# Questions tagged [motivic-homotopy]

The motivic-homotopy tag has no usage guidance.

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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 ...

**6**

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**1**answer

191 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}\...

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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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277 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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279 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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**1**answer

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### 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 ...

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265 views

### A question on the slice filtration and the slice of the motive of the projective space

In the following $k$ is an algebraically closed field of characteristic $0$.
Consider the category $SH(k)$ (the Morel-Voevodsky stable motivic homotopy category).
By the work of Voevodsky (see for ...

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**1**answer

240 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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395 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)\...

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643 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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129 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 ...

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334 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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62 views

### $\mathbb{A}^{1}$-weak equivalences $K_{n,R} \longrightarrow K(2n,n,R)$?

Theorem 2.1 in "REDUCED POWER OPERATIONS IN MOTIVIC COHOMOLOGY" written by V.Voevodsky. He said that "there are $\mathbb{A}^{1}$-weak equivalences $K_{n,R} \longrightarrow K(2n,n,R)$ which are ...

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714 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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191 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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238 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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**1**answer

339 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 ...

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950 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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81 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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299 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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183 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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231 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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139 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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213 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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234 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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132 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 ...

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128 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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### 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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248 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 ...

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266 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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296 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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### 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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### 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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### 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 ...

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319 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}(...

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### Smallness of the category of schemes of finite type

Most sources about motivic homotopy theory mention that the category of (smooth) separated schemes of finite type over a (Noetherian of finite Krull dimension) base $S$ is essentially small, which is ...

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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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517 views

### 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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148 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 $[...

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396 views

### 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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223 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 $\...

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**1**answer

447 views

### 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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**1**answer

253 views

### Doubt regarding the definition of slice filtration

Voevodsky defined the slice filtration on the motivic stable homotopy category $SH(S)$ over a Noetherian scheme $S$. In the article Open Problems in the Motivic Stable Homotopy Theory, I, Section 2, ...

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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?
...

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381 views

### 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 ...

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189 views

### Algebraic cobordism (of a point) outside the geometric diagonal

This question is about the state of current knowledge regarding Voevodsky's algebraic cobordism of a point $\mathrm{MGL}^{*,*}(\mathrm{Spec}\,k)$. That the geometric diagonal $\mathrm{MGL}^{2*,*}(\...

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### 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 ...

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1k views

### 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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242 views

### 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 ...

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500 views

### 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{...