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1answer
136 views

Chow groups of locally trivial affine fibrations

Let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic $0$. A locally trivial $\mathbf{A}^n$-fibration is a morphism $\pi \colon Y \to X$ such that $\pi^{-1}(U)\cong ...
3
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286 views

Do those manifolds atrached to L-functions give rise naturally to motives? [closed]

Edited after Will Sawin's comment: Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product ...
4
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1answer
223 views

Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...
1
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1answer
147 views

motive of the general linear group

Let $k$ be a perfect field. Let $GL_n$ be the general linear group over $k$. Does anybody know a reference for the computation of the motive $$ M(GL_n) $$ in Voevodsky's category $DM(k)$?
3
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2answers
349 views

Should the Grothendieck ring of varieties be K_0 of numerical motives?

Assuming the Standard Conjectures, should the Grothendieck ring of varieties be the $K_0$ of the abelian category of numerical motives?
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1answer
204 views

Is Scholl construction of modular motives related to Deligne's construction of $\ell$-adic representations?

first of all, I need to declare my extreme ignorance on the topic of modular forms, so , please, does not assume that I know Deligne's construction in details. In ...
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0answers
234 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" ...
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2answers
1k views

On Grothendieck's idea on his Standard Conjecture B

Let me recall the Standard Conjecture B (see [1,2] below): The $\Lambda$-operation of Hodge theory is algebraic. It more or less says that the partial inverse to “cupping with the class of a ...
8
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1answer
335 views

Why do we need localization by Leftschetz motive?

Definition of the Grothendieck group and Leftschetz motive. The Grothendieck group of varieties is a free abelian group generated by classes of algebraic varieties with the following relation: $$ ...
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338 views

Why is the CM closure of $\mathbb{Q}$ the “ultimate” coefficient field for motives?

In a rough way, a category of motives over a field $k$ with coefficients in a field $K$ gives a universal cohomology theory with coefficients in $K$ for algebraic varieties defined over $k$. I had the ...
1
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1answer
233 views

Splitting varieties of two Galois cohomology symbols

One characteristic property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$, meaning: For some $\alpha \in H^n(k,\mu_p)$ ...
3
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1answer
258 views

“Weight-mondoromy” for open varieties

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy ...
2
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1answer
325 views

Algebraic equivalence vs linear equivalence

Maybe the question is too general, but nevertheless: under what conditions on algebraic variety $X$, algebraic equivalence of divisors coincide with linear equivalence? What are typical classes of ...
3
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1answer
208 views

Etale Realization and Gysin Sequence

Ivorra defined a tensor triangulated functor from Voevodsky's triangulated category of motives to the derived category of complexes of etale sheaves of $\mathbb{Z}/n$ modules with bounded cohomology ...
10
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2answers
475 views

Are there known cases of the Mumford–Tate conjecture that do not use Abelian varieties?

(For a formulation of the Mumford–Tate conjecture, see below.) The question As far as I know, all non-trivial known cases of the Mumford–Tate conjecture more or less depend on the Mumford–Tate ...
3
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0answers
105 views

Galois group for 0-dimensional motives

$\newcommand{\M}{\mathcal{M}_0}$$\newcommand{\Q}{\mathbb{Q}}$ It is my understanding that in dimension 0, the theory of motives should just be Galois theory for fields. I am hoping to find a reference ...
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0answers
314 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 ...
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2answers
2k views

Mysterious quotes (at least for me)

I heard two quotes, one from Alain Connes and an other one from Orlov. Alain Connes was talking about noncommutative geometry and he said the following: " a noncommutative algebra creates its own ...
10
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3answers
801 views

why are motives more serious than “naive” motives?

I know my question is a bit vague, sorry for this. Let $k$ be a field of characteristic zero. Consider the Grothendieck ring of varieties over $k$, usually denoted by $K_0(Var_k)$. This is generated ...
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0answers
214 views

Any counterexamples known for the Generalized Tate conjecture?

One can state the generalized Tate conjecture over arbitrary finitely generated fields; to this end one should just define Galois representation to be effective if the eignevalues of the actions of ...
1
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0answers
100 views

relations between nori motives and pure motives

The category of Nori motives $NMM$ is defined by tools of graph category. I think one can similarly define a category $EM$ as the graph category of $P(k)$ (the graph of projective smooth k-varieties) ...
2
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1answer
169 views

About embedding pure motives into the triangulated category of mixed motives and some further questions about motivic cohomology

I have tried reading some texts about motives, mainly motivic cohomology (Bloch's " Lectures on Algebraic Cycles", Voevodsky's "Motivic Cohomology" etc). However some things confused me. I don't know ...
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0answers
116 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 ...
1
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0answers
138 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 ...
0
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0answers
47 views

Binary motives in the decomposition of a minimal Pfister neighbor

Let $\alpha \in H^n(k,\mu_2)$ and $X_\alpha$ be the respective Pfister quadric. Its well known due to Rost that the Motive $M(X_\alpha)$ decomposes as a sum of twisted Rost motives $R_\alpha$ such ...
5
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2answers
253 views

Are Anderson $T$-motives motives for the function field analogy?

this question is related to this one Geometry for Anderson's motives?, though the previous one doesn't answer exactly my question. Let $\mathbb{C}_{\infty}$ be the function field analog of ...
2
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1answer
463 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 ...
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1answer
1k views

Is there a higher Grothendieck ring?

Fix a field $k$. The Grothendieck ring $K_0(\mathrm{Var}_k)$ of varieties over $k$ is defined as the quotient of the free abelian group on isomorphism classes of algebraic varieties by the scissor ...
1
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1answer
102 views

Dimension of binary motives of a quadric

Let $Q$ be a anisotropic quadric of dimension $d$ over $k$. We work in the category of effective Chow-Motives over $k$. Let $T$ be the Tate-Motive. For a motive $M$ we write $M(l)$ for its $l$-th ...
4
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0answers
112 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 ...
1
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0answers
92 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)?
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2answers
675 views

Connections between Standard, Hodge and Tate conjectures on algebraic cycles?

What implications would a solution of the Standard Conjectures have on the Hodge and Tate Conjectures and reverse?
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1answer
298 views

Motivic L-function vs motivic zeta function

Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of $$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$ where $Fr_p$ is a ...
6
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1answer
225 views

Is the Tate conjecture known for etale covers of products of curves

Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map ...
2
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0answers
167 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 ...
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0answers
809 views

Has there been any progress on the standard conjectures on algebraic cycles?

What's the current state of these conjectures? Who is working on them? In "Standard conjectures on algebraic cycles" Grothendieck says: "They would form the basis of the so-called "theory of ...
0
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1answer
233 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 ...
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1answer
298 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 ...
16
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0answers
342 views

$\zeta(n)$ as a mixed Tate motive

I am trying to understand why there exists, for each $n \geq 2$, a mixed Tate motive $M$ over $\mathbb{Q}$ such that $M \in Ext^1_{MT(\mathbb{Q})}(\mathbb{Q}(0), \mathbb{Q}(n))$ and $\zeta(n)$, ...
4
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1answer
258 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 $$ ...
7
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2answers
424 views

Equivariant motivic sheaves

Thanks to the work of Cisinski-Deglise: http://arxiv.org/abs/0912.2110, we now have a triangulated category of `motivic sheaves' available that admits the standard yoga of the six functors. Is there ...
9
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1answer
308 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 ...
0
votes
1answer
110 views

Rost Correspondence and minimal Pfister-Neighbors

In http://www.math.uiuc.edu/K-theory/0357/ Karpenko utters the following: Conjecture 1.6. If an anisotropic quadric $X = Q$ possesses a Rost correspondence, then the quadratic form (defining ...
4
votes
1answer
271 views

motives of elliptic curves, modular forms, Hecke characters

Let $E$ be an elliptic curve over $\mathbb{Q}$. By the modularity theorem, $L(E, s)$ is the $L$-function of some modular form $f$. Now one has the following motives: (a) The Chow motive $h^1(E)$ ...
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2answers
448 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 ...
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0answers
615 views

The link between the subfactors and the motives as enriched Galois theories? [closed]

On 29th March 2007, at the "École normale supérieure" of Paris, the mathematician Vaughan Jones, gave a conference (in French) entitled "Les sous-facteurs : une théorie de Galois enrichie" (see here), ...
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0answers
196 views

tensor of motives

Let $X$ and $Y$ be two smooth projective varieties over $k$. Consider the Chow motives $M(X\times\mathbb{P}^n)\simeq M(X)\otimes M(\mathbb{P}^n)$ and $M(Y\times\mathbb{P}^n)\simeq M(Y)\otimes ...
15
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2answers
605 views

$p$-adic periods

For a variety $X$ defined over $\mathbb{Q}$, there's a (functorial) comparison isomorphism $$ H^i_{dR}(X)\otimes\mathbb{C}\to H^i_B(X,\mathbb{Q})\otimes\mathbb{C}. $$ If we pick $\mathbb{Q}$-bases for ...
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1answer
221 views

Cohomology of relative motives

Notation Let $S$ be a scheme, proper over a field $k$. Let $\mathrm{SmPr}_{S}$ denote the category of smooth projective $S$-schemes. Let $\mathcal{M}_{S}$ denote the category of relative Chow motives ...
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737 views

Motivic derived algebraic geometry

In algebraic geometry one studies spaces locally modelled on commutative algebras, i.e. commutative algebra objects in the symmetric monoidal category Ab of abelian groups. Now supposedly in the ...