Questions tagged [motivic-homotopy]
The motivic-homotopy tag has no usage guidance.
89
questions
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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 ...
4
votes
0answers
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 $k$...
4
votes
0answers
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)\...
4
votes
0answers
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 ...
2
votes
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 ...
1
vote
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
votes
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
votes
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
votes
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
votes
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 ...
1
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0answers
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
votes
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
votes
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
votes
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
votes
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 ...
5
votes
0answers
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....
7
votes
0answers
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 + ...
4
votes
0answers
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 ...
8
votes
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
$$H\mathbb{Z}\...
37
votes
2answers
4k views
Why is Voevodsky's motivic homotopy theory 'the right' approach?
Morel and Voevoedsky developed what is now called motivic homotopy theory, which aims to apply techniques of algebraic topology to algebraic varieties and, more generally, to schemes. A simple way of ...
8
votes
1answer
393 views
Motivic cohomology is universal with respect to what (co)homology theories?
I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem?
...
9
votes
2answers
344 views
A question about the vanishing of motivic cohomology in negative Tate twist
Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.
Is it true that
$$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[...
4
votes
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
votes
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 ...
12
votes
0answers
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
votes
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 ...
9
votes
0answers
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
votes
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 ...
21
votes
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 ...
4
votes
0answers
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. ...
5
votes
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{...
10
votes
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
votes
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 ...
3
votes
0answers
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 ...
4
votes
0answers
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,...
7
votes
0answers
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 $...
4
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0answers
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{...
4
votes
0answers
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" ...
4
votes
0answers
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 ...
6
votes
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 ...
8
votes
0answers
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
votes
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 ...
15
votes
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
votes
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
votes
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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0answers
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 ...
28
votes
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 = \...
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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
votes
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
votes
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}(...
