1
vote
0answers
89 views

Differences and relationships between Motivic cohomology and Universal cohomology theory? [duplicate]

Differences and relationships between Motivic cohomology (Beilinson, Lichtenbaum and Voevodsky) and Universal cohomology theory (Grothendieck)?
9
votes
1answer
292 views

Is there a yoga of effectivity for motives and their realizations?

Inside the 'usual' categories of motives (Chow, Voevodsky ones) there are the categories of effective motives. Similarly, there are effective Hodge structures; they seem to be closely related with ...
16
votes
1answer
762 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 ...
2
votes
0answers
273 views

Voevodsky's 'split standard triple' argument: an explanation; does it work with $Z/nZ$-coefficients?

For varieties over a perfect field (of some characteristic $p$ that could be 0) Voevodsky defines the notion of a 'standard triple' (see ...
1
vote
1answer
660 views

Is there a connection between the theory of motives and homotopy theory?

I have read that motives were designed to be the common part of the many homology theories, a way of unifying them. But as I understand it: homotopy is closely related to homology, there is only 1 ...
9
votes
1answer
648 views

Are two conjectural descriptions of the motivic t-structure (via cohomology and via affine varieties) known to be equivalent?

I know of two conjectural descriptions of the motivic $t$-structure for Voevodsky's motives. The first one is based on the conjecture that Weil cohomology theories should yield exact and ...
1
vote
0answers
141 views

$G_m$-cohomology of a motif (that corresponds to a stack?)

As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety I am in the following situation: $G$ is an algerbraic group, and X is a smooth ...
5
votes
0answers
195 views

Which pure categories related with Weil cohomology theories are semi-simple?

As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...
7
votes
0answers
397 views

Integral decomposition of the diagonal (Chow motives)

Let $k$ be a field of characteristic zero and let $X$ be a smooth proper varity over $k$ of dimension $d$. The K√ľnneth standard conjecture conjectures that there exist projectors $e_0, e_1, \ldots, ...
6
votes
0answers
328 views

Is singular cohomology representable by a (Voevodsky's) motivic complex?

For any $c>0$ does there exist an object $C$ of Voevodsky's $DM^{eff}_-$ (over the field of complex numbers) such that for any $i\le c$ and any smooth variety $X$ the $i$-th singular cohomology of ...
5
votes
0answers
293 views

Generalized Hodge conjecture for cohomology of smooth non-proper varieties?

Is there such a statement known i.e. does there exist a conjectural description of the coniveau filtration for singular cohomology of a smooth non-proper variety over the field of complex numbers (in ...
12
votes
2answers
1k views

Why does the Gamma function satisfy a functional equation?

In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...