Questions tagged [motivic-homotopy]

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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)\...
Tim Campion's user avatar
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12 votes
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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 ...
user40276's user avatar
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11 votes
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614 views

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 ...
David Roberts's user avatar
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9 votes
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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 ...
Jonathan Beardsley's user avatar
9 votes
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187 views

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 ...
Arcilan's user avatar
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8 votes
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335 views

$C_2$-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 $...
Drew Heard's user avatar
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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 ...
Stefano Nicotra's user avatar
7 votes
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222 views

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 + ...
user2902293's user avatar
7 votes
0 answers
445 views

$\Bbb A^1$-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 $\...
user24453's user avatar
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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 ...
Anandam Banerjee's user avatar
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538 views

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. $E$ ...
Thi's user avatar
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Bigraded endomorphisms of the motivic sphere over a field

In An introduction to $\mathbb A^1$-homotopy theory ([1]) and On the motivic $\pi_0$ of the sphere spectrum ([2]) Morel describes a computation of $\bigoplus_{n\in \mathbb Z} [S^0, \mathbb G_m^{\wedge ...
Maxime Ramzi's user avatar
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6 votes
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Topological realization of fiber sequences of motivic spaces

Consider the realization functor $t^{\mathbb{C}}:H^{mot}\to H^{top}$ from the unstable motivic homotopy category with base field $\mathbb{C}$ to the homotopy category of topological spaces. In this ...
Xing Gu's user avatar
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Is there an exponential map in $\mathbb A^1$ homotopy theory?

Let $k$ be a field, and let $Z \subset X$ be a smooth subscheme of a smooth scheme $X$. When $k = \mathbb C$, there is a distinguished isotopy class of (topological) open embeddings $N_Z \to X$. In ...
Phil Tosteson's user avatar
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230 views

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 ...
Mikhail Bondarko's user avatar
5 votes
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120 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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Is this etale motivic or motivic cohomology?

I am trying to reconcile my understanding of motivic cohomology (based on the Lecture Notes by Mazza-Voevodsky-Weibel) with the homotopic point of view. I am currently struggling to answer this ...
J.P. Gimori's user avatar
5 votes
0 answers
260 views

Stalk of motivic homotopy sheaves

In contrast to "classical" homotopy theory, in the motivic homotopy theory, we don't have homotopy group but rather homotopy sheaves in the Nisnevich topology, which is associated to the ...
curious math guy's user avatar
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176 views

What is the fibrant replacement of an Eilenberg-MacLane space in unstable motivic homotopy theory?

One can take an Eilenberg-MacLane space $K(A, n)$ for some abelian group $A$, and view it as (locally) constant simplicial sheaf on $Sch/k$, the category of schemes, or smooth schemes, over a field $k$...
user155861's user avatar
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194 views

K(1)-localization and homotopy fixed points in motivic homotopy theory

It is known in classical stable homotopy theory that there is an equivalence $$ L_{K(1)}S\simeq KU^{h\mathbb{Z}_p^\times} $$ which gives an especially convenient way to compute the K(1)-local sphere....
CWcx's user avatar
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167 views

Fulton's pullback vs. Pullback via Gersten complexes vs. Pullback coming from motivic homotopy SH(k)

Let $f: X \rightarrow Y$ be a morphism of smooth projective $k$-schemes (let's assume $f$ flat or even smooth). There is pullback in Fulton's style $f^*_{Ful}: CH^p(Y) \rightarrow CH^p(X)$ given by $[...
Thi's user avatar
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781 views

Two questions on motivic homotopy theory

I am a beginner in motivic homotopy theory and motivic cohomology and have just begun reading books by Dundas et al. and Mazza-Voevodsky-Weibel. I have two basic questions: Why is the question of $\...
Carl's user avatar
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154 views

Every stable homotopical functor factors through $\mathbf{SH}$

In this nlab page, it says that the fact that every stable homotopical functor factors through $\mathbf{SH}$ (the motivic stable homotopy category of Morel-Voevodsky) is proven in Ayoub's thesis. ...
Alexey Do's user avatar
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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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4 votes
0 answers
308 views

Universal six-functor formalism on an $\infty$-category

In the article The Universal Six-Functor Formalism by Brad Drew and Martin Gallauer it is proved that for an ordinary category $S$ with a wide subcategory $P$ of 'smooth morphisms' containing all ...
Bastiaan Cnossen's user avatar
4 votes
0 answers
109 views

Filtrations of motivic spectral sequences

I had a general question about motivic spectral sequences. In order to derive them we first begin with a filtration of the algebraic $K$-theory spectra. Something like this $\cdots \rightarrow W^2(X)\...
user127776's user avatar
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Direct image and infinite suspension

I have a basic doubt regarding infinite suspension functor and the direct image. I write it for schemes but I guess it works the same for the topological setting so I welcome answers also from the ...
Tintin's user avatar
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Unstable and stable looping and delooping

I have some basic questions on the relation of looping and delooping in the stable and unstable homotopy categories. I state them it in the motivic setting, but if somebody has an answer for an ...
Tintin's user avatar
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4 votes
0 answers
732 views

Access to a classic reference of Dold-Puppe

There is an old reference that I am unable tofind. It is Dold-Puppe´s communication at a conference. More concretely, it is cited as: A. Dold, D. Puppe: Duality, trace and transfer. Proceedings of the ...
Tintin's user avatar
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4 votes
0 answers
99 views

Defining structure maps of spectra by lifting from the homotopy category

Voevodsky's original definition of the algebraic $K$-theory spectrum, $KGL$, was given as follows: The component spaces were fibrant replacements of the infinite Grassmannian $BGL$. The structure ...
J Doe's user avatar
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0 answers
506 views

Why do motivic stacks make sense?

In the paper "Motivic model categories and motivic derived algebraic geometry", Yuki Kato, whose email-address I sadly couldn't find out, describes a procedure to "motivy" the objects of any $(\infty,...
Alexander Praehauser's user avatar
4 votes
0 answers
244 views

Effectivity and Lower Shriek for Voevodsky Motives

I am in a situation where I need a result of the following form. Suppose $X$ is a smooth $k$-variety, $U$ is a dense open subvariety with complement $Z$ a smooth divisor. Let $\pi^X:X\rightarrow\text{...
user114292's user avatar
4 votes
0 answers
171 views

Which models are available for the motivic homotopy category $SH^{S^1}(k)$

The motivic $S^1$-stable homotopy category $SH^{S^1}(k)$ (where $k$ is a field that is often assumed to be perfect; yet one can probably take quite general base schemes here) is "intermediate" ...
Mikhail Bondarko's user avatar
4 votes
0 answers
300 views

On "topological" Hopf map eta and its relation to the motivic one

Morel has defined the motivic Hopf map $\eta$ (in the motivic stable homotopy category $SH(k)$). I suspect that the following facts are valid for it and its topological "cousin"; please correct me if ...
Mikhail Bondarko's user avatar
4 votes
0 answers
227 views

Motivic Interpretation of Rationally Trivial Cycles

The Chow groups are defined by taking groups of cycles modulo rationally trivial cycles. One then has a cycle class map to etale cohomology (over the base field), and for a number field, one expects ...
David Corwin's user avatar
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3 votes
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140 views

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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3 votes
0 answers
273 views

What are some concrete applications of Grothendieck's six operations?

In Gallauer's An introduction to six-functor formalisms I read: Indeed, the language and theory of six-functor formalisms permeates much of modern algebraic geometry and beyond, and has spawned ...
user493267's user avatar
3 votes
0 answers
342 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
3 votes
0 answers
214 views

Loop spaces of motivic Eilenberg-Mac Lane spaces

Consider the unstable $\mathbb{A}^1$-homotopy category (say over $\mathbb{C}$). By the loop space $\Omega X$ of an object $X$, we mean the homotopy fiber of $pt\to X$. For an abelian group A and the ...
Xing Gu's user avatar
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3 votes
0 answers
246 views

Descent and Chow groups

One of the features of the $\mathbb{A}^1$-homotopy theory is the existence of the motivic Eilenberg-MacLane space $K(\mathbb{Z}(n),2n)$ such that for $k$-schemes $X$, we have $$[X,K(\mathbb{Z}(n),2n)]\...
curious math guy's user avatar
3 votes
0 answers
124 views

Preserves naively $\mathbb{A}^{1}$-homotopic maps

I've been studying $\mathbb{A}^{1}$-homotopy recently and would like some guidance with the question below. Thank you so much. Setup Fix $k$ a field of characteristic zero. Let $Sm_{k}$ denote the ...
Faye3's user avatar
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3 votes
0 answers
152 views

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 ...
Mikhail Bondarko's user avatar
3 votes
0 answers
173 views

"Extending scalars" for (motivic) ring spectra and for modules over them: are the corresponding Moore spectra highly structured ring objects?

Let $S$ be a (motivic symmetric) ring spectrum (more generally, one can possibly consider a commutative ring object in any symmetric stable model category); let $R$ be an associative commutative ...
Mikhail Bondarko's user avatar
2 votes
0 answers
139 views

Compatibility of different exchange structures $\operatorname{Ex}^*_{\#},\operatorname{Ex}^*_*,\operatorname{Ex}_{\# *}$

Let $\mathcal{Cat}$ denotes the $2$-category of small categories and $\mathscr{S}=\mathrm{Sch}/S$ be some category of schemes over a given scheme $S$, consider a $2$-functor $\mathscr{M}:\mathscr{S}^{...
Alexey Do's user avatar
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2 votes
0 answers
125 views

Geometric conditions on motivic fibrations

What are the geometric conditions on a map of varieties/schemes being a motivic fibration (i.e. a fibration in the motivic model structure on simplicial presheaves on affine schemes)? For example, are ...
Grisha Taroyan's user avatar
2 votes
0 answers
128 views

A basic computation with spectra

Let $\mathbb{E}=\big(E_n, \sigma_n\colon T\wedge E_n\to E_{n+1}\big)_{n\in\mathbb{N}}$ be a $T$-spectrum, either in the topological setting (with $T=S^1$) or in the algebraic setting (with $T=\mathbb{...
Tintin's user avatar
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2 votes
0 answers
207 views

$G$-torsor over $\mathbb{A}^1_S$ where characteristic of $S$ does not divide $|G|$

I am reading the paper $\mathbb{A}^1$-homotopy theory of schemes by Morel and Voevodsky from 1999. There is a proposition saying that Let $G$ be a finite étale group scheme over $S$ of order prime to ...
XT Chen's user avatar
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1 vote
0 answers
77 views

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
1 vote
0 answers
245 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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1 vote
0 answers
78 views

Example of a non-strongly $A^1$ invariant sheaf of groups

A Nisnevich sheaf of groups is called strongly $A^1$-invariant if its classifying space $BG$ is $A^1$-invariant. I would like an example that is $A^1$ invariant but not strongly $A^1$-invariant for an ...
Nicky's user avatar
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