The motives tag has no usage guidance.

**1**

vote

**0**answers

149 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 ...

**3**

votes

**1**answer

252 views

### Do $Q_l$-etale Euler characteristics of Chow motives coincide for all $l$?

I am interested in (Chow) motives over (algebraically closed) characteristic $p>0$ fields. For $H$ being $\mathbb{Q}_l$-adic cohomology, one can consider $Ch_l(M)=\sum (-1)^i\dim_{\mathbb{Q}_l} ...

**2**

votes

**0**answers

336 views

### A motivic complex

By definition, Voevodsky's motivic complex (an object of his $DM^{eff}_-$) is a complex of sheaves with transfers whose cohomology sheaves are homotopy invariant. Now, I consider the complex (of ...

**3**

votes

**1**answer

268 views

### For a G-variety, what could one say about the motif of the corresponding simplicial variety

Let G be an algerbraic group, and X be a G-variety (that I will assume to be smooth). Then one can consider a simlicial variety whose terms are $G^i\times X$. This simplicial variety yields a 'complex ...

**4**

votes

**1**answer

508 views

### Questions on standard (motivic) conjectures

Over an (algebraically closed) characteristic $p$ field, is it known that the cohomological equivalence of cycles relation (with respect to $\mathbb{Q}_l$-adic \'etale cohomology) does not depend on ...

**2**

votes

**1**answer

414 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 ...

**3**

votes

**1**answer

298 views

### Explain the relation between $K_0$ and morphisms of Chow motives

The Chern class yields an isomorphism $K_0(X)\otimes \mathbb Q\cong \bigoplus_{i\ge 0} Chow^i(X)\otimes \mathbb Q$ (for a smooth variety $X$ over a field?), whereas the latter group is isomorphic to ...

**10**

votes

**1**answer

528 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 ...

**6**

votes

**0**answers

276 views

### Does one need l to be invertible in S in order to consider the l-adic cohomology of S-schemes and motives?

When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens ...

**5**

votes

**1**answer

427 views

### The conjectural relation between mixed motivic sheaves and the perverse t-structure.

As far as I remember, there 'should exist' an exact etale realization functor from the category of mixed motivic sheaves (over a base scheme $S$) to the category of perverse $l$-adic sheaves over $S$. ...

**5**

votes

**2**answers

600 views

### Automorphic form encoding the orders of $N$ modulo $p$.

Let $N$ be a nonzero rational number. For every prime number $p$ with $v_p(N)=0$, let $a_p$ denote the index in $\mathbb Z/p\mathbb Z$ of the subgroup generated by $N$ modulo $p$. So we have $a_p=1$ ...

**1**

vote

**1**answer

335 views

### A composition of a finite morphisms with the transpose correspondence: is it the multiplication by the degree?

Let $X,Y$ be smooth varieties over a field $k$ (which in my case is perfect of finite characteristic $p$; we may also assume that $X,Y$ are connected); $s:Y\to X$ is a finite morphism of degree $d$. ...

**2**

votes

**0**answers

171 views

### A Hodge substructure with ''nice' weight factors that does not correspond to a mixed submotif?

Suppose that we have a mixed motif $M$ with only two (subsequent) weights, and have a subobject for each weight factor. How we can control whether there exists a submotif of $M$ with these two ...

**2**

votes

**0**answers

132 views

### On 'graded polarizable' triangulated categories; are there any mixed Galois module analogues known? Also on mixed realizations

There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is ...

**5**

votes

**0**answers

198 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

**0**answers

459 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 Künneth standard conjecture conjectures that there exist projectors $e_0, e_1, \ldots, ...

**8**

votes

**1**answer

370 views

### Is the injective model structure on symmetric spectra Bousfield localizable?

I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on ...

**6**

votes

**0**answers

375 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 ...

**9**

votes

**1**answer

702 views

### Motivic characterization of affine spaces

Let $k$ be a field and let $X$ be a smooth irreducible variety over $k$.
Suppose that I know that the image of $X$ in the Grothendieck group
of varieties over $k$ is equal to that of
a) ${\mathbb ...

**15**

votes

**2**answers

3k views

### Kontsevich's conjectures on the Grothendieck-Teichmüller group?

Reading Kontsevich's "Operads and Motives in Deformation Quantization", I was wondering about the state of the many conjectures concerning the Grothendieck-Teichmüller group in chapter 4. (Also, where ...

**5**

votes

**0**answers

310 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 ...

**38**

votes

**2**answers

2k views

### Langlands in dimension 2: the Yoshida conjecture

Background:
One prominent part of the Langlands program is the conjecture that
all motives are automorphic.
It is of interest to consider special cases that are more precise, if less
sweeping. ...

**6**

votes

**1**answer

708 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 ...

**2**

votes

**0**answers

158 views

### Transformation rule for motivic integration: divisors or divisors classes?

In the Kontsevich's 'change of variables for motivic integral' rule does one consider any possible choices for the corresponding canonical divisors, or is it necessary to fix certain representatives ...

**3**

votes

**1**answer

372 views

### Morava's “Motives and cell bundles”?

Hello, do you know more about, or some exposition of Morava's talk?

**6**

votes

**2**answers

685 views

### Rankin-Selberg convolutions of motivic L-series

Background:
Let $M_{f_i}, i=1,2$ be two modular motives associated to cusp forms
$f_i \in S_{w_i}(\Gamma_0(N_i))$ of weight $w_i$ and level $N_i$ respectively.
The Rankin-Selberg convolution ...

**9**

votes

**3**answers

2k views

### Are there “motivic” proofs of Weil conjectures in special cases?

This is a question meant as a first step to get into reading more on Weil conjectures and standard conjectures. It is known that the standard conjectures on vanishing of cycles would imply the Weil ...

**22**

votes

**3**answers

1k views

### t-structures and higher categories?

I'm curious to find out where the viewpoint of higher categories may be useful so here is a somewhat vague question (which may or may not have a reasonable answer).
Given a triangulated category, one ...

**16**

votes

**1**answer

3k views

### Deligne's proof of Ramanujan's conjecture

I am trying to understand Deligne's proof of the Ramanujan conjecture and more generally how one associates geometric objects (ultimately, motives) to modular forms.
As the first step, which I ...

**20**

votes

**8**answers

4k views

### What is the proper initiation to the theory of motives for a new student of algebraic geometry?

A preliminary apology is in order: I realize that most of my contributions to this site are in the form of reference requests. I understand that this makes it seem as though I do nothing more than sit ...

**6**

votes

**1**answer

726 views

### Hodge Standard Conjecture in Positive Characteristic

In the Wikipedia article on the Hodge Standard Conjecture it is written (note: it has since been fixed):
In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge ...

**16**

votes

**2**answers

1k views

### Unipotency in realisations of the motivic fundamental group

Deligne, in his 1987 paper on the fundamental group of $\mathbb{P}^1 \setminus \{0,1,\infty\}$ (in "Galois Groups over $\mathbb{Q}$"), defines a system of realisations for a motivic fundamental group. ...

**3**

votes

**0**answers

612 views

### Gauss--Manin connection for de Rham realisation

Let $X$ and $S$ be smooth schemes of finite type over a field $k$ and let $\pi:X\to S$ be a smooth morphism of finite type. The relative de Rham cohomology $H^i_{dR}(X/S)$ of $X$ over $S$ is a ...

**7**

votes

**0**answers

1k views

### What are “fractional motives”?

Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them?
Edit: Further cases of "fractional motives" as ...

**10**

votes

**0**answers

965 views

### Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...

**17**

votes

**3**answers

1k views

### What do you lose when passing to the motive?

I contemplated about what information about a scheme we lose when passing to its motive. I came up with the following examples:
The projective bundle of a vector bundle does only depend on the rank ...

**14**

votes

**4**answers

1k 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, ...

**26**

votes

**4**answers

2k views

### Understanding the definition of the Lefschetz (pure effective) motive

For all those who are unlikely to have answers to my questions, I provide some
Background:
In some sense, pure motives are generalisations of smooth projective varieties. Every Weil cohomology ...

**13**

votes

**2**answers

2k 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 ...

**16**

votes

**1**answer

911 views

### constants in Gamma factors in functional equation for zeta functions.

Usually the Riemann zeta function $\zeta(s)$ gets multiplied by a "gamma factor" to give a function $\xi(s)$ satisfying a functional equation $\xi(s)=\xi(1-s)$. If I changed this gamma factor by a ...

**8**

votes

**2**answers

1k views

### Artin motives, References for.

I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...

**23**

votes

**4**answers

3k views

### difference between equivalence relations on algebraic cycles

For the definitions of the equivalence relations on algebraic cycles see http://en.wikipedia.org/wiki/Adequate_equivalence_relation.
I want to know how far away from each other the equivalence ...

**4**

votes

**1**answer

419 views

### Is the scalar extension functor for Chow motives conservative?

Denote $CHM(F)$ to be the category of Chow motives over a field $F$.
Let's consider an algebraic exension $E/F$, then
there is a natural extension of scalars functor $CHM(F) \to CHM(E)$.
I was ...

**9**

votes

**6**answers

1k views

### Kunneth formula for motivic cohomology

I was wondering when the Kunneth formula holds for motivic cohomology:
$$
H^p(X,A(\alpha)) = \bigoplus_{i+j=p;\beta+\gamma = \alpha} H^j(X,A(\beta)) \otimes H^i(X,A(\gamma))
$$
where ...

**6**

votes

**2**answers

560 views

### Néron theory for motives of arbitrary weight

SGA 7, tome 1, exp. IX, contains in its introduction and in section 13.4 remarks about ideas and conjectures of Deligne on a “théorie de Néron pour motifs de poids quelconque”. Would someone please ...

**70**

votes

**7**answers

7k views

### What is the field with one element?

I've heard of this many times, but I don't know anything about it.
What I do know is that it is supposed to solve the problem of the fact that the final object in the category of schemes is ...

**9**

votes

**3**answers

2k views

### ubiquitous quantum cohomology

Manin stressed that every projective scheme should have a quantum-cohomology structure. I'd like to know more about that. And since the varieties considered in texts about monodromy resp. vanishing ...

**47**

votes

**3**answers

7k views

### What's the “Yoga of Motives”?

There are some things about geometry that show why a motivic viewpoint is deep and important. A good indication is that Grothendieck and others had to invent some important and new ...

**5**

votes

**3**answers

517 views

### Solving “a, b, a+b have given divisors” problem

I've read an interesting article, math.NT/0409456 where you're just trying to solve a simple problem:
For a given (finite) set of primes S find all solutions to an equation ...

**10**

votes

**0**answers

656 views

### Kato's log motives

What are they and what are their intended uses? Does anyone have notes/slides of this talk?
I am curious about "log motives" because there seems to exist a "log motivic yoga" among experts in ...