Questions tagged [motivic-homotopy]

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10
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0answers
363 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 ...
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137 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 $k$...
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58 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)\...
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104 views

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 ...
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1answer
85 views

Grayson filtration and weight filtration

I had a question that might be well-known but I'm not sure where to find it. Grayson defined a filtration on the algebraic $K$-theory of affine regular rings via commuting automorphisms which you can ...
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1answer
158 views

Dimension of $\ell$-adic Eilenberg-Maclane space

I'm currently studying the $\ell$-adic cohomology functor, i.e. the functor $$F:X \rightarrow H^i_{ét}(X,\mathbb{Q}_{\ell}).$$ In some sense, this is a representable functor, i.e. there exists an $\...
7
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1answer
231 views

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 ...
6
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1answer
300 views

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:\...
3
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1answer
180 views

Infinite loop space of ring spectra: the cup product

I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory. Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an ...
4
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1answer
220 views

Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete?

Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the ...
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59 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 ...
5
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1answer
185 views

Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization

Let $L_{Nis}(sPre(Sm_S))$ be the Nisnevich localization of the category of simplicial presheaves, how to see that whether $\mathbb{A}^1$-projections $\mathbb{A}^1\times_S X\to X$ are closed under ...
3
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1answer
253 views

Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers

Let $M$ be a model topos and $S$ a set of morphisms, there exists a set of morphism $\bar{S}$ which is generated by the $S$-local equivalences which is closed under homotopy pullbacks in $M$. Suppose ...
3
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1answer
119 views

Basic question on the cobordism spectrum

I am reading a little about cobordism and I have a basic question, which makes sense both in the topological and motivic setting. Let $\mathrm{Gr}_{n,\infty}$ denote the infinite $n$-Grassmanian and ...
7
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2answers
286 views

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 ...
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113 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....
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169 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 + ...
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80 views

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 ...
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1answer
301 views

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}\...
37
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2answers
4k views

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 ...
8
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1answer
393 views

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? ...
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2answers
344 views

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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1answer
105 views

Are the fibrations between $\mathbb{A}^{1}$-local objects $\mathbb{A}^{1}$-fibration?

Let $S$ be Noetherian scheme and $(Sm/S)_{Nis}$ is the Nisnevich site of smooth schemes over $S$. The category of simplicial sheaves on $(Sm/S)_{Nis}$ is denoted $Spc(S)$ and this category has two ...
3
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1answer
262 views

Basic questions on spectra

I have a basic question on Voevodsky's stable homotopy category of spectra $\mathbf{SH}(S)$, where $S$ is a finite dimensional noetherian scheme. Let $E$ be an $\Omega$-spectrum and $\varphi \colon ...
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452 views

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)\...
11
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2answers
878 views

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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166 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 ...
11
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1answer
390 views

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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1answer
1k views

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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322 views

Access to a classic reference of Dold-Puppe

There is an old reference that I am unable to easily find. It is Dold-Puppe´s communication at a conference. More concretely, it is cited as: A. Dold, D. Puppe: Duality, trace and transfer. ...
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1answer
277 views

Blowup formula for motivic cohomology

If $X$ is a smooth projective variety over a field, $Z\subset X$ a smooth closed subvariety of codimension $d$, $X'\to X$ the blowup of $X$ along $Z$, there's the blowup formula $$H^j(X'_{et},\mathbf{...
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1answer
395 views

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 ...
27
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2answers
1k views

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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83 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 ...
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371 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,...
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227 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 $...
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239 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{...
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155 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" ...
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231 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 ...
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1answer
277 views

More on categories of modules over the algebraic cobordism spectrum

I have the following questions on monoidal model structure(s) for the motivic stable homotopy category $SH(k)$ (where $k$ is a field); certainly, I am also interested in general statements concerning ...
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157 views

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 ...
3
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0answers
137 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 ...
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1answer
443 views

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 ...
5
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1answer
264 views

Expression of morphisms in motivic homotopy categories in terms of Nisnevich cohomology?

For a perfect field $k$ there is a collection of stable motivic homotopy categories equipped with the corresponding Morel's (homotopy) $t$-structures: $SH^{S^1}(k)$, $SH(k)$, $DA(k)$, and also modules ...
2
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1answer
287 views

Dualizability and motivic cohomology

Suppose $k$ is an algebraically closed field of characteristic $p$. Let $A=\mathbb{Z}/\ell\mathbb{Z}$, $\ell$ a prime coprime to $p$. Denote by $MA$ the motivic Eilenberg-Maclane spectrum over $k$. Is ...
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338 views

Rational homotopy and l-adic cohomology

In rational homotopy theory there is a basic construction which, given a prime number $l$ and a $CW$-complex $X$, produces a localized space $X_l$ equipped with a map $X\rightarrow X_l$ that induces ...
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2answers
1k views

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 = \...
12
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0answers
631 views

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 ...
6
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1answer
215 views

Localization, Slice Tower, and Motivic Spectra

Suppose $k$ is an algebraically closed field of characteristic $p>0$. There is an $\infty$-category of motivic spectra over $k$, denoted $\mathcal{S}pt(k)$. As in algebraic topology, there are ...
4
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1answer
338 views

Proof of Theorem 6.8 in the paper “Singular homology of abstract algebraic varieties”

Background: Theorem 6.8 in Suslin and Voevodsky's article "Singular homology of abstract algebraic varieties" states that there is an isomorphism between effective relative zero cycles $z_0^c(Z)^{eff}(...