{"id":1827,"date":"2022-06-22T18:14:49","date_gmt":"2022-06-22T21:14:49","guid":{"rendered":"http:\/\/localhost\/?page_id=1827"},"modified":"2022-06-22T22:09:01","modified_gmt":"2022-06-23T01:09:01","slug":"dimensao","status":"publish","type":"page","link":"http:\/\/localhost\/index.php\/ensino\/algebra-comutativa\/dimensao\/","title":{"rendered":"Dimens\u00e3o"},"content":{"rendered":"
Assuma que \\(k\\)<\/span> \u00e9 um corpo e \\(R\\)<\/span> \u00e9 um \\(\\mathbb K\\)<\/span>-dom\u00ednio finitamente gerado. Assuma que \\(\\mathfrak p_1\\subset\\mathfrak p_2\\subset\\cdots\\subset \\mathfrak p_r\\)<\/span> \u00e9 uma cadeia em \\(\\mbox{Spec}(R)\\)<\/span> e seja \\(d\\)<\/span> o grau de transcend\u00eancia de \\(\\mbox{Frac}(R)\\)<\/span> sobre \\(k\\)<\/span>. Ent\u00e3o \\(r\\leq d\\)<\/span> com igualdade se e somente se a cadeia \\((\\mathfrak p_i)\\)<\/span> \u00e9 saturada no sentido que ela n\u00e3o \u00e9 refinamento de uma cadeia mais longa.<\/div>\n
Use o Teorema de Normaliza\u00e7\u00e3o de Noether para escolher \\(t_1,\\ldots,t_n\\in R\\)<\/span> algebricamente independentes tal que \\[\\mathfrak p_i\\cap P=(t_1,\\ldots,t_{h_i})_P\\]<\/span> para \\(i\\in\\{1,\\ldots,r\\}\\)<\/span> onde \\(P=k[t_1,\\ldots,t_n]\\)<\/span>. Ponha \\(\\mathbb L=\\mbox{Frac}(R)\\)<\/span> e \\(\\mathbb K=\\mbox{Frac}(P)\\)<\/span>. Se \\(a\/b\\in\\mathbb L\\)<\/span>, ent\u00e3o \\(a\\)<\/span> e \\(b\\)<\/span> s\u00e3o integrais sobre \\(R\\)<\/span> e \\(a\\)<\/span> e \\(b\\)<\/span> s\u00e3o obviamente alg\u00e9bricos sobre \\(\\mathbb K=\\mbox{Frac}(R)\\)<\/span>. Assim \\(a\/b\\)<\/span> \u00e9 alg\u00e9brico sobre \\(\\mathbb K\\)<\/span>. Logo, a extens\u00e3o \\(\\mathbb K\\subseteq \\mathbb L\\)<\/span> \u00e9 alg\u00e9brica e assim \\(d=n\\)<\/span>. O primo \\(\\mathfrak p_i\\)<\/span> est\u00e1 acima de \\(\\mathfrak p_i\\cap P\\)<\/span> e o lema de incomparabilidade implica que \\(\\mathfrak p_i\\cap P\\neq \\mathfrak p_{i+1}\\cap P\\)<\/span> para todo \\(i\\)<\/span> e \\(h_1<h_2<\\cdots<h_r\\)<\/span>. Logo \\(r\\leq n=d\\)<\/span>. Em particular, \\(d\\)<\/span> \u00e9 o maior valor poss\u00edvel para \\(r\\)<\/span> e neste caso a cadeia \u00e9 saturada.<\/p>\n

Assuma que a cadeia \\((\\mathfrak p_i)\\)<\/span> \u00e9 saturada e tem comprimento \\(r\\)<\/span>. Neste caso \\(\\mathfrak p_1=0\\)<\/span> e \\(h_1=0\\)<\/span> e \\(\\mathfrak p_r\\)<\/span> \u00e9 maximal. Como \\(P\\subseteq R\\)<\/span> \u00e9 integral, temos que \\(\\mathfrak p_r\\cap P\\)<\/span> \u00e9 maximal e \\(h_r=n\\)<\/span>. Se \\(r<n\\)<\/span>, precisa existir algum \\(i\\)<\/span> tal que \\(h_{i}+2\\leq h_{i+1}\\)<\/span> temos que \\[\\mathfrak p_i\\cap P=(t_1,\\ldots,t_{h_i})_P\\subset (t_1,\\ldots,t_{h_{i}},t_{h_i+1})_P \\subset
\n (t_1,\\ldots,t_{h_{i+1}})_P=\\mathfrak p_{i+1}\\cap P.\\]<\/span> Ponha \\(\\mathfrak r= (t_1,\\ldots,t_{h_{i}},t_{h_i+1})_P\\)<\/span>. Ent\u00e3o \\(P\/(\\mathfrak p_i\\cap P)\\cong k[t_{h_i+1},\\ldots,t_n]\\)<\/span> \u00e9 um anel de polin\u00f4mios sobre \\(k\\)<\/span>, e \u00e9 um dom\u00ednio normal. Al\u00e9m disso, o mapa \\(P\/(\\mathfrak p_i\\cap P)\\to R\/\\mathfrak p_i\\)<\/span> definido por \\(p+(\\mathfrak p_i\\cap P)\\mapsto p+\\mathfrak p_i\\)<\/span> \u00e9 injetiva e \\(P\/(\\mathfrak p_i\\cap P)\\subseteq R\/\\mathfrak p_i\\)<\/span> \u00e9 uma extens\u00e3o integral, pois \\(P\\subseteq R\\)<\/span> \u00e9 integral. Pelo Teorema de Going-down, existe um primo \\(\\mathfrak p\\in \\mbox{Spec}(R)\\)<\/span> tal que \\(\\mathfrak p\/\\mathfrak p_i\\in\\mbox{Spec}(R\/\\mathfrak p_i)\\)<\/span> est\u00e1 acima de \\(\\mathfrak r\/(\\mathfrak p_i\\cap P)\\)<\/span> e \\(\\mathfrak p\/\\mathfrak p_i\\subset \\mathfrak p_{i+1}\/\\mathfrak p_i\\)<\/span>. Como \\(\\mathfrak r\/(\\mathfrak p_i\\cap P)\\neq 0\\)<\/span>, \\(\\mathfrak p\\neq \\mathfrak p_i\\)<\/span> e \\(\\mathfrak p_i\\subset \\mathfrak p\\subset \\mathfrak p_{i+1}\\)<\/span>. Obtivemos uma contradi\u00e7\u00e3o, pois a cadeia \\((\\mathfrak p_i)\\)<\/span> foi assumida saturada.<\/div>\n

Seja \\(R\\)<\/span> um anel. A dimens\u00e3o de Krull de \\(R\\)<\/span> \u00e9 o supremo dos comprimentos das cadeias \\[\\mathfrak p_1\\subset\\mathfrak p_2\\subset \\cdots \\subset \\mathfrak p_k\\]<\/span> de ideais primos em \\(R\\)<\/span>. A dimens\u00e3o de Krull est\u00e1 denotado por \\(\\dim R\\)<\/span>.<\/div>\n
A dimens\u00e3o de um corpo \u00e9 zero; $\\dim \\Z=1$, $\\dim\\F[x_1,\\ldots,x_k]=k$ onde os $x_i$ s\u00e3o inc\u00f4gnitas.<\/div>\n
Se \\(R\\)<\/span> \u00e9 um \\(k\\)<\/span>-dom\u00ednio finitamente gerado, ent\u00e3o \\(\\dim R\\)<\/span> \u00e9 igual ao grau de transcend\u00eancia de \\(\\mbox{Frac}(R)\\)<\/span> sobre \\(k\\)<\/span>.<\/div>\n
Seja \\(R\\)<\/span> um \\(k\\)<\/span>-dom\u00ednio finitamente gerado, seja \\(\\mathfrak p\\in\\mbox{Spec}(R)\\)<\/span> e seja \\(\\mathfrak m\\)<\/span> um ideal maximal em \\(R\\)<\/span>. Ent\u00e3o \\[\\dim R_{(\\mathfrak p)}+\\dim R\/\\mathfrak p=\\dim R\\quad\\mbox{e}\\quad \\dim R_{(\\mathfrak m)}=\\dim R.\\]<\/span><\/div>\n
Assuma que \\[\\mathfrak p_1\\subset \\mathfrak p_2\\subset \\cdots\\subset \\mathfrak p=\\mathfrak p_i\\subset\\cdots \\subset \\mathfrak p_r\\]<\/span> \u00e9 uma cadeia saturada dos primos que cont\u00e9m \\(\\mathfrak p\\)<\/span>. Ent\u00e3o \\(r\\)<\/span> \u00e9 igual ao grau de transcend\u00eancia de \\(R\\)<\/span> sobre \\(k\\)<\/span> e \\(r=\\dim R\\)<\/span>. Por outro lado \\[\\label{eq:c1}
\n \\mathfrak p_i\/\\mathfrak p\\subset \\cdots\\subset \\mathfrak p_r\/\\mathfrak p_i\\]<\/span> \u00e9 uma cadeia em \\(\\mbox{Spec}(R\/\\mathfrak p)\\)<\/span> e \\[\\label{eq:c2}
\n \\mathfrak p_1 R_{(\\mathfrak p)}\\subset \\cdots \\subset\\mathfrak p_i R_{(\\mathfrak p)}\\]<\/span> \u00e9 uma cadeia em \\(\\mbox{Spec}(R_{(\\mathfrak p)})\\)<\/span>. Assim \\(\\dim R\/\\mathfrak p+\\dim R_{(\\mathfrak p)}\\geq \\dim R\\geq r\\)<\/span>. Por outro lado, cadeias como as duas acima resultam em cadeias em \\(\\mbox{Spec}(R)\\)<\/span> e assim \\(\\dim R\\geq
\n\\dim R\/\\mathfrak p+\\dim R_{(\\mathfrak p)}\\)<\/span>. Logo \\[\\dim R_{(\\mathfrak p)}+\\dim R\/\\mathfrak p=\\dim R.\\]<\/span> A segunda afirma\u00e7\u00e3o segue da primeira observando que \\(\\dim R\/\\mathfrak m=0\\)<\/span> como \\(R\/\\mathfrak m\\)<\/span> \u00e9 um corpo.<\/div>\n
\nSeja $R$ um anel e $\\mathfrak p\\in\\mbox{Spec}\\, R$. A altura (height) de $\\mathfrak p$ \u00e9 definido como $\\dim R_{(\\mathfrak p)}$. A altura de $\\mathfrak p$ coincide com o supremo dos comprimentos das cadeias
\n\\[
\n\\mathfrak p_0\\subset \\mathfrak p_1\\subset \\cdots \\subset \\mathfrak p
\n\\]
\nem $\\mbox{Spec}\\, R$.
\nA altura de $\\mathfrak p$ \u00e9 denotada por $\\mbox{ht}\\,\\mathfrak p$.\n<\/div>\n
\nSe $R$ \u00e9 um anel e $\\mathfrak p\\in\\mbox{Spec}\\,R$, ent\u00e3o
\n\\[
\n\\mbox{ht}\\,\\mathfrak p+\\dim R\/\\mathfrak p\\leq \\dim R.
\n\\]\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Assuma que \\(k\\) \u00e9 um corpo e \\(R\\) \u00e9 um \\(\\mathbb K\\)-dom\u00ednio finitamente gerado. Assuma que \\(\\mathfrak p_1\\subset\\mathfrak p_2\\subset\\cdots\\subset \\mathfrak p_r\\) \u00e9 uma cadeia em \\(\\mbox{Spec}(R)\\) e seja \\(d\\) o grau de transcend\u00eancia de \\(\\mbox{Frac}(R)\\) sobre \\(k\\). Ent\u00e3o \\(r\\leq d\\) com igualdade se e somente se a cadeia \\((\\mathfrak p_i)\\) \u00e9 saturada no sentido que … Continue reading Dimens\u00e3o<\/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\/1827"}],"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=1827"}],"version-history":[{"count":5,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1827\/revisions"}],"predecessor-version":[{"id":1842,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1827\/revisions\/1842"}],"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=1827"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}