# Questions tagged [motivic-homotopy]

The motivic-homotopy tag has no usage guidance.

130
questions

4
votes

1
answer

275
views

### Homotopy coherence datum for composition of Becker-Gottlieb transfers

I have a question about certain detail in following answer by Denis Nardin adressing the concept of presheaves with transfer (mostly known in constructions in motivic homotopy theory) from viewpoint ...

4
votes

0
answers

162
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. ...

12
votes

1
answer

2k
views

### 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 $...

2
votes

0
answers

142
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}^{...

2
votes

0
answers

126
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 ...

2
votes

1
answer

200
views

### Removing quasi-projective assumption in the formalism of four operations

In Ayoub's thesis, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I), Ayoub proved that given a stable homotopical $2$-functor (Definition 1.4.1) $\...

3
votes

0
answers

154
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 ...

9
votes

0
answers

558
views

### 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 ...

4
votes

0
answers

410
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. ...

3
votes

1
answer

277
views

### Pushout homotopy squares in motivic homotopy theory

I am using Vladimir Voevodsky's notes on motivic homotopy theory and I am having trouble understanding the proofs of Corollary 2.20 and Lemma 2.21. Both of these deal with homotopy pushout squares and ...

3
votes

0
answers

276
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 ...

5
votes

0
answers

124
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 ...

4
votes

1
answer

224
views

### Functoriality conjectures on the slice filtration

Voevodsky wrote on his paper "Open Problems in the Motivic Stable Homotopy Theory, I" that
Three other groups of conjectures in motivic homotopy theory, not included in to this paper, seem ...

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^{\...

3
votes

0
answers

359
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 $\mathop{\mathrm{Fun}}(\mathrm{Cor}_k^{\mathrm{op}}, \mathcal S)$ ...

6
votes

1
answer

351
views

### 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 ...

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{...

2
votes

0
answers

208
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 ...

8
votes

1
answer

497
views

### 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 ...

5
votes

1
answer

483
views

### $\mathbf{A}^1$-invariance of Brauer groups and $H^2_{\mathrm{et}}(-;\mathbb{G}_m)$

The $\mathbf{A}^1$-invariance of vector bundles have been discussed in, for example, this paper by Asok, Hoyois and Wendt. This of course implies storng $\mathbf{A}^1$-invariance results for the first ...

1
vote

0
answers

248
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 ...

0
votes

0
answers

145
views

### A fibration is a map which has the right lifting property with respect to injections that are weak equivalences

As mentioned in Motivic Homotopy Theory, an alternative criterion for $X\rightarrow Y$ to be a fibration is that the diagram
$$\require{AMScd}
\begin{CD}
A @>>> X;\\
@VV{i}V @VVV \\
B @>&...

2
votes

1
answer

150
views

### The multiplicativity of the (complex) geometric realization of motivic cohomology

Consider the (complex) geometric realization of the motivic cohomology theory on simplicial presheaves over complex smooth schemes, which is a functorial homomorphism of $R$-modules, where $R$ is the ...

4
votes

1
answer

178
views

### (Algebraic) cobordism and the rank function

I write the question for algebraic cobordism but I have the analogue question for classic cobordism.
The spectrum representing algebraic cobordism
$$
\mathbf{MGL}=(*, \mathrm{Th}(1) , \ldots , \mathrm{...

6
votes

1
answer

418
views

### $\pi_0^{\mathbb{A}^1}(\mathbb{A}^m-0/\mu_n)\simeq\mathbb{G}_m/\mathbb{G}_m^n$

I have been reading Asok and Doran - $\mathbb A^1$-homotopy groups, excision, and solvable quotients. In the Example 2.17, page 1155, there is a claim that for $m>1$ and $n\geq 1$, $\pi_0^{\mathbb{...

0
votes

0
answers

169
views

### $\mathbb{A}^1$ connectedness of open subsets of $\mathbb{A}^n$

Let $k$ be an infinite field. There is a claim in the article Remark 2.16, page 1155, that if $U\subset \mathbb{A}^n_k$ is an open subset such that the complement of $U$ in $\mathbb{A}^n_k$ is of ...

5
votes

0
answers

465
views

### 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 ...

4
votes

0
answers

316
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 ...

6
votes

1
answer

536
views

### Representable cohomology theories in motivic homotopy theory

I am reading Mazza's, Voevodsky's and Weibel's book Lecture Notes on Motivic Cohomology and have grown curious about the following question:
Which cohomology theories on $Sm/k$ are representable, i.e. ...

6
votes

0
answers

235
views

### 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 ...

26
votes

1
answer

1k
views

### 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 ...

6
votes

0
answers

170
views

### 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 ...

3
votes

1
answer

292
views

### A class in the motivic cohomology group $H^{0,1}(\operatorname{Spec}k;\mathbb{Z}/p)$

In the following paper
N. Yagita, Examples for the Mod p Motivic Cohomology of Classifying Spaces,
on the first page, below display (1.1), it says "It is known that there is an element $\tau\in H^...

3
votes

0
answers

216
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 ...

3
votes

0
answers

257
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)]\...

6
votes

0
answers

298
views

### 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 ...

4
votes

1
answer

234
views

### Suspension Theorem in $\mathbb{A}^1$-homotopy

In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have
$$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$
So I'm wondering if this has an analogue in ...

5
votes

0
answers

262
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 ...

3
votes

1
answer

401
views

### Whitehead Theorem in $\mathbb{A}^1$-homotopy theory

I'm reading "UNSTABLE MOTIVIC HOMOTOPY THEORY" by Kirsten Wickelgren and Ben Williams (https://arxiv.org/pdf/1902.08857.pdf). There they have a version of Whitehead's Theorem, namely Prop ...

4
votes

0
answers

155
views

### Relation between $\mathbb{A}^1$-homotopy theory and derived algebraic geometry [duplicate]

I've often heard that one of the benefits of derived algebraic geometry, next to a cleaner intersection theory, is that "provides natural settings " for the $\mathbb{A}^1$-homotopy theory (...

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 ...

11
votes

0
answers

616
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 ...

5
votes

0
answers

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 $k$...

4
votes

0
answers

110
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

0
answers

192
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

1
answer

236
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

1
answer

262
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 $\...

9
votes

1
answer

278
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

1
answer

390
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

1
answer

289
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 ...