All Questions
Tagged with ag.algebraic-geometry motives
359 questions
3
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1
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295
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Motives and birational invariance
One can construct non-isomorphic smooth projective varieties which define the same motive by blowing up $\mathbb{P}^2$ at five points. I think I learned this here at MathOverflow. But these examples ...
- 205
2
votes
0
answers
144
views
Picard group of the category of numerical motives
Is anything known about the Picard group of $Chow_{Num}(k, \mathbb{F}_{p})$ (numerical Chow motives with $\mathbb{F}_{p}-$coefficients)?
Perhaps the Picard groups of some other categories of pure ...
- 87
3
votes
0
answers
389
views
Status of motives in higher category theory: motives and algebraic cycles through a higher categorical perspective
A while ago this interesting question was asked Derived Algebraic Geometry and Chow Rings/Chow Motives.
Primary question:
Have there been any recent developments/advances on the above question? If not,...
3
votes
0
answers
166
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Étale descent of étale motives for algebraic spaces
Let $X$ be a (sufficiently nice) algebraic space, one can define the category of étale motives $\mathbf{DA}(X,R)$ (with $R$ a ring) like the case of schemes (see for instance, La réalisation étale et ...
- 883
2
votes
0
answers
278
views
Why is the weight monodromy hard in mixed characteristics?
I know very little about the conjecture, beyond Grothendieck's monodromy theorem perhaps (a dense open subgroup of inertia acting unipotently on pure motives). But I heard that it was completely ...
- 2,907
2
votes
0
answers
169
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Reference for facts used in Bloch, "Algebraic cycles and L-functions II"
The proof of lemma 1.1 in [1] does not give references for a few statements it uses.
In the setting of the proof, $X$ is a smooth projective variety over a number field $k$ with a fixed embedding to $\...
- 531
8
votes
0
answers
333
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Triple comparison of cohomology in algebraic geometry
Let $X$ be a smooth proper variety over $\mathbb{Q}$ and $p$ a prime number. For an integer $k$, we have:
a finitely-generated abelian group $H^k(X^{\mathrm{an}}(\mathbb{C});\mathbb{Z})$
a finitely-...
- 15.4k
0
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0
answers
123
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Roots of weight of a characteristic polynomial of Frobenius
We are expected to solve a conjecture of the title. Reference is Jean-Pierre Serre — Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques.
Precisely;
Conjecture A:...
- 153
3
votes
1
answer
357
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Elementary questions on motives
Motives are objects that appear in algebraic geometry, which is closely related to algebraic cycles. It was considered by Grothendieck, and developed notably by Deligne, Voevodsky etc.
I have the ...
- 153
3
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0
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109
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Chow-Künneth conjecture and Galois base change
Consider $K'|K$ a finite galois extension of degre $m$ and galois group $G$. Recall the Chow-Künneth conjecture :
Conjecture
For any (smooth projective) variety over a field $k$ and $H$ a Weil ...
3
votes
1
answer
370
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Bloch–Beilinson conjecture for varieties over function fields of positive characteristic
Is there a version of the Bloch–Beilinson conjecture for smooth projective varieties over global fields of positive characteristic? The conjecture I’m referring to is the “recurring fantasy” on page 1 ...
- 531
5
votes
1
answer
468
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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-...
- 24.9k
12
votes
1
answer
1k
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Relation between motives and geometric Langlands
When working over a number field (or a function field over a finite field), one predicts that the Langlands program is related to the theory of motives over this field. There are several ways I have ...
13
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1
answer
2k
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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 $...
- 705
1
vote
0
answers
374
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Amitsur's theorem for generalized Severi–Brauer varieties
Let $k$ be a field of characteristic zero and assume that $A$ is a central simple algebra of index $2^n > 2$. We denote by $\operatorname{SB}_i(A)$ the $i$-th (generalized) Severi–Brauer variety of ...
- 1,479
29
votes
4
answers
3k
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What is the status of the theory of motives?
It has been almost 60 years since Grothendieck conceived the conjectural theory of motives in order to grasp the common behavior of the most important (Weil) cohomology theories.
But what is the ...
- 4,547
2
votes
0
answers
151
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Compatibility of system of $\ell$-adic representations associated to Voevodsky motives
Let $M$ be an object of Voevodsky's category $DM_{gm}(K,\mathbb{Q})$ for a number field $K$. For each prime number $\ell$, there is an $\ell$-adic realization $M_{\ell}$ in the bounded derived ...
- 15.4k
3
votes
0
answers
167
views
Simplicial resolution for commutative group scheme
Let $X$ be a quasi-projective $k$-variety. In this case the symmetric power $S^d(X)$ is well-defined. If $S^\bullet(X)=\bigsqcup_{n>0}S^d(X)$, where we suppose $S^0(X)=\operatorname{spec}(k)$, then ...
- 41
2
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1
answer
221
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Removing quasi-projective assumption in the formalism of four operations
In Ayoub's thesis, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I), Ayoub proved that given a stable homotopical $2$-functor (Definition 1.4.1) $\...
- 883
3
votes
0
answers
186
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Direct images commute with homotopy colimits
In Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Ayoub defined the notion of a stable homotopical algebraic derivators; roughly, for a ...
- 883
2
votes
1
answer
248
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Motives of resolutions of singularities
Suppose $X'$ is a resolution of singularities of a projective variety $X$ over a field $k$ of characteristic 0 that is functorial for smooth morphisms.
How are the (mixed) motives of $X$ related to (...
- 21
3
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0
answers
148
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Grothendieck ring of varieties in positive characteristic, away from the characteristic
In "The universal Euler characteristic for varieties of characteristic zero", Bittner shows that over a field $k$ of characteristic zero, the Grothendieck ring $K_{0}(Var_{k})$ of varieties ...
- 2,197
3
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0
answers
159
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Applications of the theory of derivators to constructing cone functors
One of main reasons that the theory of derivators was introduced is to fix the non-functoriality of the cone construction of triangulated categories. I know that today derivator theory is broad and ...
- 883
8
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0
answers
533
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Has Grothendieck's motivic vision been realised?
Apparently (https://twitter.com/stewartbrand/status/1635057392814821376) Bing's AI search thinks that "the full theory of motives remains elusive". My impression was that the current ...
- 56.9k
6
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0
answers
265
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Rank $2$ motivic local systems on a curve
This question is about the article "Motivic local systems on curves and Maeda's conjecture" by Yeuk Hay Joshua Lam.
In the proof of Theorem 1.1 it is claimed (on lines 4-5 of p. 7) that any ...
- 10.5k
6
votes
0
answers
221
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Motives in tropical geometry
Is there a notion of motives in tropical geometry? Similar like the notion introduced by Grothendieck in algebraic geometry.
- 163
3
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2
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429
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Functor between categories of motives
$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Motives{Motives}$Let us assume for the moment that we have a "nice" category of motives, that is for fields $k$ we have a contravariant ...
- 4,418
4
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0
answers
426
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In which "sense" unramified Milnor-Witt K-groups are unramified
Let $X$ be an integral locally noetherian smooth
scheme over base field $k$. Then for every $x \in X^{(1)}$ point of codimension $1$, the stalk $\mathcal{O}_{X,x}$ is a discrete valuation
ring. ...
- 6,028
12
votes
2
answers
1k
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What exactly do the standard conjectures in characteristic zero refer to?
As the title suggest it seems standard conjectures mean different things depending on the context. I had the impression that in characteristic 0 they are a list of conjectures about varieties over an ...
- 5,901
3
votes
0
answers
175
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Boundedness indices in Voevodsky's smash nilpotence conjecture in family
Let $X$ be a smooth projective variety over an algebraically closed field $k$. Voevodsky introduced the following notion : an algebraic cycle $Z$ in $X$ is smash nilpotent if there exist $N>0$ such ...
- 7,300
7
votes
1
answer
512
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Is there any theory of "étale cohomology" with algebraic coefficients?
For simplicity, I will restrict attention to untwisted coefficients.
Let $k$ be a finite field of characteristic $p$, and $\ell\ne p$ prime. Can one define a cohomology theory with $\mathbb{Q}_\ell^{\...
- 371
4
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0
answers
342
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Voevodsky's motives and Deligne's systems of realizations
$\newcommand{\gm}{\mathrm{gm}}$Let $\mathbf{DM}_{\gm}(\mathbb{Q},\mathbb{Z})$ be Voevodsky's category of geometric motives over $\mathbb{Q}$ with coefficients in $\mathbb{Z}$ (e.g. as on p.124 of ...
- 15.4k
5
votes
1
answer
397
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Which varieties are sums of tensor powers of the Lefschetz motive?
Any smooth projective variety $X$ gives an object $h(X)$ in the category of pure Chow motives. If $X$ is a generalized flag variety, i.e. a quotient $G/P$ where $G$ is semisimple linear algebraic ...
- 22.3k
23
votes
1
answer
1k
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Can we state the Riemann Hypothesis part of the Weil conjectures directly in terms of the count of points?
For algebraic curves we can state the Riemann hypothesis part of the Weil conjectures directly as a formula for the number of points on the curves, sidestepping the zeta function. Namely, given a ...
- 22.3k
4
votes
1
answer
238
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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 ...
- 2,871
3
votes
1
answer
533
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Homotopy invariance of $\ell$-adic cohomology
In the end of the Voevodsky’s lectures on cross functors, P. Deligne considers a couple of axioms which define (using the vocabulary of Ayoub's thesis) a stable homotopical 2-functor. Among them, we ...
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1
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472
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Why is the category of motives generated by varieties?
I'm reading Ayoub's paper Motifs des varietes analytiques rigides, but I'm not quite familiar with motives. In this paper, he defines the category of motives to be $\mathbf{RigDM}^{\rm eff}_{\rm Nis}(...
- 91
3
votes
0
answers
433
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Stable $\infty$-category of motives
In nLab motive, it defines the derived category of motives as the full sub-$\infty$-category of the $\infty$-category of functors $\mathop{\mathrm{Fun}}(\mathrm{Cor}_k^{\mathrm{op}}, \mathcal S)$ ...
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11
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1
answer
1k
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How should I think about 1-motives?
By definition, a 1-motive over an algebraically closed field $k$ is the data
$$
M = [X\stackrel{u}{\to}G]
$$
where $X$ is a free abelian group of finite type, $G$ is a semi-abelian variety over $k$, ...
user474983
4
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0
answers
219
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Generate periods only by smooth varieties
Like explained in this passage that a period is a complex number whose real and imaginary parts are integrations of rational functions over $\mathbb{Q}$ on some $\mathbb{Q}$-semi-algebra set in $\...
- 275
1
vote
0
answers
213
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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 ...
- 349
1
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0
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260
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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 ...
- 19
4
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0
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488
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Deligne's letter to Soulé from 1985
There is a famous letter of Deligne to C. Soulé in which, apparently, Deligne first formulated the conjecture on the existence of an abelian category of mixed motives, extending Grothendieck's pure ...
6
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0
answers
439
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Cohomology theories for algebraic varieties over number fields
There is a standard line which is repeated by anyone writing/talking about motives and cohomology of algebraic varieties over number fields: namely, there are many such cohomologies and then the ...
- 2,751
5
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0
answers
176
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Can a Chow motif be isomorphic to its own direct summand?
Let $M$ be a $R$-linear Chow motif over a field $k$ that is perfect but not necessarily algebraically closed. Can one prove that $M$ is not a direct summand of itself (that is, $M\not\cong M\bigoplus ...
- 16.9k
3
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0
answers
331
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Motives (and examples) of projective bundles over projective spaces
If a projective bundle $Y$ over a variety $X$ is obtained from a vector bundle $E/X$ then the cohomology and the motif of $Y$ is known to be closely related to that of $X$. Now, what can one say in ...
- 16.9k
5
votes
1
answer
300
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A question on motivic zeta-function
It's well-known that over $\mathbb F_q$ every smooth projective conic $C$ is isomorphic to a projective line. So the formula for the motivic zeta-function $Z_{mot}(C)$ is evident since $S^n\mathbb P^1 ...
- 111
3
votes
0
answers
206
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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 ...
- 5,901
1
vote
1
answer
383
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Grothendieck rings and the Tannakian formalism
I understand that the Tannakian formalism (which I only "know" extremely superficially) is very important for the theory of motives. I guess "the" conjectural category of motives ...
- 4,547
8
votes
0
answers
587
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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 ...
- 5,901