of an Eilenberg-MacLane space in unstable motivic homotopy theory?
One can take an Eilenberg-MacLane space $K(A, n)$ for some abelian group $A$, and view it as (locally) constant simplicial sheaf on $Sch/k$, the category of schemes, or smooth schemes, over a field $k$...
- 51
5
votes
0
answers
512
views
Poincaré duality for motivic cohomology
Is Poincaré duality for étale motivic cohomology known for projective regular (not necessarily smooth, just regular) schemes over Dedekind rings?
More precisely, two questions. Let $f: \mathcal{X}\to\...
user92332
5
votes
0
answers
172
views
Fulton's pullback vs. Pullback via Gersten complexes vs. Pullback coming from motivic homotopy SH(k)
Let $f: X \rightarrow Y$ be a morphism of smooth projective $k$-schemes (let's assume $f$ flat or even smooth). There is pullback in Fulton's style $f^*_{Ful}: CH^p(Y) \rightarrow CH^p(X)$ given by $[...
- 326
5
votes
0
answers
801
views
Two questions on motivic homotopy theory
I am a beginner in motivic homotopy theory and motivic cohomology and have just begun reading books by Dundas et al. and Mazza-Voevodsky-Weibel. I have two basic questions:
Why is the question of $\...
- 805
4
votes
2
answers
375
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\\(...
- 918
4
votes
1
answer
912
views
Standard conjectures on positive characteristic
In this MO answer of M. Bondarko, he says:
"the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..."
and in Remarks on Grothendieck's ...
- 545
4
votes
1
answer
238
views
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
4
votes
1
answer
260
views
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
4
votes
1
answer
248
views
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
4
votes
1
answer
580
views
Voevodsky's proof in any characteristic (for motivic and Chow)
Regarding the published version of "Motivic cohomology groups are isomorphic to higher Chow groups in any characterstic" (IMRN) available at:
http://imrn.oxfordjournals.org/content/2002/7/351.full....
- 650
4
votes
1
answer
276
views
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
4
votes
1
answer
182
views
Explicit description of Verdier quotient of effective motives
Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory ...
- 41