All Questions
Tagged with motives kt.k-theory-and-homology
19 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
26
votes
1
answer
4k
views
Voevodsky's counterexample to the existence of a motivic t-structure
I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the literature quite hard to follow -"from experts, for experts".
Voevodsky in "...
- 982
18
votes
4
answers
2k
views
Why does one invert $G_m$ in the construction of the motivic stable homotopy category?
Morel and Voevodsky construct the motivic stable homotopy category, a category through which all cohomology theories factor and where they are representable, by starting with a category of schemes, ...
- 12.3k
14
votes
1
answer
1k
views
Motivic cohomology vs. K-theory for singular varieties
As far as I understand, for a smooth variety $X$ its motivic cohomology could be described as the corresponding piece of the $\gamma$-filtration of (Quillen's) $K^*(X)$; this is completely true for $\...
- 16.9k
14
votes
2
answers
1k
views
Is Deligne cohomology the motivic cohomology of analytic spaces?
Let $X$ be a smooth projective complex analytic space.
We can cook up a complex analytic version of Bloch's cycle complex by declaring
$z^n(X^{\rm an}, m)$
is the free abelian group on all ...
user92332
12
votes
3
answers
2k
views
Motivic vs Deligne cohomology
Where can I find the construction of the cycle class map from motivic cohomology to Deligne cohomology of smooth projective varieties over the complex numbers?
It should be a construction by Bloch ...
user95222
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
10
votes
0
answers
340
views
Geometric vs combinatorial motives over Spec Z
Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" ...
- 5,637
9
votes
0
answers
463
views
Beilinson regulators and Bloch's mythological algebraic intermediate Jacobians
In the paper introducing his motivic cycle complexes, Bloch outlines a project he says he was going to return to in the future:
Towards the end of page 270, he says, given a smooth projective variety ...
user95222
6
votes
1
answer
1k
views
When is the K-theory presheaf a sheaf?
Let $F$ be a Deligne-Mumford stack that is of finite type, smooth and proper over $\mathrm{Spec~}k$ for a perfect field $k$. Consider $K_m$, the presheaf of $m$-th $K$-groups on $F_{et}$, the etale ...
- 562
6
votes
0
answers
242
views
Are Tate twists of t-positive motives positive with respect to the Voevodsky's homotopy t-structure?
Let $X$ be a Voevodsky's motif (over a perfect field) that belongs to the positive part of the homotopy $t$-structure (i.e. its cohomology as an object of $D^-(ShSmCor)$ is zero in negative degrees). ...
- 16.9k
5
votes
3
answers
2k
views
Motivation for Karoubi envelope/ idempotent completion
This is the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In the first part we considered ...
- 6,018
4
votes
1
answer
182
views
Explicit description of Verdier quotient of effective motives
Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory ...
- 41
4
votes
1
answer
564
views
Borel regulator and Bloch-Beilinson regulators
Is there any relation, either conjectural or known, between the Borel regulator for the Quillen $K$-theory of algebraic number rings, and the Bloch-Beilinson regulator from motivic cohomology to real ...
user119470
4
votes
0
answers
426
views
In which "sense" unramified Milnor-Witt K-groups are unramified
Let $X$ be an integral locally noetherian smooth
scheme over base field $k$. Then for every $x \in X^{(1)}$ point of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation
ring. ...
- 6,018
4
votes
0
answers
205
views
$\mathbf{A}^1$- contractibility
Suppose $U$ is an $\mathbf{A}^1$-contractible smooth scheme over a field $k$, that is, it is isomorphic to a point in the $\mathbf{A}^1$-homotopy category of smooth schemes over $k$.
Does motivic ...
user119470
4
votes
0
answers
218
views
The 'most general' papers on rational Borel-Moore motivic homology and K'-theory?
There are two ways to define Borel-Moore motivic homology (of schemes) with rational coefficients: one should either consider certain complexes of algebraic cycles, or the $\gamma$-filtrations of ...
- 16.9k
4
votes
0
answers
197
views
The vanishing of homotopy invariant $K$-theory of dg-categories
In my previous question The vanishing of non-connective K-theory in negative degrees
I asked when one can be sure that the negative non-connective $K$-groups of a differential graded category vanish. ...
- 16.9k
1
vote
0
answers
140
views
The right (not the left 'Suslin complex' one) adjoint to the embedding of $ DM^{eff} $ into $D(ShvTr) $?
Inside the derived category of Nisnevich sheaves with transfers there is the category $DM^{eff} $ of Voevodsky's effective motivic complexes (actually, Voevodsky only considered bounded above ...
- 16.9k