All Questions
37 questions
3
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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,...
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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
3
votes
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 ...
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3
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0
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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 $\mathop{\mathrm{Fun}}(\mathrm{Cor}_k^{\mathrm{op}}, \mathcal S)$ ...
- 988
4
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0
answers
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 ...
- 12.9k
31
votes
2
answers
3k
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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 ...
CommunityBot
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9
votes
0
answers
276
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Which field extensions do not affect Chow groups?
Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?
...
- 16.9k
3
votes
1
answer
174
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Are "strongly finite dimensional" homotopy invariant sheaves with transfers (locally) constant?
Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\...
- 12.9k
1
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0
answers
304
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Does the pure motive determine the Voevodsky motive?
I do not quite understand the construction of Voevodsky motives yet. Let $k$ be a field (possibly not algebraically closed), $X$ be a connected smooth projective $k$-scheme. Does the motive of $X$ in ...
CommunityBot
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20
votes
1
answer
902
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Is there any publication of Bombieri about the standard conjectures on algebraic cycles?
In "Standard conjectures of algebraic cycles" Grothendieck says:
"... These [Standard conjectures] are not really new, and they were worked out about three years ago independently by Bombieri and ...
- 188k
12
votes
1
answer
608
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Reference - motives of curves
There is a really interesting comment in this question that I was unable to find a reference...
Under the "Tate conjectures, then every motive belongs to the tensor category generated by motives of ...
- 13.6k
2
votes
1
answer
326
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Lefschetz standard conjecture under specialization/generization
Let $S$ be a smooth connected noetherian scheme (not necessarily over a field) with residue fields that are all of finite type over their prime field.
Let $f: \mathcal{X}\to S$ be a smooth projective ...
CommunityBot
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3
votes
1
answer
568
views
Absolute Hodge cycles
Let $X$ is a smooth projective variety defined over a finite extension $K/\mathbf{Q}$, $\sigma : K\to\mathbf{C}$ any of the finitely many field embeddings of $K$ into the complex numbers, and call $X^{...
- 35.2k
8
votes
1
answer
414
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Sha finiteness vs $\ell$-primary torsion
Where do I find a proof of the fact that over global function fields of characteristic $p>0$, finiteness of the Tate-Shafarevich group of an abelian variety is equivalent to finiteness of its $\ell$...
CommunityBot
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4
votes
1
answer
564
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Borel regulator and Bloch-Beilinson regulators
Is there any relation, either conjectural or known, between the Borel regulator for the Quillen $K$-theory of algebraic number rings, and the Bloch-Beilinson regulator from motivic cohomology to real ...
- 17.4k
12
votes
3
answers
1k
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Chow Groups of varieties over number fields
I believe that there is a conjecture that for any smooth projective variety $X$ over a number field $K$, its Chow groups $CH^i(X)$ (or at least $CH^i(X)\otimes_{\mathbf Z} \mathbf Q$) are finitely ...
- 37k
7
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1
answer
710
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A question on Voevodsky´s categories
I want to try to understand the Voevodsky´s big triangulated categories of motives $DM$ and $DM^{eff}$. Unfortunately, I am being not able to find answers to the following, too vague, questions:
1.- ...
CommunityBot
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11
votes
3
answers
2k
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Reference for Nori motives
I would like to study Nori motives and I am a complete outsider of the subject. I do, however, have background on Chow motives, Voevodsky motives $\mathrm{DM}$ and his stable homotopy category $\...
- 13.6k
3
votes
0
answers
154
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Connecting Quillen functors between motivic homotopy categories (of different "types"): references?
For a perfect base field $k$ there exists the following collection of "motivic homotopy" categories related to it:
(a) the homotopy category of simplicial presheaves (from smooth $k$-varieties); here ...
- 16.9k
6
votes
4
answers
2k
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References - Voevodsky motives are the derived category of Nori motives?
First I would like to know if this has been worked out, and if the answer is affirmative I would like to know some references.
CommunityBot
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2
votes
1
answer
334
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Reference - Generalized Hodge conjecture for triangulated motives
GHC for triangulated motives: The Hodge conjecture holds and an object $\rm M \in Dmg$ is effective if and only if its Hodge realization is effective.
I would like to know some references on GHC ...
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8
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1
answer
567
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What are Motivic homotopy types?
There are suggestions that says that Grothendieck developed (in some sense) a theory of Motivic homotopy types or at least named it.
I would like to know the reference in which Grothendieck did it, ...
- 1,720
10
votes
1
answer
947
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Derived version of equivalence between motives and representations of Motivic galois groups?
A slight variant $\tilde Mot_{num}(k,\mathbb{Q})$ of the category of pure motives $Mot_{num}(k,\mathbb{Q})$ is a Tannakian category equivalent to a category of representations of some algebraic group $...
- 1,720
11
votes
2
answers
1k
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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 $...
- 16.9k
1
vote
0
answers
116
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On "splitting off small weights" from Chow motives
I am interested in certain properties Chow motives that seem to be (more or less) "classical" (so, I mostly need references; the base field may be assumed to be algebraically closed).
So, consider ...
- 16.9k
4
votes
1
answer
438
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About the decomposition of a Chow group of a variety
I would like that someone helps me to find an article on the net treating the following decomposition : $ \mathrm{CH}^k (X)_{ \mathbb{Q} } = \displaystyle\bigoplus_{ i + j = k } \mathrm{CH}^{i,j} (X) $...
- 40.2k
7
votes
0
answers
313
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Any counterexamples known for the Generalized Tate conjecture?
One can state the generalized Tate conjecture over arbitrary finitely generated fields; to this end one should just define Galois representation to be effective if the eignevalues of the actions of ...
- 16.9k
1
vote
0
answers
179
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Which "concrete" morphisms of varieties and motives induce bijections of their lower Chow groups?
This question is a continuation of Varieties with Chow groups supported in positive codimension: examples and properties?
What examples are known of morphisms of varieties and Chow motives (say, over ...
CommunityBot
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0
votes
1
answer
388
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Articles about Weil cohomology theory by Grothendieck and Artin
In "The Standard Conjectures" Kleiman says that the following properties of Weil cohomology theory were proved in 1963 for étale cohomology by Artin and Grothendieck, except for the last one that it ...
CommunityBot
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4
votes
1
answer
451
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$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 ...
- 326
6
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0
answers
230
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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 ...
- 16.9k
3
votes
1
answer
619
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"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 ...
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14
votes
1
answer
746
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 ...
- 16.9k
2
votes
1
answer
555
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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 ...
- 40.2k
3
votes
1
answer
284
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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 ...
- 16.9k
3
votes
1
answer
767
views
Is there a 'classical' definition for the support of a perverse sheaves.
I would like to define the support of a mixed motivic sheaf. This should be something similar to the support of a perverse sheaf.:) Is there any 'classical' definition for the latter?
I suspect that ...
- 16.9k
40
votes
8
answers
12k
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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 ...
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