All Questions
Tagged with motivic-cohomology homotopy-theory
10 questions
26
votes
1
answer
1k
views
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 ...
- 1,514
22
votes
3
answers
3k
views
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?
- 15.1k
14
votes
1
answer
1k
views
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 ...
- 2,227
12
votes
1
answer
1k
views
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 ...
- 15.1k
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
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
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
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 ...
- 4,316
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 ...
- 631
1
vote
0
answers
260
views
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