Completion of DVR












0












$begingroup$


Let $R$ be a DVR with a uniformizing parameter $t$, that is, $(R,(t))$ is a local ring.
Let $hat{R}$ to be the $(t)$-adic completion of $R$.
Then it seems that every element $alphainhat{R}$ can be written in
$$alpha=sum_{i=0}^infty a_i t^i=a_0+a_1t+a_2t^2+cdotsquad(1)$$
where $a_inotin(t)$ so that $a_i$'s are unit.
$t$ just works as it is an indeterminate w.r.t $R^times$.
Indeed, if $t$ satisfies an algebraic equation $c_nt^n+cdots+c_1t=c_0$ for some $c_iin R^times$, then by comparing the valuation of each sides, it can be shown that all $c_i=0$.
So this is my question: is it really possible to write the elements of $R^times$ as in (1)?






abstract-algebra commutative-algebra







share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    Let $R$ be a DVR with a uniformizing parameter $t$, that is, $(R,(t))$ is a local ring.
    Let $hat{R}$ to be the $(t)$-adic completion of $R$.
    Then it seems that every element $alphainhat{R}$ can be written in
    $$alpha=sum_{i=0}^infty a_i t^i=a_0+a_1t+a_2t^2+cdotsquad(1)$$
    where $a_inotin(t)$ so that $a_i$'s are unit.
    $t$ just works as it is an indeterminate w.r.t $R^times$.
    Indeed, if $t$ satisfies an algebraic equation $c_nt^n+cdots+c_1t=c_0$ for some $c_iin R^times$, then by comparing the valuation of each sides, it can be shown that all $c_i=0$.
    So this is my question: is it really possible to write the elements of $R^times$ as in (1)?






    abstract-algebra commutative-algebra







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let $R$ be a DVR with a uniformizing parameter $t$, that is, $(R,(t))$ is a local ring.
      Let $hat{R}$ to be the $(t)$-adic completion of $R$.
      Then it seems that every element $alphainhat{R}$ can be written in
      $$alpha=sum_{i=0}^infty a_i t^i=a_0+a_1t+a_2t^2+cdotsquad(1)$$
      where $a_inotin(t)$ so that $a_i$'s are unit.
      $t$ just works as it is an indeterminate w.r.t $R^times$.
      Indeed, if $t$ satisfies an algebraic equation $c_nt^n+cdots+c_1t=c_0$ for some $c_iin R^times$, then by comparing the valuation of each sides, it can be shown that all $c_i=0$.
      So this is my question: is it really possible to write the elements of $R^times$ as in (1)?






      abstract-algebra commutative-algebra







      share|cite|improve this question









      $endgroup$




      Let $R$ be a DVR with a uniformizing parameter $t$, that is, $(R,(t))$ is a local ring.
      Let $hat{R}$ to be the $(t)$-adic completion of $R$.
      Then it seems that every element $alphainhat{R}$ can be written in
      $$alpha=sum_{i=0}^infty a_i t^i=a_0+a_1t+a_2t^2+cdotsquad(1)$$
      where $a_inotin(t)$ so that $a_i$'s are unit.
      $t$ just works as it is an indeterminate w.r.t $R^times$.
      Indeed, if $t$ satisfies an algebraic equation $c_nt^n+cdots+c_1t=c_0$ for some $c_iin R^times$, then by comparing the valuation of each sides, it can be shown that all $c_i=0$.
      So this is my question: is it really possible to write the elements of $R^times$ as in (1)?






      abstract-algebra commutative-algebra




      abstract-algebra commutative-algebra






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 17 '18 at 5:11









      user190964user190964

      755




      755






















          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
          and we may represent elements of this inverse limit concretely as compatible sequences
          $$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
          Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
          $$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
          So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)



          Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.



          Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Really thanks for indicating references.
            $endgroup$
            – user190964
            Dec 21 '18 at 7:27











          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: "69"
          };
          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
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3043553%2fcompletion-of-dvr%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









          0












          $begingroup$

          Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
          and we may represent elements of this inverse limit concretely as compatible sequences
          $$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
          Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
          $$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
          So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)



          Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.



          Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Really thanks for indicating references.
            $endgroup$
            – user190964
            Dec 21 '18 at 7:27
















          0












          $begingroup$

          Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
          and we may represent elements of this inverse limit concretely as compatible sequences
          $$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
          Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
          $$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
          So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)



          Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.



          Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Really thanks for indicating references.
            $endgroup$
            – user190964
            Dec 21 '18 at 7:27














          0












          0








          0





          $begingroup$

          Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
          and we may represent elements of this inverse limit concretely as compatible sequences
          $$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
          Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
          $$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
          So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)



          Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.



          Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).






          share|cite|improve this answer











          $endgroup$



          Indeed, elements of $hat{R}$ may be written as power series in $t$ with coefficients in some set of representatives of $R/t.$ By definition, $$hat{R} := varprojlim(dotsto R/(t^i)to R/(t^{i-1})todotsto R/(t^2)to R/t),$$
          and we may represent elements of this inverse limit concretely as compatible sequences
          $$(a_1, a_2, a_3,dots)inprod_{i = 1}^infty R/(t^i).$$
          Concretely, this means that if $i > j,$ $a_i - a_j = a_{i,j}t^j$ (or $a_i = a_j + a_{i,j} t^j$) for some $a_{i,j}in R.$ Inductively, our sequence looks like
          $$(a_1, a_1 + a_{1,2}t, a_1 + a_{1,2}t+a_{2,3}t^2, dots).$$
          So the data of our sequence $(a_1,a_2,a_3,dots)$ is equivalent to the data of the sequence $(a_1, a_{1,2}, a_{2,3},dots),$ which gives you the translation that lets you write elements of $hat{R}$ as power series in $t.$ That you can take each $a_{i,i+1}$ to be a unit in $R$ or $0$ follows because each nonzero element $ain R$ can be written uniquely as $u t^i$ for some $uin R^times,$ $iinBbb Z_{geq 0}.$ (See here or here for more details.)



          Now, $hat{R}$ is also a DVR with uniformizer $t,$ and so units are precisely elements of $hat{R}$ which are not in the maximal ideal $that{R}$ - when written as power series, these are precisely the power series with nonzero leading coefficient.



          Some good references include Serre's "Local Fields," although I couldn't find this proposition. Neukirch's "Algebraic Number Theory" (in particular, chapter 2) is another good source, and has a detailed proof of the above in chapter II, section 4 (proposition 4.4).







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 17 '18 at 6:39

























          answered Dec 17 '18 at 5:24









          StahlStahl

          16.7k43355




          16.7k43355












          • $begingroup$
            Really thanks for indicating references.
            $endgroup$
            – user190964
            Dec 21 '18 at 7:27


















          • $begingroup$
            Really thanks for indicating references.
            $endgroup$
            – user190964
            Dec 21 '18 at 7:27
















          $begingroup$
          Really thanks for indicating references.
          $endgroup$
          – user190964
          Dec 21 '18 at 7:27




          $begingroup$
          Really thanks for indicating references.
          $endgroup$
          – user190964
          Dec 21 '18 at 7:27


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • 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%2fmath.stackexchange.com%2fquestions%2f3043553%2fcompletion-of-dvr%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

          Bressuire

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more

          Cabo Verde

          Image
            Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
          Read more

          Gyllenstierna

          Image
          .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
          Read more