All Questions
13 questions
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
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
20
votes
3
answers
2k
views
Voevodsky's Triangulated Categories of Motives and their Relationships
As we know, Voevodsky constructed several candidates for the triangulated category of motives using different constructions and topologies (h, qfh, etale, and Nisnevich).
I would like to know what ...
- 203
17
votes
1
answer
1k
views
Why presheaves with transfer?
Presheaves with transfer are a main technical ingredient in Voevodsky's construction of his category of mixed motives. From the perspective of motivic homotopy theory, the only difference between SH ...
- 171
7
votes
1
answer
552
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
1
answer
474
views
Motivic $\mathbf{Z}(1)$
I know that the Bloch higher Chow complex, $\mathbf{Z}(i)_{\mathcal{M}}$, on a smooth scheme over a field $k$, reads, in degree $1$:
$$\mathbf{Z}(1)_{\mathcal{M}}\simeq\mathbf{G}_m[-1].$$
How to see ...
user113393
5
votes
0
answers
513
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
4
votes
1
answer
628
views
why Borel's computation implies Beilinson-Soulé?
Let $k$ be a field of characteristic zero and $DM(k)_{\mathrm{Q}}$ Voevodsky's category of motives over $k$ with rational coefficients. The Beilinson-Soulé conjecture says
$$
\mathrm{Hom}_{DM(k)_{\...
- 41
4
votes
1
answer
237
views
What are the consequences of the finite generation of $\operatorname{Ext}^1_{\mathcal{O}_F}(\mathbb{1},M)$?
Let $F$ be a number fields. Conjecturally, there is a rigid $\mathbb{Q}$-linear abelian category of mixed motives over $F$. Let $\mathbb{1}$ denotes the unit object of this category. Given a mixed ...
- 1,479
4
votes
0
answers
220
views
Passing motivic decompositions from rational to algebraic equivalence
It is well known that there are several adequate equivalence relations for algebraic cycles (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions).
The ...
- 1,479
4
votes
0
answers
306
views
What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?
Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures?
What is the ...
user55909
3
votes
0
answers
206
views
Generalization of conjectures involving Beilinson regulators
I had some questions about the Beilinson conjectures as mentioned in this page. I have to admit I do not know much about Deligne cohomology. The conjectures involve some form of comparison map between ...
- 5,901
0
votes
1
answer
388
views
Articles about Weil cohomology theory by Grothendieck and Artin
In "The Standard Conjectures" Kleiman says that the following properties of Weil cohomology theory were proved in 1963 for étale cohomology by Artin and Grothendieck, except for the last one that it ...
user55909