All Questions
14 questions
32
votes
4
answers
3k
views
Spectrum of the Grothendieck ring of varieties
Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...
- 4,713
3
votes
1
answer
533
views
Homotopy invariance of $\ell$-adic cohomology
In the end of the Voevodsky’s lectures on cross functors, P. Deligne considers a couple of axioms which define (using the vocabulary of Ayoub's thesis) a stable homotopical 2-functor. Among them, we ...
- 148k
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 ...
- 63.9k
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 ...
- 3,065
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 ...
- 13.6k
3
votes
0
answers
178
views
Finiteness results in the category of schemes up to $\mathbb{A}^1$-homotopy
In algebraic geometry, we know that there exist geometrical conditions on a scheme $X/k$ for having finitely many rational points when $k$ is a number field. Namely for curves there is the Mordell ...
- 4,408
3
votes
0
answers
182
views
Where is smoothness used in Voevodsky's homotopy theory of schemes? [duplicate]
Let $S$ be a smooth noetherian scheme, and let $Sm/S$ be the category of smooth schemes over $S$. Voevodsky constructs the homotopy category of motives (resp. the stable homotopy category of motives) $...
- 3,019
13
votes
2
answers
2k
views
What is the best reference for motives?
I want to learn about homotopy theory on number fields, and I heard that the theory of motives made it possible, so I want to know what is a good textbook for motive theory.
To be honest, I don’t ...
- 5,504
4
votes
0
answers
255
views
Quotient of a motive by a finite group
Given a smooth scheme $X$ over a field $k$, we can consider its motive $M(X)$. It is an object in Voevodsky's triangulated category of motives $DM(k,\Lambda)$ where $\Lambda$ is the ring of ...
- 2,670
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. ...
- 4,189
12
votes
0
answers
1k
views
Use of derivators to the theory of motives?
This is a rather imprecise question but i think this could become a interesting pool of ideas and comments.
The theory of motives has evolved to a complex field of research the moment Voevodsky (and ...
- 9,650
10
votes
1
answer
477
views
Equivalence of rational Voevodsky motives: partial Converse to Conjecture of Orlov
There is a conjecture of Orlov stating that if $X$ and $Y$ are smooth projective complex varieties that are derived equivalent (equivalent bounded derived categories of coherent sheaves), then their ...
- 39.3k
18
votes
3
answers
1k
views
Is the Hodge Conjecture an $\mathbb{A}^1$-homotopy invariant?
Let $X$ and $Y$ be two nonsingular projective varieties defined over the complex numbers.
If $X$ and $Y$ are $\mathbb{A}^1$-homotopy equivalent, then does $X$ satisfying the Hodge conjecture imply ...
- 16.9k
4
votes
1
answer
451
views
$T$-stable vs. $S^1$-stable motivic homotopy category: which sorts of transfers are available?
I would like to have some sort of transfers in motivic stable homotopy categories in order to adapt Voevodsky's split standard triple argument to cohomology theories that can be factorized through ...
- 326