Questions tagged [motivic-homotopy]
The motivic-homotopy tag has no usage guidance.
129
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Why is Voevodsky's motivic homotopy theory 'the right' approach?
Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of ...
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Why is the motivic category defined over the site of smooth schemes only?
Fix a base scheme $S$. Stable and unstable motivic categories over $S$ are defined as certain categories of higher stacks on the Nisnevich site $Sm_S$ of smooth schemes over $S$. Why smooth?
As a ...
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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 ...
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Is there a geometric realization of $\mathbf{C}((t))$-varieties?
Let $MV_F$ be the $\mathbf{A}^1$-homotopy category over the field $F$. Let $H$ be the homotopy category of spaces, and let $H_{/S^1}$ be the homotopy category of spaces over the circle.
When $F = \...
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Voevodsky's counterexample to the existence of a motivic t-structure
I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the literature quite hard to follow -"from experts, for experts".
Voevodsky in "...
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What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences?
I come today to mathoverflow to showcase some genuine confusion about the motivic world. I want to ask some questions before actually starting to study the subject, to build some sense of direction.
I ...
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What is the status of the Friedlander-Milnor conjecture today?
For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense:
Conjecture ...
21
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Is algebraic $K$-theory a motivic spectrum?
I've received conflicting messages on this point -- on the one hand, I've been told that "forming a natural home for algebraic $K$-theory" was one motivation for the development of motivic homotopy ...
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Voevodsky's Triangulated Categories of Motives and their Relationships
As we know, Voevodsky constructed several candidates for the triangulated category of motives using different constructions and topologies (h, qfh, etale, and Nisnevich).
I would like to know what ...
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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 ...
16
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Representability of Weil Cohomology Theories in Stable Motivic Homotopy Theory
My understanding is that one purpose of stable motivic homotopy theory is to emulate classical stable homotopy theory. In particular, we would like Weil cohomology theories to be representable by ...
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Relation between motivic homotopy category and the derived category of motives
What's the relation between the pointed motivic homotopy category $\mathcal{H}_*(k)$ and the derived category of motives $\mathbf{DM}^-_{eff}(k)$ besides the representability of motivic cohomology in ...
15
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What are the advantages of various "models" for the motivic stable homotopy category
People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask ...
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What is the best reference for motives?
I want to learn about homotopy theory on number fields, and I heard that the theory of motives made it possible, so I want to know what is a good textbook for motive theory.
To be honest, I don’t ...
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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 $...
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Simple question: different definitions of Bousfield localization
I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them.
First definition: Let $\mathbf{...
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What's wrong with the obvious argument that the unstable motivic category is an $\infty$-topos?
Fix a base scheme $S$, and let $Sm_S$ be the (ordinary) category of smooth schemes over $S$.
Denote by $Psh(Sm_S)$ the $\infty$-category of $\infty$-presheaves on $Sm_S$.
Let $Sh_{Nis}(Sm_S)\...
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What is missing in the current constructions of pure and mixed motives?
Yo! Maybe this question is too broad, so maybe it should be community wiki? In summary, I want to known all the known comparisons between all the constructions of pure and mixed motives and what make ...
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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 $\...
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Which motivic spectra are dualizable?
Let $S$ be a scheme, and $SH(S)$ the stable motivic category over $S$. Which objects of $SH(S)$ are dualizable with respect to the smash product?
All I can find on this question is an old abstract of ...
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Reference for Manin's idea on algebraic geometry over the symmetric monoidal model category of Motives
Reference for Y. Manin's idea of "algebraic geometry over the symmetric monoidal model category of motives."
Has been sugested to me that this was made in a Manin's letter. There is an escaned copy?
...
11
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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 $...
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Does Merkurjev's argument help Voevodsky's program?
In the talk
Unimath - its present and its future, July 10, 2017. Video and slides of a talk, Isaac Newton Institute for Mathematical Sciences, Cambridge. (abstract)
Voevodsky mentioned that he was ...
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A question about the vanishing of motivic cohomology in negative Tate twist
Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.
Is it true that
$$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[...
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When is the Thom spectrum of a virtual vector bundle effective?
Remark: My question is valid in the classic setting of the stable homotopy category of spectra of CW-complexes. An answer on that setting will also be valid.
Denote as $SH(X)$ Voevodsky's stable ...
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Category of motivic spectra
When the survey Axiomatic Stable Homotopy, Neil Strickland, 2004 was written the category of motivic spectra was not investigated from the point of view of axiomatic stable homotopy, as considered e.g....
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Realization Functor From $SH$ to Derived Category of $Gal$-Modules
Let $k$ be a field. I would like a reference for realization functors from Morel-Voevodksy's stable category $SH(k)$ to the derived categories of $Gal(\bar{k}/k)$-modules. Has something like this been ...
9
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What is the etale sheafification of the (unramified) Milnor-Witt $K$-theory
I would like a reference/argument for the truth/falsity of the following statement:
The etale sheafification of the unramified Milnor-Witt K-theory (Nisnevich) sheaves are the (etale sheafification ...
9
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Is the $\mathbb A^1$ homotopy type of a punctured curve independent of the choice of puncture?
Let $C$ be a smooth connected algebraic curve over $\mathbb C$. Assume that $C$ admits no automorphisms. Let $p_1, p_2 \in C$ be two distinct points.
Question: Are the $\mathbb A^1$ homotopy ...
9
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Motivic cohomology is universal with respect to what (co)homology theories?
I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem?
...
9
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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 ...
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The relation between the motivic Galois group and the motivic Steenrod algebra
There is a point of view on the Steenrod algebra that goes something like the following: the functor $-\otimes H\mathbb{F}_p\colon Mod_{\mathbb{S}}\to Mod_{H\mathbb{F}_p}$ corresponds to pulling back ...
9
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Motivic homotopy theory and Noether problem
Let $G$ be a finite group, and let $V$ be a faithful representation of $G$. The Noether problem asks whether $V/G$ is rational (stably rational, retract rational) or not.
To construct ...
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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, ...
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Inverting objects in a symmetric monoidal category
In Voevodsky’s ICM address:
https://www.uio.no/studier/emner/matnat/math/MAT9580/v18/documents/voevodsky-a1-homotopy-theory-icm-1998.pdf
In theorem 4.3 it is claimed that given a symmetric monoidal ...
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Are there non trivial maps from $H\mathbb{Z}$ to $MGL$?
Let $k$ be a field of characteristic $0$. Let us denote by $\mathbf{1}_{k}$ the sphere spectrum. Let $MGL$ be the algebraic cobordism spectrum.
We have the following diagram
$$H\mathbb{Z}\...
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$C_2$-equivariant Betti realization of MGL
Let $MGL$ denote the motivic spectrum representing algebraic cobordism. Over $\mathop{Spec}(\mathbb{C})$ there is a Betti realization functor $\mathop{SH}(\mathbb{R}) \to \mathop{SH}$, which takes $...
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Unaugmentable cosimplicial simplicial sheaves and realization functor
I'm studying the construction of the $\mathrm{Sing}$ functor in Morel-Voevodsky ``$\mathbb{A}^1$-homotopy theory of schemes'' and I was trying to understand the properties of its left adjoint, the ...
7
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2
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Model category structure on spectra
I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.
Let $S$ be a finite dimensional Noetherian scheme ...
7
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1
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Etale and Algebraic K-theory with rational coefficients
We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\...
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Is $MGL$ an $H\mathbb{Z}$-algebra?
Let $\mathrm{MGL}$ be the $\mathbb{P}^1$-ring spectrum over a field $k$ representing algebraic cobordism. Suppose, for simplicity, that $k$ is of characteristic 0. Let $H\mathbb{Z}$ be the motivic ...
7
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Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)
The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper?
Since weak ...
7
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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 (...
7
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Motivic cohomology of $n$-sphere
All motivic cohomology groups are taken with $\mu_2$ coefficient and $k$ has characteristic different from $2$.
Consider the affine variety $X$ with coordinate ring $k[x_1,\ldots,x_n]/(x_1^2+\ldots + ...
7
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$\Bbb A^1$-Localisation of Schemes, and $\Bbb A^1$-Rigid Schemes
Question 1: Are there some publications or preprints that provide $\Bbb A^1$-fibrant replacements of certain classes of (smooth) schemes?
Of course, smooth schemes that are $\Bbb A^1$-fibrant are $\...
7
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Understanding homotopy t-structure
The following question came up while reading Hoyois'
From algebraic cobordism to motivic cohomology.
Let $S$ be a Noetherian scheme of finite Krull dimension and let $SH(S)$ denote the homotopy ...
7
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Motivic homotopy spectral sequence
I would like to have a question about the re-index convention. Let us consider a spectrum $E$ (I am mainly interested in motivic setting, however let's consider the simplicial case firstly, i.e. $E$ ...
6
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Could a motivic spectrum have a "zeta function"?
I'm currently learning about zeta functions, so I apologize in advance if this is riddled with nonsense. Suppose you have a sequence $E=(E_0,E_1,...)$ of motivic spaces along with structure maps $s_i:\...
6
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Smash product of spheres in $\mathbf{SH}$ and product in cohomology
I have two very concrete and simple question. Just in case I write downwards what led me into this.
My questions: Let $\mathbf{SH}(X)$ be the stable homotopy category of Voevodsky. Denote $S^n$ the ...
6
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1
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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 ...