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Any "motive"(-like) theory which can catch that cusp $y^2=x^3$ (and similar) are non-trivial?

Consider cusp $y^2=x^3$ which can also be described as $k[z]~without~z$ , taking $x=z^3,y=z^2$. Algebraically its $Spec$ is quite different from $k$. For example: it has plenty non-trivial "line-...
Alexander Chervov's user avatar
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 $...
Ola Sande's user avatar
  • 705
4 votes
1 answer
238 views

Functoriality conjectures on the slice filtration

Voevodsky wrote on his paper "Open Problems in the Motivic Stable Homotopy Theory, I" that Three other groups of conjectures in motivic homotopy theory, not included in to this paper, seem ...
Tintin's user avatar
  • 2,871
1 vote
0 answers
213 views

Algebraic correspondence as morphisms in Betti cohomology

$\newcommand{\sing}{\mathrm{sing}}$Take a commutative ring $R$ and smooth projective complex varieties $X$ and $Y$. An element $\alpha\in CH^*(X\times Y)_R$ induces the algebraic correspondence for ...
OOOOOO's user avatar
  • 349
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 ...
Arpith's user avatar
  • 19
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 ...
user127776's user avatar
  • 5,901
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 ...
user127776's user avatar
  • 5,901
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 ...
display llvll's user avatar
3 votes
1 answer
477 views

Arc space & formal loops in motivic integration

One of the most essential ingredients in the theory of motivic integration are the space of arcs of a given $k$-variety $X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points ...
user267839's user avatar
  • 6,028
1 vote
1 answer
245 views

Pseudo-Abelian Completion in the constrution of Motifs (by Y. Manin)

I reading Yuri Manin's famous paper on "CORRESPONDENCES, MOTIFS AND MONOIDAL TRANSFORMATIONS" and struggle with his definition for so called pseudo-abelian completion given on page 453 by a reason I ...
user267839's user avatar
  • 6,028
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 ...
Daniil Rudenko's user avatar
10 votes
2 answers
534 views

A question about the vanishing of motivic cohomology in negative Tate twist

Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$. Is it true that $$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[...
user438991's user avatar
8 votes
0 answers
574 views

Reference request: Motivic Cohomology and Cycle class maps

For a smooth projective variety $X$ over any field $K$, Voevodsky showed in his paper ``Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic" that the motivic ...
Jesse Silliman's user avatar
5 votes
1 answer
426 views

Blowup formula for motivic cohomology

If $X$ is a smooth projective variety over a field, $Z\subset X$ a smooth closed subvariety of codimension $d$, $X'\to X$ the blowup of $X$ along $Z$, there's the blowup formula $$H^j(X'_{et},\mathbf{...
user avatar
7 votes
1 answer
689 views

Two motivic complexes, compared

Bloch defines the motivic complexes $\mathbf{Z}(n)$ in his paper "Algebraic Cycles and Higher K-Theory" (1986). Some references (that I currently am unable to track down) use $$\check{\mathbf{Z}}(n) :...
user avatar
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 ...
user avatar
11 votes
1 answer
967 views

How to think about infinite generatedness of motivic cohomology

In this question I previously asked how to think about the motivic complex $\mathbf{Z}(1)_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^*((-)\wedge S^{2n}...
user avatar
8 votes
1 answer
989 views

How to think about $\mathbf{Z}(n)_{\mathcal{M}}$

One definition of motivic cohomology for smooth schemes $X$ over a field, is via Friedlander-Suslin complexes. A refresher (you may skip to the question at the bottom) One defines (1) $z_n(X,d) :=$...
user avatar
9 votes
1 answer
643 views

Torsion in Deligne cohomology

Let $X$ be a smooth projective complex analytic space, $i,p\ge 0$ integers, $\mathbf{Z}(p)_{\mathcal{D}}$ the Deligne complex of $X$, $H^i_{\mathcal{D}}(X,\mathbf{Z}(p))$ its hypercohomology. What ...
user avatar
5 votes
1 answer
482 views

Around algebraic equivalence of cycles

Let $k$ be a finitely generated field, $X$ a smooth projective $k$-variety, $\ell$ a prime number, $\ell\in k^{\times}$, $r\ge 0$ an integer. The Tate conjecture asserts surjectivity of the cycle ...
user avatar
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 ...
user avatar
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 ...
user avatar
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 ...
user avatar
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\...
user avatar
7 votes
1 answer
710 views

A question on Voevodsky´s categories

I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions: 1.- ...
user438991's user avatar
4 votes
0 answers
244 views

Effectivity and Lower Shriek for Voevodsky Motives

I am in a situation where I need a result of the following form. Suppose $X$ is a smooth $k$-variety, $U$ is a dense open subvariety with complement $Z$ a smooth divisor. Let $\pi^X:X\rightarrow\text{...
user114292's user avatar
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 $X$ ...
user avatar
6 votes
0 answers
400 views

Hodge Realisation of Mixed Tate Motives

For a field $k$ which satisfies Beilinson-Soule vanishing conjecture, the from Levine's paper, https://www.uni-due.de/~bm0032/publ/TateMotives.pdf There exists an abelian category of mixed Tate ...
Wenzhe's user avatar
  • 2,971
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 ...
user111652's user avatar
2 votes
1 answer
367 views

Dualizability and motivic cohomology

Suppose $k$ is an algebraically closed field of characteristic $p$. Let $A=\mathbb{Z}/\ell\mathbb{Z}$, $\ell$ a prime coprime to $p$. Denote by $MA$ the motivic Eilenberg-Maclane spectrum over $k$. Is ...
user106838's user avatar
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 ...
Fabian Carlström's user avatar
12 votes
0 answers
726 views

What is missing in the current constructions of pure and mixed motives?

Yo! Maybe this question is too broad, so maybe it should be community wiki? In summary, I want to known all the known comparisons between all the constructions of pure and mixed motives and what make ...
user40276's user avatar
  • 2,227
7 votes
1 answer
920 views

Intuition behind the definition of finite correspondences

Finite correspondences were introduced by Suslin-Voevodsky (if I am not wrong) to define motivic complexes that compute motivic cohomology. Let $X$ and be smooth separated schemes of finite type over ...
Anandam Banerjee's user avatar
11 votes
1 answer
917 views

Motivic cohomology and pushforward maps

I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field. According to Mazza--Voevodsky--Weibel "...
David Loeffler's user avatar
0 votes
0 answers
288 views

What can one say about zero-cycle groups for products of Chow motives

What can one say about the Chow group of zero-cycles (up to rational equivalence) for a product of smooth projective varieties and Chow motives (so, I am interested in the kernel $Chow_0(P)\otimes ...
Mikhail Bondarko's user avatar
1 vote
0 answers
179 views

Which "concrete" morphisms of varieties and motives induce bijections of their lower Chow groups?

This question is a continuation of Varieties with Chow groups supported in positive codimension: examples and properties? What examples are known of morphisms of varieties and Chow motives (say, over ...
Mikhail Bondarko's user avatar
4 votes
1 answer
912 views

Standard conjectures on positive characteristic

In this MO answer of M. Bondarko, he says: "the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..." and in Remarks on Grothendieck's ...
M. Carmona's user avatar
1 vote
0 answers
176 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)?
user avatar
8 votes
1 answer
467 views

Known norm varieties and the Bloch-Kato conjecture

The Bloch-Kato conjecture states that $K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$. A important part in the proof of the Bloch-Kato conjecture is to ...
nxir's user avatar
  • 1,479
18 votes
4 answers
2k views

Applications of homotopy purity theorem of Morel-Voevodsky

One of the most important theorems in motivic homotopy theory is the homotopy purity theorem of Morel-Voevodsky which says that the motivic Thom space of the normal bundle $\mathcal N_{Z/X}$ of a ...
Carl's user avatar
  • 805
5 votes
0 answers
801 views

Two questions on motivic homotopy theory

I am a beginner in motivic homotopy theory and motivic cohomology and have just begun reading books by Dundas et al. and Mazza-Voevodsky-Weibel. I have two basic questions: Why is the question of $\...
Carl's user avatar
  • 805
6 votes
0 answers
351 views

Correspondences of schemes induced by a finite Galois extension

I am trying to do the first exercises of the Mazza-Voevodsky-Weibel book Lecture Notes on Motivic Cohomology, which are some elementary results about correspondences between schemes. Recall that if $...
Yohann Ségalat's user avatar
1 vote
1 answer
266 views

Definition of Hurewicz map relating $SH(k)$ with $DM_\_^{eff}(k)$

In "Motivic Homotopy Theory" on page 153 it is stated that there exists a canonical Hurewicz map relating the motivic stable homotopy category with the category $\operatorname{DM}_\_^{eff}(k)$. ...
Felix Wellen's user avatar
2 votes
0 answers
326 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 http://books.google.ru/books?id=TzUmk87bN9cC&pg=PA85&...
Mikhail Bondarko's user avatar
14 votes
1 answer
746 views

Can there exist Chow motives/motivic cohomology for compact Kähler manifolds?

Can there exist a 'reasonable' extension of the (higher) Chow groups of complex smooth projective algebraic varieties to functors on the category of compact Kähler manifolds? Are there any ...
Mikhail Bondarko's user avatar
71 votes
1 answer
8k views

What is the relationship between motivic cohomology and the theory of motives?

I will begin by giving a rough sketch of my understanding of motives. In many expositions about motives (for example, http://www.jmilne.org/math/xnotes/MOT102.pdf), the category of motives is defined ...
Makhalan Duff's user avatar
3 votes
0 answers
638 views

Weak Lefschetz for torsion motivic cohomology (or torsion Chow groups)?

I am trying to prove the following Weak-Lefschetz-type statement for motivic cohomology: if $Z$ is a smooth hyperplane section of a smooth projective $X$ of dimension $d$ (say, over the field of ...
Mikhail Bondarko's user avatar
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 ...
Tintin's user avatar
  • 2,871
10 votes
1 answer
743 views

Stable motivic cohomology with finite coefficients?

In this question, which attracted no responces so far, I've asked whether it is possible to extend the Beilinson-Lichtenbaum etale descent rule for motivic cohomology to singular varieties, in ...
Leonid Positselski's user avatar