{"id":1833,"date":"2022-06-22T18:21:08","date_gmt":"2022-06-22T21:21:08","guid":{"rendered":"http:\/\/localhost\/?page_id=1833"},"modified":"2022-06-22T21:48:04","modified_gmt":"2022-06-23T00:48:04","slug":"normalizacao-de-noether","status":"publish","type":"page","link":"http:\/\/localhost\/index.php\/ensino\/algebra-comutativa\/normalizacao-de-noether\/","title":{"rendered":"Normaliza\u00e7\u00e3o de Noether"},"content":{"rendered":"
Seja \\(\\mathbb F\\subseteq\\mathbb K\\)<\/span> uma extens\u00e3o de corpos. Um subconjunto \\(\\Omega\\subseteq \\mathbb K\\)<\/span> \u00e9 dito algebricamente dependente<\/em><\/span> (sobre \\(\\mathbb F\\)<\/span>) se existe um polin\u00f4mio \\(f\\in\\mathbb F[t_1,\\ldots,t_n]\\setminus\\{0\\}\\)<\/span> e elementos \\(\\alpha_1,\\ldots,\\alpha_n\\in\\Omega\\)<\/span> distintos tais que \\[f(\\alpha_1,\\ldots,\\alpha_n)=0.\\]<\/span> Caso contr\u00e1rio, \\(\\Omega\\)<\/span> \u00e9 dito algebricamente independente<\/em><\/span>. O subconjunto \\(\\emptyset\\)<\/span> \u00e9 considerado algebricamente independente.<\/div>\n

Usando o Lema de Zorn, pode-se provar que \\(\\mathbb K\\)<\/span> possui subconjuntos maximais algebricamente independentes. Tal conjunto \u00e9 dito base transcendental<\/em><\/span> de \\(\\mathbb K\\)<\/span> sobre \\(\\mathbb F\\)<\/span>. Segue da defini\u00e7\u00e3o que se \\(B\\)<\/span> \u00e9 uma uma base transcendental de \\(\\mathbb K\\)<\/span> sobre \\(\\mathbb F\\)<\/span>, ent\u00e3o \\(\\mathbb K\\)<\/span> \u00e9 alg\u00e9brico sobre \\(\\mathbb F(B)\\)<\/span>.<\/p>\n

Seja \\(\\mathbb F\\subseteq\\mathbb K\\)<\/span> uma extens\u00e3o de corpos.<\/p>\n
    \n
  1. \n

    Toda base transcendental de \\(\\mathbb K\\)<\/span> sobre \\(\\mathbb F\\)<\/span> t\u00eam a mesma cardinalidade.<\/p>\n<\/li>\n

  2. \n

    As seguintes s\u00e3o equivalentes para um conjunto \\(\\Omega\\subseteq \\mathbb K\\)<\/span>.<\/p>\n

      \n
    1. \n

      \\(\\Omega\\)<\/span> \u00e9 uma base transcendental de \\(\\mathbb K\\)<\/span> sobre \\(\\mathbb F\\)<\/span>.<\/p>\n<\/li>\n

    2. \n

      \\(\\Omega\\)<\/span> \u00e9 algebricamente indepdentente e \\(\\mathbb F(\\Omega)\\subseteq \\mathbb K\\)<\/span> \u00e9 uma extens\u00e3o alg\u00e9brica.<\/p>\n<\/li>\n<\/ol>\n<\/li>\n

    3. \n

      Sejam \\(X,Y\\subseteq \\mathbb K\\)<\/span> tais que<\/p>\n

        \n
      1. \n

        \\(X\\subseteq Y\\)<\/span>;<\/p>\n<\/li>\n

      2. \n

        \\(X\\)<\/span> \u00e9 algebricamente independente;<\/p>\n<\/li>\n

      3. \n

        the extens\u00e3o \\(\\mathbb F(Y)\\subseteq \\mathbb K\\)<\/span> \u00e9 alg\u00e9brica.<\/p>\n<\/li>\n<\/ol>\n

        Ent\u00e3o existe uma base transcendental \\(B\\)<\/span> de \\(\\mathbb K\\)<\/span> sobre \\(\\mathbb F\\)<\/span> tal que \\(X\\subseteq B\\subseteq Y\\)<\/span>.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n

        Assuma que \\(\\mathbb F[x_1,\\ldots,x_n]\\)<\/span> \u00e9 um anel de polin\u00f4mios em \\(n\\)<\/span> vari\u00e1veis e sejam \\(t_1,\\ldots,t_n\\)<\/span> tais que a extens\u00e3o de an\u00e9is \\(\\mathbb F[t_1,\\ldots,t_n]\\subseteq \\mathbb F[x_1,\\ldots,x_n]\\)<\/span> \u00e9 integral. Ent\u00e3o \\(t_1,\\ldots,t_n\\)<\/span> s\u00e3o algebricamente independentes.<\/div>\n
        Sejam \\(\\mathbb K=\\mathbb F(t_1,\\ldots,t_n)\\)<\/span> e \\(\\mathbb L=\\mathbb F(x_1,\\ldots,x_n)\\)<\/span> os corpos de fra\u00e7\u00f5es de \\(\\mathbb F[t_1,\\ldots,t_n]\\)<\/span> e \\(\\mathbb F[x_1,\\ldots,x_n]\\)<\/span>, respetivamente. Como \\(x_i\\)<\/span> \u00e9 integral sobre \\(\\mathbb F[t_1,\\ldots,t_n]\\)<\/span>, \\(x_i\\)<\/span> \u00e9 raiz de um polin\u00f4mio m\u00f4nico com coeficientes em \\(\\mathbb F[t_1,\\ldots,t_n]\\)<\/span> e assim \\(x_i\\)<\/span> \u00e9 alg\u00e9brico sobre \\(\\mathbb K\\)<\/span>. Logo, \\(\\mathbb L\\)<\/span> \u00e9 uma extens\u00e3o alg\u00e9brica de \\(\\mathbb K\\)<\/span> e aplicando o teorema acima para \\(X=\\emptyset\\)<\/span> e \\(Y=\\{t_1,\\ldots,t_n\\}\\)<\/span>, um subconjunto \\(Y'\\subseteq \\{t_1,\\ldots,t_n\\}\\)<\/span> \u00e9 base transcendental de \\(\\mathbb L\\)<\/span> sobre \\(\\mathbb F\\)<\/span>. Por outro lado, \\(\\{x_1,\\ldots,x_n\\}\\)<\/span> \u00e9 base transcendental de \\(\\mathbb L\\)<\/span> sobre \\(\\mathbb K\\)<\/span> e toda base transcendental tem \\(n\\)<\/span> elementos. Assim \\(Y'=Y\\)<\/span> e, em particular, \\(\\{t_1,\\ldots,t_n\\}\\)<\/span> \u00e9 um conjunto algebricamente independente.<\/div>\n
        Assuma que \\(A\\subseteq B\\)<\/span> \u00e9 uma extens\u00e3o de an\u00e9is e seja \\(x\\in B\\)<\/span>. Se \\(B\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(A\\)<\/span>, ent\u00e3o \\(B[x]\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(A[x]\\)<\/span>.<\/div>\n
        Seja \\(B=(b_1,\\ldots,b_k)_A\\)<\/span>. Ent\u00e3o \\(B[x]=(b_1,\\ldots,b_k)_{A[x]}\\)<\/span>.<\/div>\n
        Assuma que \\(\\mathbb F[x_1,\\ldots,x_n]\\)<\/span> \u00e9 uma \u00e1lgebra de polin\u00f4mios em \\(n\\)<\/span> vari\u00e1veis e sejam \\(X,Y\\subseteq \\{x_1,\\ldots,x_n\\}\\)<\/span>. Ent\u00e3o \\[(X)_{\\mathbb F[x_{x_1,\\ldots,x_n}]}\\cap \\mathbb F[Y]\\cong
        \n (X\\cap Y)_{ \\mathbb F[Y]}.\\]<\/span> Em particular, se \\(X\\cap Y=\\emptyset\\)<\/span>, ent\u00e3o \\((X)_{\\mathbb F[x_{x_1,\\ldots,x_n}]}\\cap \\mathbb F[Y]=0\\)<\/span>.<\/div>\n
        (Lema 1)
        \nSeja \\(R=\\mathbb F[x_1,\\ldots,x_n]\\)<\/span> uma \u00e1lgebra de polin\u00f4mios e seja \\(t_1\\in R\\setminus\\{0\\}\\)<\/span> tal que \\(t_1R\\neq R\\)<\/span>. Existem \\(t_2,\\ldots,t_n\\)<\/span> tais que<\/p>\n
          \n
        1. \n

          \\(t_1,t_2,\\ldots,t_n\\)<\/span> s\u00e3o algebricamente independentes sobre \\(\\mathbb F\\)<\/span>;<\/p>\n<\/li>\n

        2. \n

          \\(R\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P=\\mathbb F[t_1,\\ldots,t_n]\\)<\/span>;<\/p>\n<\/li>\n

        3. \n

          \\(t_1R\\cap P = t_1P\\)<\/span>.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n

          Seja \\(\\ell\\)<\/span> um n\u00famero natural a ser determinado mais precisamente depois. Defina para \\(i\\in\\{2,\\ldots,n\\}\\)<\/span>, \\(t_i=x_i-x_1^{\\ell^{i-1}}\\)<\/span>. Ponha \\(P=\\mathbb F[t_1,\\ldots,t_n]\\)<\/span> e observe que \\(P[x_1]=R\\)<\/span>. Afirmamos que \\(x_1\\)<\/span> \u00e9 integral sobre \\(\\mathbb F[t_1,\\ldots,t_n]\\)<\/span>. Como \\(t_1\\in\\mathbb F[x_1,\\ldots,x_n]\\)<\/span>, temos que \\[\\label{eq:nagata}
          \n t_1=\\sum_v \\alpha_vx_1^{v_1}\\cdots x_n^{v_n}=
          \n \\sum_v \\alpha_v x_1^{v_1}(t_2+x_1^{\\ell})^{v_2}\\cdots(t_n+x_1^{\\ell^{n-1}})^{v_n}.\\]<\/span> Usando a nota\u00e7\u00e3o \\[\\label{eq:n}
          \n n(v)=v_1+v_2\\ell+v_3\\ell^2+\\ldots+v_n\\ell^{n-1},\\]<\/span> cada parcela na soma anterior tem a forma \\[\\alpha_v x_1^{v_1}(t_2+x_1^{\\ell})^{v_2}\\cdots(t_n+x_1^{\\ell^{n-1}})^{v_n}=
          \n x_1^{n(v)}+\\mbox{(termos de grau menor em $t_1$).}\\]<\/span> Ora assuma que \\(\\ell\\)<\/span> \u00e9 maior que todo expoente \\(v_i\\)<\/span> aparecendo em\u00a0na express\u00e3o para \\(t_1\\)<\/span> acima. Neste caso \\(n(v)\\)<\/span> pode ser visto como a expans\u00e3o de um n\u00famero natural na base \\(\\ell\\)<\/span>. Em particular, \\(n(v)\\neq n(v')\\)<\/span> se \\(v\\neq v'\\)<\/span>. Isso quer dizer que os termos \\(\\alpha_v x_1^{n(v)}\\)<\/span> n\u00e3o se cancelam. Seja \\(w\\)<\/span> o vetor com \\(\\alpha_w\\neq 0\\)<\/span> e \\(n(w)\\)<\/span> maximal. Neste caso \\[t_1=\\alpha_wx_1^{n(w)}+\\mbox{(termos de menor grau em $x_1$)}.\\]<\/span> Assim, obtemos a equa\u00e7\u00e3o \\[x_1^{n(w)}-t_1+\\mbox{(termos de menor grau em $x_1$)}=0\\]<\/span> na qual o lado esquerdo \u00e9 um polin\u00f4mio na vari\u00e1vel \\(x_1\\)<\/span> com coeficinetes em \\(\\mathbb F[t_1,\\ldots,t_n]\\)<\/span>. Portanto \\(x_1\\)<\/span> \u00e9 integral sobre \\(P=\\mathbb F[t_1,\\ldots,t_n]\\)<\/span> e \\(R\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(\\mathbb F[t_1,\\ldots,t_n]\\)<\/span>. Ora, o lema\u00a0acima implica tamb\u00e9m que \\(t_1,\\ldots,t_n\\)<\/span> s\u00e3o algebricamente independentes sobre \\(\\mathbb F\\)<\/span>.<\/p>\n

          Nos resta provar (3). Temos que \\(t_1\\in P\\)<\/span> e \\(t_1\\in t_1R\\)<\/span> e assim \\(t_1P\\subseteq t_1R\\cap P\\)<\/span>. Seja \\(x\\in t_1R\\cap P\\)<\/span> e escreva \\(x=t_1y\\)<\/span> onde \\(y\\in R\\)<\/span>. Considere a cadeia de extens\u00f5es \\[P\\subseteq R\\cap \\mbox{Frac}(P)\\subseteq R\\]<\/span> e observe que \\(y\\in R\\cap \\mbox{Frac}(P)\\)<\/span>. Como \\(R\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P\\)<\/span>, temos que \\(R\\cap \\mbox{Frac}(P)\\)<\/span> (sendo um subm\u00f3dulo de um m\u00f3dulo finitamente gerado sobre um anel noetheriano) \u00e9 m\u00f3dulo-finito sobre \\(P\\)<\/span>. Agora considerando \\(R\\cap \\mbox{Frac}(P)\\)<\/span> em \\(\\mbox{Frac}(P)\\)<\/span>, temos que \\(R\\cap \\mbox{Frac}(P)\\)<\/span> \u00e9 integral sobre \\(P\\)<\/span>, ou seja \\(R\\cap \\mbox{Frac}(P)\\)<\/span> est\u00e1 contido na normaliza\u00e7\u00e3o \\(\\widetilde P\\)<\/span> de \\(P\\)<\/span> em \\(\\mbox{Frac}(P)\\)<\/span>. Mas \\(P\\)<\/span>, sendo \u00e1lgebra de polin\u00f4mios sobre um corpo, \u00e9 DFU, e assim \\(\\widetilde P=P\\)<\/span>. Isso implica que \\(y\\in P\\)<\/span>, ou seja \\(x\\in t_1P\\)<\/span>.<\/div>\n

          (Lema 2) Seja \\(R=\\mathbb F[x_1,\\ldots,x_n]\\)<\/span> uma \u00e1lgebra de polin\u00f4mios e seja \\(\\mathfrak a\\subsetneqq R\\)<\/span> um ideal. Existem \\(t_1,\\ldots,t_n\\)<\/span> tais que<\/p>\n
            \n
          1. \n

            \\(t_1,t_2,\\ldots,t_n\\)<\/span> s\u00e3o algebricamente independentes sobre \\(\\mathbb F\\)<\/span>;<\/p>\n<\/li>\n

          2. \n

            \\(R\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P=\\mathbb F[t_1,\\ldots,t_n]\\)<\/span>;<\/p>\n<\/li>\n

          3. \n

            \\(\\mathfrak a\\cap P = (t_1,\\ldots,t_h)\\)<\/span> com algum \\(h\\in\\{1,\\ldots,n\\}\\)<\/span>.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n

            Primeiro, se $\\mathfrak a=0$, ent\u00e3o podemos tomar $t_i=x_i$ para todo $i$. Assuma logo que $\\mathfrak a\\neq 0$ e use indu\u00e7\u00e3o por \\(n\\)<\/span>. Quando \\(n=1\\)<\/span>, ent\u00e3o \\(R=\\mathbb F[x_1]\\)<\/span> e \\(\\mathfrak a=t_1R\\)<\/span> com algum \\(t_1\\in R\\)<\/span> e o resultado est\u00e1 v\u00e1lido pelo Lema 1. Assuma que o resultado est\u00e1 v\u00e1lido para an\u00e9is de polin\u00f4mios com \\(n-1\\)<\/span> vari\u00e1veis. Considere \\(\\mathfrak a\\subseteq \\mathbb F[t_1,\\ldots,t_n]\\)<\/span> como no enunciado. Lembrando que \\(\\mathfrak a\\neq 0\\)<\/span>, tomemos \\(t_1\\in\\mathfrak a\\)<\/span> arbitr\u00e1rio n\u00e3o nulo. Use Lema\u00a01 para obter elementos \\(u_2,\\ldots,u_n\\)<\/span> tais que<\/p>\n
              \n
            1. \n

              \\(t_1,u_2,\\ldots,u_n\\)<\/span> s\u00e3o algebricamente independentes.<\/p>\n<\/li>\n

            2. \n

              \\(R\\)<\/span> \u00e9 m\u00f3dulo finito sobre \\(P_1=\\mathbb F[t_1,u_2,\\ldots,u_n]\\)<\/span>;<\/p>\n<\/li>\n

            3. \n

              \\(t_1R\\cap P_1=t_1P_1\\)<\/span>.<\/p>\n<\/li>\n<\/ol>\n

              Ora, usando a Hip\u00f3tese de Indu\u00e7\u00e3o para o anel \\(R_0=\\mathbb F[u_2,\\ldots,u_n]\\)<\/span> e o ideal \\(\\mathfrak b=\\mathfrak a\\cap R_0\\)<\/span>, ache \\(t_2,\\ldots,t_n\\in R_0\\)<\/span> tais que<\/p>\n

                \n
              1. \n

                \\(t_2,\\ldots,t_n\\)<\/span> s\u00e3o algebricamente independentes sobre \\(\\mathbb F\\)<\/span>;<\/p>\n<\/li>\n

              2. \n

                \\(R_0\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P_0=\\mathbb F[t_2,\\ldots,t_n]\\)<\/span>;<\/p>\n<\/li>\n

              3. \n

                \\(\\mathfrak b\\cap P_0=\\mathfrak a\\cap P_0=(t_2,\\ldots,t_h)_{P_0}\\)<\/span> com algum \\(h\\leq n\\)<\/span>.<\/p>\n<\/li>\n<\/ol>\n

                Seja \\(P=\\mathbb F[t_1,t_2,\\ldots,t_n]\\)<\/span>. Lembre que \\(R\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P_1\\)<\/span> e \\(\\mathbb F[u_2,\\ldots,u_n]\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P_0\\)<\/span>. Por um lema acima, \\(P_1=\\mathbb F[t_1,u_1,\\ldots,u_n]\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P=\\mathbb F[t_1,t_2,\\ldots,t_n]\\)<\/span>. Por transitividade, \\(R\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P\\)<\/span>. Um outro lema\u00a0acima implica tamb\u00e9m que \\(t_1,\\ldots,t_n\\)<\/span> s\u00e3o algebricamente independentes.<\/p>\n

                Nos resta provar afirma\u00e7\u00e3o \\((3)\\)<\/span>. Claramente \\(t_1,\\ldots,t_h\\in \\mathfrak a\\cap P\\)<\/span> e assim \\((t_1,\\ldots,t_h)_{P}\\subseteq \\mathfrak a\\cap P\\)<\/span>. Assuma que \\(x\\in\\mathfrak a\\cap P\\)<\/span>. Escreva \\[x=f_0+f_1t_1+f_2t_1^2+\\cdots +f_kt_1^k\\]<\/span> com \\(f_i\\in\\mathbb F[t_2,\\ldots,t_n]\\)<\/span> e note que \\(f_0=x-\\sum_{i\\geq 1}f_it_1^i\\in \\mathfrak a\\cap \\mathbb F[t_2,\\ldots,t_n]\\)<\/span>. Por outro lado, \\[\\mathfrak a\\cap \\mathbb F[t_2,\\ldots,t_n]=(t_2,\\ldots,t_h)_{P_0}\\subseteq (t_2,\\ldots,t_h)_{P}.\\]<\/span> Logo \\(f_0\\in (t_2,\\ldots,t_h)_{P}\\)<\/span> e \\(x\\in (t_1,t_2,\\ldots,t_h)_{P}\\)<\/span>.<\/div>\n

                (Lema 3) Seja \\(R=k[x_1,\\ldots,x_n]\\)<\/span> uma \u00e1lgebra de polin\u00f4mios e seja \\(\\mathfrak a_1\\subseteq \\mathfrak a_2\\subseteq \\cdots\\subseteq \\mathfrak a_r\\subsetneqq R\\)<\/span> uma cadeia de ideais. Existem \\(t_1,\\ldots,t_n\\)<\/span> tais que<\/p>\n
                  \n
                1. \n

                  \\(t_1,t_2,\\ldots,t_n\\)<\/span> s\u00e3o algebricamente independentes;<\/p>\n<\/li>\n

                2. \n

                  \\(R\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P=k[t_1,\\ldots,t_n]\\)<\/span>;<\/p>\n<\/li>\n

                3. \n

                  \\(\\mathfrak a_i\\cap P = (t_1,\\ldots,t_{h_i})\\)<\/span> com algum \\(h_i\\in\\{1,\\ldots,n\\}\\)<\/span>.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n

                  Indu\u00e7\u00e3o por \\(r\\)<\/span>. O caso \\(r=1\\)<\/span> \u00e9 o Lema\u00a02. A hip\u00f3tese da indu\u00e7\u00e3o \u00e9 que o lema \u00e9 verdadeiro para uma cadeia de \\(r-1\\)<\/span> ideais.<\/p>\n

                  Assuma que temos uma cadeia de \\(r\\)<\/span> ideais como no enunciado. Usando a hip\u00f3tese da indu\u00e7\u00e3o para a cadeia \\(\\mathfrak a_1\\subseteq \\cdots \\subseteq \\mathfrak a_{r-1}\\)<\/span> obtemos elementos \\(u_1,\\ldots,u_n\\in R\\)<\/span> tais que<\/p>\n

                    \n
                  1. \n

                    \\(u_1,\\ldots,u_n\\)<\/span> s\u00e3o algebricamente independentes;<\/p>\n<\/li>\n

                  2. \n

                    \\(R\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P_1=k[u_1,\\ldots,u_n]\\)<\/span>;<\/p>\n<\/li>\n

                  3. \n

                    \\(\\mathfrak a_i\\cap P_1=(u_1,\\ldots,u_{h_i})_{P_1}\\)<\/span> com \\(1\\leq h_1\\leq h_2\\leq \\cdots\\leq h_{r-1}\\)<\/span>.<\/p>\n<\/li>\n<\/ol>\n

                    Seja \\(h=h_{r-1}\\)<\/span>. Ora, usando Lema\u00a02 para \\(S=k[u_{h+1},\\ldots,u_n]\\)<\/span> e \\(\\mathfrak b=\\mathfrak a_r\\cap S\\)<\/span>, obtenha \\(t_{h+1},\\ldots,t_n\\)<\/span> tais que<\/p>\n

                      \n
                    1. \n

                      \\(t_{h+1},\\ldots,t_n\\)<\/span> s\u00e3o algebricamente independentes;<\/p>\n<\/li>\n

                    2. \n

                      \\(S\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P_2=k[t_{h+1},\\ldots,t_n]\\)<\/span>;<\/p>\n<\/li>\n

                    3. \n

                      \\((\\mathfrak b\\cap P_2)= \\mathfrak a\\cap P_2=(t_{h+1},\\ldots,t_{h_r})_{P_2}\\)<\/span> com algum \\(h_r\\in\\{h+1,\\ldots,n\\}\\)<\/span>.<\/p>\n<\/li>\n<\/ol>\n

                      Agora ponha \\(t_i=u_i\\)<\/span> para \\(i\\in\\{1,\\ldots,h\\}\\)<\/span>. Afirmamos que os elementos \\(t_1,\\ldots,t_h,t_{h+1},\\ldots,t_n\\)<\/span> satisfazem as afirma\u00e7\u00f5es do Teorema. Primeiro temos que as extens\u00f5es \\[k[t_{h+1},\\ldots,t_n] \\subseteq k[u_{h+1},\\ldots,u_n]\\quad \\mbox{e}\\quad k[u_1,\\ldots,u_n]\\subseteq R\\]<\/span> s\u00e3o m\u00f3dulo-finitas. Portanto \\[k[t_1,\\ldots,t_{h},t_{h+1},\\ldots,t_n]= k[u_1,\\ldots,u_{h},t_{h+1},\\ldots,t_n]\\subseteq R\\]<\/span> \u00e9 m\u00f3dulo-finito. Isso tamb\u00e9m implica que \\(t_1,\\ldots,t_n\\)<\/span> s\u00e3o algebricamente independentes.<\/p>\n

                      Seja \\(i\\in\\{1,\\ldots,r\\}\\)<\/span>. Temos que \\(t_1,\\ldots,t_{h_i}\\in\\mathfrak a_i\\cap P\\)<\/span> ent\u00e3o \\[(t_1,\\ldots,t_{h_i})_P\\subseteq \\mathfrak a_i\\cap P.\\]<\/span> Para provar a outra inclus\u00e3o, assuma que \\(x=\\mathfrak a_i\\cap P\\)<\/span> e seja \\(m=h_i\\)<\/span>. Ent\u00e3o \\[x=\\sum_v f_vt_1^{v_1}\\cdots t_m^{v_m}\\quad\\mbox{com}\\quad f_i\\in k[t_{m+1},\\ldots,t_n].\\]<\/span> Note, para \\(i\\in\\{1,\\ldots,r-1\\}\\)<\/span>, que \\[\\begin{aligned}
                      \n f_0\\in\\mathfrak a_i\\cap k[t_{m+1},\\ldots,t_n]&=
                      \n (\\mathfrak a_i\\cap k[u_1,\\ldots,u_n])\\cap k[t_{m+1},\\ldots,t_n]\\\\
                      \n &\\subseteq (\\mathfrak a_i\\cap k[u_1,\\ldots,u_n])\\cap k[u_{m+1},\\ldots,u_n]\\\\&
                      \n =(u_1,\\ldots,u_m)_{P_1}\\cap k[u_{m+1},\\ldots,u_n]=0.\\end{aligned}\\]<\/span> Se \\(i=r\\)<\/span>, \\[\\begin{aligned}
                      \n f_0\\in\\mathfrak a_r\\cap k[t_{m+1},\\ldots,t_n]&=(\\mathfrak a_r\\cap k[t_h,\\ldots,t_n])\\cap k[t_{m+1},\\ldots,t_n]\\\\
                      \n &=(t_1,\\ldots,t_m)_{P_2}\\cap k[t_{m+1},\\ldots,t_n]=0.\\end{aligned}\\]<\/span> Nos dois casos, \\(f_0=0\\)<\/span> e \\(x\\in (t_1,\\ldots,t_h)_P\\)<\/span>.<\/div>\n

                      (Teorema de Normaliza\u00e7\u00e3o de Noether) Seja \\(R=k[y_1,\\ldots,y_n]\\)<\/span> uma \\(k\\)<\/span>-\u00e1lgebra finitamente gerada e seja \\(\\mathfrak a_1\\subseteq \\mathfrak a_2\\subseteq \\cdots\\subseteq \\mathfrak a_r\\subsetneqq R\\)<\/span> uma cadeia de ideais. Existem \\(t_1,\\ldots,t_m\\)<\/span> tais que<\/p>\n
                        \n
                      1. \n

                        \\(t_1,t_2,\\ldots,t_m\\)<\/span> s\u00e3o algebricamente independentes;<\/p>\n<\/li>\n

                      2. \n

                        \\(R\\)<\/span> \u00e9 m\u00f3dulo-finito sobre \\(P=k[t_1,\\ldots,t_m]\\)<\/span>;<\/p>\n<\/li>\n

                      3. \n

                        \\(\\mathfrak a_i\\cap P = (t_1,\\ldots,t_{h_i})\\)<\/span> com algum \\(h_i\\in\\{1,\\ldots,n\\}\\)<\/span>.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n

                        Seja \\(R_0=k[x_1,\\ldots,x_n]\\)<\/span> a \u00e1lgebra de polin\u00f4mios e considere o mapa \\(\\psi:R_0\\to R\\)<\/span> definido como \\(\\psi(x_i)=y_i\\)<\/span> para todo \\(i\\)<\/span>. Seja \\(\\mathfrak b_0=\\ker\\psi=\\psi^{-1}(0)\\)<\/span> e \\(\\mathfrak b_i=\\psi^{-1}(\\mathfrak a_i)\\)<\/span> para \\(i\\geq 1\\)<\/span>. Aplique Lema\u00a03 para a \u00e1lgebra \\(R_0\\)<\/span> e a cadeia \\(\\mathfrak b_0\\subseteq \\mathfrak b_1\\subseteq \\cdots\\subseteq \\mathfrak b_r\\)<\/span> e obtenha uma sequ\u00eancia \\(u_{-h},\\ldots,u_0,u_1,\\ldots,u_m\\in R_0\\)<\/span> que satisfaz a afirma\u00e7\u00e3o do Lema\u00a03 com \\((\\ker \\psi)\\cap P_1=(u_{-h},\\ldots,u_0)_{P_1}\\)<\/span> com $P_=k[u_{-h},\\ldots,u_m]$. Seja \\(t_i=\\psi(u_i)\\)<\/span> para \\(i\\in\\{1,\\ldots,m\\}\\)<\/span> e ponha $P=k[t_1,\\ldots,t_m]$.<\/p>\n

                        Primeiro \\[k[t_1,\\ldots,t_m]\\cong P_1\/(\\ker(\\psi)\\cap P_1)=k[u_1,\\ldots,u_n].\\]<\/span> Logo, \\(t_1,\\ldots,t_m\\)<\/span> s\u00e3o algebricamente independentes.<\/p>\n

                        Al\u00e9m disso, \\[P=k[t_1,\\ldots,t_m]\\cong (P_1+\\ker\\psi)\/\\ker\\psi\\quad\\mbox{e}\\quad
                        \n R\\cong R_0\/\\ker\\psi.\\]<\/span> Como a extens\u00e3o \\(P_1\\subseteq R_0\\)<\/span> \u00e9 m\u00f3dulo finita, a extens\u00e3o \\(P\\subseteq R\\)<\/span> tamb\u00e9m \u00e9.<\/p>\n

                        Finalmente, nos resta provar que \\(\\mathfrak a_i\\cap P=(t_1,\\ldots,t_{h_i})_P\\)<\/span>. Primeiro, note que \\[\\begin{aligned}
                        \n \\psi(\\mathfrak b_i\\cap P_0)&=\\psi((u_{-h},\\ldots,u_{h_i})_{\\mathbb F[u_{-h},\\ldots,u_m]})=
                        \n \\psi((u_{-h},\\ldots,u_{h_i}))_{\\psi(\\mathbb F[u_{-h},\\ldots,u_m])}\\\\&=
                        \n (t_1,\\ldots,t_{h_i})_{P}.
                        \n \\end{aligned}\\]<\/span> Logo \\[\\mathfrak a_i\\cap P=\\psi(\\psi^{-1}(\\mathfrak a_i\\cap P))=\\psi(\\psi^{-1}(\\mathfrak a_i)\\cap \\psi^{-1}(P))=
                        \n \\psi(\\mathfrak b_i\\cap P_0)=(t_1,\\ldots,t_{h_i})_{P}.\\]<\/span><\/div>\n","protected":false},"excerpt":{"rendered":"

                        Seja \\(\\mathbb F\\subseteq\\mathbb K\\) uma extens\u00e3o de corpos. Um subconjunto \\(\\Omega\\subseteq \\mathbb K\\) \u00e9 dito algebricamente dependente (sobre \\(\\mathbb F\\)) se existe um polin\u00f4mio \\(f\\in\\mathbb F[t_1,\\ldots,t_n]\\setminus\\{0\\}\\) e elementos \\(\\alpha_1,\\ldots,\\alpha_n\\in\\Omega\\) distintos tais que \\[f(\\alpha_1,\\ldots,\\alpha_n)=0.\\] Caso contr\u00e1rio, \\(\\Omega\\) \u00e9 dito algebricamente independente. O subconjunto \\(\\emptyset\\) \u00e9 considerado algebricamente independente. Usando o Lema de Zorn, pode-se provar que … Continue reading Normaliza\u00e7\u00e3o de Noether<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":1808,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1833"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/comments?post=1833"}],"version-history":[{"count":6,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1833\/revisions"}],"predecessor-version":[{"id":1840,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1833\/revisions\/1840"}],"up":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1808"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=1833"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}