The motives tag has no usage guidance.

**4**

votes

**1**answer

294 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 $h^1(E)$ ...

**12**

votes

**2**answers

459 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 ...

**2**

votes

**0**answers

632 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), ...

**4**

votes

**1**answer

303 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**

votes

**2**answers

631 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 ...

**8**

votes

**1**answer

228 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 ...

**11**

votes

**0**answers

793 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 ...

**12**

votes

**3**answers

642 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 ...

**3**

votes

**0**answers

388 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 ...

**11**

votes

**1**answer

784 views

### Reasons for the use of Nisnevich topology in motivic homotopy theory

The objects of interest in motivic homotopy theory are "spaces"-which are simplicial sheaves of sets on the big Nisnevich site $Sm/k$ of smooth schemes of finite type over a field $k$. I understand ...

**1**

vote

**1**answer

230 views

### Rost-Motive for n > 2

Have a look on the paper
F. Morel, Voevodsky's proof of Milnor's conjecture, Bull. Amer. Math. Soc. 35 (1998), 123-143, doi:10.1090/S0273-0979-98-00745-9.
and go to example 6.5 please.
In this ...

**1**

vote

**0**answers

115 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 ...

**10**

votes

**0**answers

351 views

### The status of automorphic induction

Background: Let $K/F$ be a degree $r$ extension of number fields. It is conjectured that an automorphic representation of GL$_n$ associated to $K$ induces an automorphic representation of GL$_{rn}$ ...

**4**

votes

**0**answers

164 views

### Finer motivic decomposition in a bigger motivic category

In his Ph.D. thesis, Semenov shows that the motivic decomposition of a variety in general is not unique. He works in the category of Chow-Motives and not in a bigger category of motives.
Is there an ...

**16**

votes

**1**answer

1k 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 ...

**6**

votes

**0**answers

165 views

### Nice references for injective model structures and Quillen functors between motivic homotopy categories

It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...

**8**

votes

**0**answers

195 views

### Stack of Tannakian categories? Galois descent?

I'm having trouble finding a reference for something that I'm guessing the experts worked out long ago. Let's take a local or global field $F$ for this post, and fix a separable algebraic closure ...

**11**

votes

**1**answer

626 views

### Moduli space of motives vs moduli space of varieties

A (projective) abelian variety $A$ over the complex numbers is determined by $H^1(A,\mathbb{Z})$ together with its Hodge structure and polarization. This miracle means that one can parametrise ...

**2**

votes

**1**answer

274 views

### Why is the Category of Correspondences not pseudo abelian?

i've just gotten into the theory of motives.I understand the construction of the Karoubian envelope (pseudo-abelian completion) to ensure that morphisms have kernels and images in order to get certain ...

**4**

votes

**1**answer

273 views

### $T$-stable vs. $S^1$-stable motivic homotopy category: which sorts of transfers are available?

I would like to have some sort of transfers in motivic stable homotopy categories in order to adapt Voevodsky's split standard triple argument to cohomology theories that can be factorized through ...

**5**

votes

**2**answers

502 views

### On triangulated categories of pro-objects

Which term is used for model categories whose homotopy categories are triangulated? Stable proper model categories?
I want $Ho(Pro-M)$ to be triangulated ($Pro-M$ is the category of pro-objects of M) ...

**4**

votes

**0**answers

165 views

### Is there a motive attached to half weight modular forms?

The question is kind of contained in the title but let me add a few words.
If $f$ is a cusp form of weight $k$ for $SL(2, \mathbb{Z})$ then Scholl constructed a Grothendieck motive $M(f)$. In this ...

**24**

votes

**2**answers

1k views

### Most significant results in motivic integration theory?

I am thinking about reading a course on motivic integration. I have already read certain introductions to the subject; yet I am not sure that they mention all the significant parts of the theory. So, ...

**19**

votes

**0**answers

1k views

### Grothendieck's letter to Faltings: On the yoga of motives and the degeneration of Leray spectral sequence

In Grothendieck's letter to Faltings, he writes
There exists at this time a kind of “yoga des motifs”, which is familiar to a handful of initiates, and in some situations provides a ﬁrm support ...

**7**

votes

**1**answer

452 views

### Motives over the complex numbers versus mixed Hodge structures

Let $\mathsf{MM}(\mathbf C)$ be the hypothetical category of mixed motives over the complex numbers, and consider the realization functor $\Phi : \mathsf{MM}( \mathbf C) \to \mathsf{MHS}$ to integral ...

**7**

votes

**1**answer

587 views

### On Deligne's determinant of motives

This is a question about Deligne's conjecture on special values of L-functions. I have to confess that I've never understood the definition of the determinant which is supposed to give the right ...

**7**

votes

**1**answer

343 views

### motive of CM elliptic curve and modular forms

I am trying to get some insight into Deligne/Scholl construction of the motive of a modular form. First of all I would like to understand the case of weight two, especialy when there is complex ...

**1**

vote

**0**answers

188 views

### Virtual Lefschetz motive

Hi there,
I have a question which popped up while reading papers on motives.
Let $V_k$ be the category of (projective) k-varieties, and let $K_0(V_k)$ be the Grothendieck ring of $V_k$; then ...

**3**

votes

**1**answer

509 views

### “Motivic structure on higher homotopy of non-nilpotent spaces” ?

Has anyone an idea where one can read more about Deepam Patel's talk "Motivic structure on higher homotopy of non-nilpotent spaces" http://www.ihes.fr/~abbes/SGA/patel.html ?
Edit/Answer: The video ...

**25**

votes

**2**answers

1k views

### What are the possible motivic Galois groups over $\mathbb Q$?

Let $E$ be a motive over $\mathbb Q$. (I should precise, that by a motive I mean
here a pure motive over $\mathbb Q$, with coefficients in $\mathbb Q$, that I see here as a conjectural object which ...

**9**

votes

**2**answers

952 views

### Status of Beilinson conjectures?

(I hesitate to post that question here, but I received on answer on FB:)
Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks ...

**1**

vote

**0**answers

160 views

### Chow-Künneth decomposition for hypersurfaces

Short version: is the Chow-Künneth motivic decomposition known for $X \hookrightarrow \mathbb{P}^n_k$ a hypersurface over a field $k$?
Long version: let $M(X)$ be the Chow motive of $X$ with ...

**3**

votes

**2**answers

241 views

### critical values of motives

Hi friends,
I have some questions concerning the critical values of motives, in the sense of Deligne. I will only look at motives of the form $h^i(X)$ where $X$ is a smooth projective algebraic ...

**9**

votes

**1**answer

620 views

### Analogue of Tate or Hodge conjecture for varieties over $\mathbb Q_p$

I've been learning about p-adic Hodge theory recently (I'm a beginner), and I've been wondering about the following question the past couple of weeks. Sorry for the long setup, it's mainly ...

**3**

votes

**0**answers

164 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 ...

**0**

votes

**1**answer

191 views

### Pull-back of algebraic cycles under holomorphic maps

Let $f:X \to Y$ be a holomorphic map between two smooth complex projective manifolds. Is there a good notion of pull-back of algebraic cycles by $f$ which preserves degree in the following sense: ...

**0**

votes

**1**answer

270 views

### Realisation functor

Let k be a field. Is there a realization functor
$DM_{gm}(k,\mathbb{Z}/n)^{op} \to D^b_c(k, \mathbb{Z}/n)$
from category of motives to category of complexes of étale sheaves of $\mathbb{Z}/n$ ...

**1**

vote

**1**answer

189 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 $\operatorname{DM}_\_^{eff}(k)$. ...

**5**

votes

**1**answer

575 views

### Should the etale cohomology of a smooth projective variety (over rationals) be semi-simple; why?

$\DeclareMathOperator{\char}{char}\DeclareMathOperator{\gal}{Gal}$
Let $P$ be a smooth projective variety over a field $K$ (one may certainly assume that $K$ is perfect; the case $K=\mathbb{Q}$ ...

**7**

votes

**1**answer

355 views

### Ring structure for the motivic spectrum/complex that represents singular cohomology?

As the discussion here Is singular cohomology representable by a (Voevodsky's) motivic complex?
shows, the singular cohomology of (smooth) complex varieties is represented by a motivic complex ...

**8**

votes

**0**answers

190 views

### Corresponding notion of unramified for motives (or de Rham cohomology)

The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if ...

**1**

vote

**0**answers

170 views

### chow kunneth motivic decomposition for dummies

Hi everybody,
I have just realized that, if I had to explain Chow-Kunneth decomposition for Chow motives to someone who is a newbie of the subject, I would hardly find a way which is far from the ...

**2**

votes

**0**answers

280 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 ...

**3**

votes

**0**answers

335 views

### Good morphisms of distinguished triangles: can Neeman's method be applied to the motivic stable homotopy category?

It is well known that non-uniqueness of a cone for a morphism in a triangulated category $C$ makes constructing exact functors (of triangulated categories) a difficult task. In section 3 of his "Some ...

**8**

votes

**1**answer

355 views

### The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?

I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one.
For the algebraic cobordism theory ...

**12**

votes

**1**answer

451 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 ...

**5**

votes

**1**answer

876 views

### (Mixed) Tate motives

Hi there,
in recent times I was reading texts about motives, and I want to ask
something about Tate motives which is not clear to me (as I came across
different definitions in different texts).
Let ...

**2**

votes

**1**answer

315 views

### Chow group of a (particular) motive [+ reference request]

I have two (not unrelated) questions. Let me first give a short introduction.
Introduction
For a general overview of the setup I refer to the introduction (§1) of [Zhang].
Let $k$ be a number field ...

**15**

votes

**2**answers

694 views

### Motivic generalisation of Neron-Ogg-Shaferevich criterion

Given a variety $X$ over $\mathbb{Q}$ with good reduction at $p$, proper smooth base change tells us that its $l$-adic cohomology groups are unramified at $p$ (and I'd guess some $p$-adic Hodge theory ...

**15**

votes

**1**answer

640 views

### Grothendieck ring of “varieties carrying a function”

Fix a base ring $R$, and consider pairs $(X,f)$ where $X$ is a scheme
of finite type over $R$ and $f:X\to R$ is an $R$-valued algebraic (not
constructible!) function on $X$.
I want to consider a ...