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.










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












    $begingroup$


    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







      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.











      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.









      share|cite|improve this question




      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.









      asked yesterday









      rorirori

      533




      533




      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.





      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.






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






















          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





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "504"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });






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










          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f324715%2fdo-higher-etale-homotopy-groups-of-spectrum-of-a-field-always-vanish%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          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
















          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.










          draft saved

          draft discarded


















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













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












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
















          Thanks for contributing an answer to MathOverflow!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f324715%2fdo-higher-etale-homotopy-groups-of-spectrum-of-a-field-always-vanish%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          El tren de la libertad Índice Antecedentes "Porque yo decido" Desarrollo de la...

          Puerta de Hutt Referencias Enlaces externos Menú de navegación15°58′00″S 5°42′00″O /...

          Castillo d'Acher Características Menú de navegación