All Questions
Tagged with motives at.algebraic-topology
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 $...
- 705
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
7
votes
0
answers
181
views
In what sense do the real and complex places correspond to setting q equal to 1 or -1?
It often happens that if we have a scheme $X/\mathbb Z$ (or an open subset thereof) and we denote by $p(q) = X(\mathbb F_q)$, then $p(1)$ and $p(-1)$ compute the euler characteristic of $X(\mathbb C)$ ...
- 7,746
3
votes
1
answer
758
views
Motives and topological data analysis
Here is some meta mathematics question.
During the last decade there has been some progress in the field of applied maths, called topological data analysis.
The setup starts with some set of points in ...
- 1,479
1
vote
0
answers
270
views
Eilenberg-Steenrod cohomological theory versus Weil cohomological theory [closed]
Can someone enlighten me what is the difference between an Eilenberg-Steenrod cohomological theory ( See here, https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms ), and a Weil ...
- 595
8
votes
1
answer
603
views
How much of the category of motives can be recovered from automorphisms of the Betti functor
Say we are working with schemes over a field $k\subset \mathbb{C}.$ A motive in the sense of Voevodsky is a functor $Sch\to D^bVect$ from (an appropriate category of) schemes to the DG category of ...
- 7,952
6
votes
1
answer
302
views
Homotopy equivalence between two basepoints of the etale homotopy type of the one-torus
Let $T = \mathbb{G}_m$ be the torus, and let $\tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $T$ and $\tilde{T}$ have a well-defined étale homotopy type. ...
- 7,952
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 $X$ ...
user87684
9
votes
0
answers
699
views
Motivic Galois theory and Betti realizations?
Why Motivic Galois groups are defined with Betti realizations? (In fact Absolute Galois groups can be defined in this way (with Betti realizations), why they are so related?).
- 99
35
votes
2
answers
3k
views
Equivalent descriptions of Hodge conjecture?
I would like to know equivalent descriptions of the Hodge conjecture (with references).
Dan Freed's Version:
Consider a topological cycle (boundary less chains that are free to deform) on a ...
5
votes
2
answers
2k
views
Connections between Standard, Hodge and Tate conjectures on algebraic cycles?
What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?
user56558
4
votes
1
answer
1k
views
Grothendieck's letter to Serre on the Standard Conjectures
Is it the letter dated in 27/08/1965 of Grothendieck where he presents to Serre the Standard conjectures on algebraic cycles?
- 61
12
votes
2
answers
794
views
What is the coefficient ring of algebraic K theory of the discrete $\mathbb{C}$?
Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete ...
- 7,952
31
votes
1
answer
4k
views
For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?
My apologies if this question is too naive.
Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...
- 118k
5
votes
2
answers
713
views
On triangulated categories of pro-objects
Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?
I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) ...
- 16.9k
7
votes
1
answer
498
views
Ring structure for the motivic spectrum/complex that represents singular cohomology?
As the discussion here Is singular cohomology representable by a (Voevodsky's) motivic complex?
shows, the singular cohomology of (smooth) complex varieties is represented by a motivic complex (...
- 16.9k
141
votes
0
answers
13k
views
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 7,527
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
37
votes
1
answer
3k
views
Morava on Shafarevich conjecture
$\DeclareMathOperator\Q{\mathbf{Q}}$Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture.
The Shafarevich Conjecture states: ...
- 2,734
3
votes
1
answer
408
views
Morava's "Motives and cell bundles"?
Hello, do you know more about, or some exposition of Morava's talk?
- 10.8k