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      Questions tagged [motivic-homotopy]

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      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 ...
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      1answer
      166 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 ...
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      1answer
      224 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
      105 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 ...
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      2answers
      267 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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      0answers
      96 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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      0answers
      153 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
      75 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
      291 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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      votes
      2answers
      3k 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
      354 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
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      2answers
      317 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
      102 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
      258 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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      0answers
      435 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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