All Questions
Tagged with at.algebraic-topology motivic-cohomology
20 questions
13
votes
1
answer
2k
views
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 $...
- 12.9k
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
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 ...
- 12.9k
2
votes
1
answer
175
views
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 ...
9
votes
4
answers
1k
views
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 ...
- 1,589
8
votes
0
answers
587
views
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
3
votes
1
answer
294
views
A class in the motivic cohomology group $H^{0,1}(\operatorname{Spec}k;\mathbb{Z}/p)$
In the following paper
N. Yagita, Examples for the Mod p Motivic Cohomology of Classifying Spaces,
on the first page, below display (1.1), it says "It is known that there is an element $\tau\in H^...
- 17.4k
1
vote
0
answers
133
views
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 ...
- 5,901
3
votes
0
answers
224
views
Loop spaces of motivic Eilenberg-Mac Lane spaces
Consider the unstable $\mathbb{A}^1$-homotopy category (say over $\mathbb{C}$). By the loop space $\Omega X$ of an object $X$, we mean the homotopy fiber of $pt\to X$.
For an abelian group A and the ...
- 935
4
votes
1
answer
249
views
Suspension Theorem in $\mathbb{A}^1$-homotopy
In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have
$$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$
So I'm wondering if this has an analogue in ...
- 30.3k
1
vote
0
answers
107
views
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 ...
- 935
3
votes
1
answer
139
views
Proof that $Sing^IX$ is $I$-invariant for an interval object in a site by "simplicial decomposition"
I just saw a proof showing that $Sing^{A^1}X$ is $A^1$ invariant where they use an algebraic topological analogue of ``simplcial decomposition'' defined as follow:
The argument works by showing that ...
- 37.8k
6
votes
1
answer
511
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 ...
- 1,243
4
votes
1
answer
560
views
Thom class in motivic cohomology
Let $E$ be a vector bundle over a smooth scheme $X$. The Thom space of $E$ is $Th(E)=E/E-i(X)$ where $i\colon X \longrightarrow E$ is the zero section. This space is
$\mathbb{A}^{1}$ isomorphic to $\...
- 23.3k
21
votes
1
answer
2k
views
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 $X$ ...
- 2,327
15
votes
3
answers
3k
views
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 \...
- 651
3
votes
1
answer
350
views
Do there exist nontrivial motivic cohomology operations preserving weights?
Suppose that for each field $F$ a linear map $X(F): H_M^{p,q}(F, \mathbb{Q}) \longrightarrow H_M^{p,q}(F,\mathbb{Q})$ is given, such that $X$ commutes with inclusions of fields and transfers for ...
- 17.4k
2
votes
1
answer
589
views
Are finite correspondances flat?
In Voevodsky&Mazza&Weibel's book on motivic cohomology they define "an elementary correspondance from X (Smooth connected scheme over $k$) to Y (separated scheme over $k$) as an irreducible ...
- 2,871
21
votes
0
answers
1k
views
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 $RO(\mathbb{Z}/p)$-...
- 865
18
votes
1
answer
853
views
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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