Questions tagged [motives]

for questions about motives in algebraic geometry, including constructions of categories of motives and motivic sheaves, and aspects of the standard conjectures.

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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 ...
JustLikeNumberTheory's user avatar
12 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 $...
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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 ...
nxir's user avatar
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26 votes
4 answers
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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 ...
THC's user avatar
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2 votes
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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 ...
David Corwin's user avatar
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3 votes
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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 ...
Sam's user avatar
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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) $\...
Alexey Do's user avatar
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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 ...
Alexey Do's user avatar
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Correspondence between motives and automorphic representations

What I know: I understand motives via its realization; in Coates' and Perrin-Riou's paper On $p$-adic L-functions Attached to Motives over $\mathbb{Q}$ (see http://doi.org/10.2969/aspm/01710023), the ...
Maty Mangoo's user avatar
2 votes
1 answer
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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 (...
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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 ...
Piotr Pstrągowski's user avatar
3 votes
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151 views

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 ...
Alexey Do's user avatar
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Map between Mordell-Weil group and Ext of (Mixed) Motives

We know that the motivic cohomology of an abelian variety $A$ over a number field $k$ computes the Mordell-Weil group up to torsion, and so if we were to grant the existence and nice behaviour of ...
curious math guy's user avatar
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A Galois equivariant Weil cohomology theory with coefficients in the rational numbers and a variation of the Tate/Hodge conjecture

A well-known example of Serre shows that there can be no Weil cohomology theory with $\mathbb Q$ coefficients for schemes over $\mathbb F_{p^2}$. However, this example is no obstruction to a Weil ...
Asvin's user avatar
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8 votes
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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 ...
Neil Strickland's user avatar
6 votes
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221 views

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 ...
naf's user avatar
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4 votes
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What are the modularity conjectures for Artin motives?

Classically, singular cohomology is an important tool for studying topological spaces, in particular, complex varieties. In the mid-twentieth century it was realized that there are many analogues of ...
David Schwein's user avatar
6 votes
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203 views

Motives in tropical geometry

Is there a notion of motives in tropical geometry? Similar like the notion introduced by Grothendieck in algebraic geometry.
Raoul's user avatar
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When is a subspace of the cohomology of a smooth projective scheme on $k$ a motive?

Let $X$ be a smooth projective scheme over a number field $k$, and $V_{p}$ (resp. $V_{\text{dR}}, V_{\text{B}}$) a sub-space of $H_{et,p}^{\ast}(X)$ (resp. $H^{\ast}_{\text{dR}}(X), H^{\ast}_{\text{B} ...
Marsault Chabat's user avatar
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87 views

Manifestation of Hecke operator on the category of abelian varieties (or motives)

If we are given some postivie integer $N$ and a prime $p$, then we have the Hecke operator $T_p$ on modular forms, which is a cohomological manifestation of the Hecke correspondence $$X_0(N)\leftarrow ...
curious math guy's user avatar
3 votes
2 answers
414 views

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 ...
curious math guy's user avatar
4 votes
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406 views

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. ...
user267839's user avatar
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12 votes
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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 ...
user127776's user avatar
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7 votes
0 answers
166 views

Has anyone written about filtered Tannakian categories?

tl;dr Is there any source that discusses the concept of a filtered Tannakian category? I'm writing a paper with this notion and want to know if it's ever been discussed. The original book by Saavedra-...
David Corwin's user avatar
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3 votes
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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 ...
Libli's user avatar
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7 votes
1 answer
465 views

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^{\...
Curious's user avatar
  • 301
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321 views

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 ...
David Corwin's user avatar
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7 votes
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Finite generation of motivic cohomology of number fields

Let $F$ be a number field ($F=\mathbb Q$ is fine for my purposes) and let $n\geq2$ be an integer. Is it known whether the first motivic cohomology groups $$\mathrm H^1(\mathrm{Spec}(F),\mathbb Z(n))$$ ...
Alexander Betts's user avatar
5 votes
1 answer
217 views

Motive associated to a cuspidal representation of $GSp_{4}$

In the paper by L.Clozel in this book (a French text), there is this conjecture (conjecture 4.5 p139) Conjecture: Given $\pi$ an algebraic cuspidal representation of $Gl(n)$ of weight $w$ and denote ...
Marsault Chabat's user avatar
5 votes
0 answers
119 views

Examples of comonoids (coalgebras) in the stable homotopy category $\mathbf{SH}$

My question is both for the topological and for the algebraic/motivic version of the stable homotopy category $\mathbf{SH}$. It is well known that most cohomologies are represented in $\mathbf{SH}$ by ...
Tintin's user avatar
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4 votes
1 answer
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What is motivic sheaf intuitively?

I am not very familiar with motif theory, but I do know a little about Hodge theory. I view (mixed) motif theory as an enhancement of (mixed) Hodge structures. Q1. Is (mixed) motivic sheaf theory an ...
D.Namrebod's user avatar
5 votes
1 answer
356 views

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 ...
John Baez's user avatar
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22 votes
1 answer
1k views

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 ...
John Baez's user avatar
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4 votes
1 answer
217 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
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3 votes
1 answer
479 views

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 ...
Gabriel's user avatar
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1 vote
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Localization with or without transfers

Let $Sh_{Nis}^{tr}$ be the category of Nisnevich sheaves with transfers of abelian groups over a perfect field. Let $u\colon Sh_{Nis}^{tr}\to Sh_{Nis}$ be the functor “forget transfers” and let $h_0^{\...
user197402's user avatar
9 votes
1 answer
440 views

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}(...
Chen Zekun's user avatar
3 votes
0 answers
338 views

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 $\mathop{\mathrm{Fun}}(\mathrm{Cor}_k^{\mathrm{op}}, \mathcal S)$ ...
Aoi Koshigaya's user avatar
6 votes
1 answer
333 views

Nisnevich topology inspired by Adeles

I'm quite a newbe in the field of motives & A1 homotopy theory, so please forgive me if the question is too elementary: In the intro from wikipedia on Nisnevish topology is remarked that it's ...
user267839's user avatar
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11 votes
1 answer
944 views

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 $k$, ...
user avatar
4 votes
0 answers
224 views

Motives based on Hodge cycles vs algebraic cycles

I am not a specialist of motives. I am afraid my questions are rather naive. We have the category of (pure) motives based on Hodge cycles by Deligne. In his articles with Milne, morphisms between ...
Takahiro Matsuda's user avatar
4 votes
0 answers
218 views

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 $\...
CO2's user avatar
  • 235
1 vote
0 answers
201 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
  • 357
1 vote
0 answers
120 views

Motivic homotopy categories closed under subobjects and quotients

It is well known that the category $\mathbf{HI}_{\rm Nis}^{\rm tr}(k)$ of $\mathbb{A}^1$-local Nisnevich sheaves with transfers is closed under subobjects and quotients, from the highly nontrivial ...
user197402's user avatar
1 vote
0 answers
244 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
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4 votes
0 answers
459 views

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 ...
DeligneLetterHunter's user avatar
6 votes
0 answers
430 views

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 ...
Dr. Evil's user avatar
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5 votes
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175 views

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 ...
Mikhail Bondarko's user avatar
2 votes
0 answers
239 views

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 ...
Mikhail Bondarko's user avatar
5 votes
1 answer
281 views

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 ...
John S.'s user avatar
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