Do higher etale homotopy groups of spectrum of a field always vanish?Fibration sequences in étale homotopy...



Do higher etale homotopy groups of spectrum of a field always vanish?


Fibration sequences in étale homotopy theory arising from geometric fibresThe etale fundamental group of a fieldEtale topos as a classifyng topos ?“Étalification” of a schemealternate interpretations of Galois action on Tate modulequestion about the induced homomorphism of etale fundamental groupsClassifying Spaces and Eilenberg-Maclane objects in the category of simplicial ringsHow is a universal deformation ring in a 1-dimensional DM stack related to the complete etale local ring of coarse moduli scheme?Galois descent for absolute Galois groupLines in upper half-space













4












$begingroup$


Let $k$ be a field. In what generality is it true that higher etale homotopy groups
of $mathrm{Spec},k$ vanish?



If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.



I am not sure what happens for general fields.










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    4












    $begingroup$


    Let $k$ be a field. In what generality is it true that higher etale homotopy groups
    of $mathrm{Spec},k$ vanish?



    If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.



    I am not sure what happens for general fields.










    share|cite|improve this question







    New contributor




    rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$















      4












      4








      4


      2



      $begingroup$


      Let $k$ be a field. In what generality is it true that higher etale homotopy groups
      of $mathrm{Spec},k$ vanish?



      If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.



      I am not sure what happens for general fields.










      share|cite|improve this question







      New contributor




      rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      Let $k$ be a field. In what generality is it true that higher etale homotopy groups
      of $mathrm{Spec},k$ vanish?



      If the absolute Galois group is finite, we have a universal cover $mathrm{Spec},k^{sep}rightarrowmathrm{Spec},k$ which, I believe, is the initial object of the category of etale hypercovers. If we apply $pi_0$ to the simplicial $k$-scheme associated to this cover, the result coincides on the nose with the bar construction for the classifying space of $Gal(k^{sep}/k)$.



      I am not sure what happens for general fields.







      ag.algebraic-geometry etale-covers






      share|cite|improve this question







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      rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question







      New contributor




      rori is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      asked yesterday









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          1 Answer
          1






          active

          oldest

          votes


















          10












          $begingroup$

          The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.



          Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            yesterday






          • 2




            $begingroup$
            @DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
            $endgroup$
            – Denis Nardin
            yesterday








          • 1




            $begingroup$
            I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
            $endgroup$
            – David Roberts
            yesterday










          • $begingroup$
            @DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
            $endgroup$
            – Denis Nardin
            yesterday








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            yesterday











          Your Answer





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          1 Answer
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          1 Answer
          1






          active

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          active

          oldest

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          active

          oldest

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          10












          $begingroup$

          The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.



          Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            yesterday






          • 2




            $begingroup$
            @DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
            $endgroup$
            – Denis Nardin
            yesterday








          • 1




            $begingroup$
            I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
            $endgroup$
            – David Roberts
            yesterday










          • $begingroup$
            @DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
            $endgroup$
            – Denis Nardin
            yesterday








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            yesterday
















          10












          $begingroup$

          The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.



          Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.






          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            yesterday






          • 2




            $begingroup$
            @DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
            $endgroup$
            – Denis Nardin
            yesterday








          • 1




            $begingroup$
            I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
            $endgroup$
            – David Roberts
            yesterday










          • $begingroup$
            @DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
            $endgroup$
            – Denis Nardin
            yesterday








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            yesterday














          10












          10








          10





          $begingroup$

          The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.



          Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.






          share|cite|improve this answer









          $endgroup$



          The étale topos of a field $k$ is just the topos of sets with a continuous $mathrm{Gal}(k)$-action (here continuous is equivalent to all stabilizers being open), hence it is the colimit (in the ∞-category of topoi) of the topos of $mathrm{Gal}(k)/H$-sets where $H$ ranges through the open subgroups of $mathrm{Gal}(k)$.



          Since the étale homotopy type commutes with (homotopy) colimits, we have that the étale homotopy type of $(mathrm{Spec},k)_{ét}$ is the homotopy colimit of $Bmathrm{Gal}(k)/H$, and so it is the profinite space usually written $Bmathrm{Gal}(k)$ or $K(mathrm{Gal}(k),1)$. In particular it has no higher homotopy groups.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered yesterday









          Denis NardinDenis Nardin

          8,64723562




          8,64723562








          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            yesterday






          • 2




            $begingroup$
            @DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
            $endgroup$
            – Denis Nardin
            yesterday








          • 1




            $begingroup$
            I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
            $endgroup$
            – David Roberts
            yesterday










          • $begingroup$
            @DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
            $endgroup$
            – Denis Nardin
            yesterday








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            yesterday














          • 1




            $begingroup$
            No need for $infty$-categories here :-)
            $endgroup$
            – David Roberts
            yesterday






          • 2




            $begingroup$
            @DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
            $endgroup$
            – Denis Nardin
            yesterday








          • 1




            $begingroup$
            I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
            $endgroup$
            – David Roberts
            yesterday










          • $begingroup$
            @DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
            $endgroup$
            – Denis Nardin
            yesterday








          • 2




            $begingroup$
            I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
            $endgroup$
            – David Roberts
            yesterday








          1




          1




          $begingroup$
          No need for $infty$-categories here :-)
          $endgroup$
          – David Roberts
          yesterday




          $begingroup$
          No need for $infty$-categories here :-)
          $endgroup$
          – David Roberts
          yesterday




          2




          2




          $begingroup$
          @DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
          $endgroup$
          – Denis Nardin
          yesterday






          $begingroup$
          @DavidRoberts Well sure. But there's no need to avoid them either :). Also I'm not sure if the étale homotopy type commutes with strict colimit (what would they even be in the target?), so I erred on the safe side. I know you can avoid ∞-cats by arguing that it is a cofinal family of hypercovers, but I decided that it wasn't worth the contorsions
          $endgroup$
          – Denis Nardin
          yesterday






          1




          1




          $begingroup$
          I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
          $endgroup$
          – David Roberts
          yesterday




          $begingroup$
          I mean: the technology to compute colimits in the 2-category of toposes has been around a lot longer than colimits in $infty$-toposes (SGA4?), and easier to grasp and work with.
          $endgroup$
          – David Roberts
          yesterday












          $begingroup$
          @DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
          $endgroup$
          – Denis Nardin
          yesterday






          $begingroup$
          @DavidRoberts Let's agree to disagree on what's easier to grasp and work with :). No discussion about it being earlier though (I think in this case they are essentially equivalent, my throwing the $infty$ there was mainly a little attempt to demistify the image of $infty$-cats as something esoteric).
          $endgroup$
          – Denis Nardin
          yesterday






          2




          2




          $begingroup$
          I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
          $endgroup$
          – David Roberts
          yesterday




          $begingroup$
          I'm an Australian, 2-categories are my bag. Ah, well, I hope your attempt works :-)
          $endgroup$
          – David Roberts
          yesterday










          rori is a new contributor. Be nice, and check out our Code of Conduct.










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          rori is a new contributor. Be nice, and check out our Code of Conduct.
















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