\n<\/p>\n
\u4ece\u7c07\u5230\u6982\u5f62<\/span>\n05\/22\/2025<\/span>\nXu Yuan<\/span>\n\n\u76ee\u5f55<\/span> 2. \u4eff\u5c04\u6982\u578b<\/a><\/span>\n\n\u8bbe$R$\u662f\u4e00\u4e2a\u73af, \u5b9a\u4e49$R$\u7684\u7d20\u8c31$\\spec(R)$\u4e3a$R$\u4e2d\u6240\u6709\u7d20\u7406\u60f3\u6784\u6210\u7684\u96c6\u5408. \u8be5\u96c6\u5408\u4e5f\u88ab\u79f0\u4e3a$R$\u914d\u6210\u7684\\textbf{\u4eff\u5c04\u6982\u5f62}. \u4f5c\u4e3a\u4eff\u5c04\u7c07\u7684\u63a8\u5e7f, $R$\u4e8b\u5b9e\u4e0a\u5e94\u8be5\u770b\u6210$X=\\spec(R)$\u7684\u591a\u9879\u5f0f\u73af.<\/p>\n e.g.\u8bbe$R=A(X)$, \u5176\u4e2d$X$\u662f\u4ee3\u6570\u95ed\u57df$K$\u4e0a\u7684\u4eff\u5c04\u7c07. \u90a3\u4e48\u5b58\u5728\u5982\u4e0b\u7684\u4e00\u4e00\u5bf9\u5e94 \u4ec0\u4e48\u662f\u6982\u5f62? \u6982\u5f62\u63d0\u70bc\u51fa\u4e86\u7c07\u7684\u54ea\u4e9b\u4fe1\u606f? \u672c\u6587\u4ece\u4eff\u5c04\u6982\u5f62\u51fa\u53d1, \u9010\u6e10\u7ed9\u51fa\u6982\u5f62\u7684\u62bd\u8c61\u5b9a\u4e49. \u7ee7\u7eed\u9605\u8bfb \u4ece\u7c07\u5230\u6982\u5f62<\/span>→<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[11,8],"class_list":["post-574","post","type-post","status-publish","format-standard","hentry","category-notes","tag-11","tag-8"],"_links":{"self":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/574","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=574"}],"version-history":[{"count":17,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/574\/revisions"}],"predecessor-version":[{"id":596,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/574\/revisions\/596"}],"wp:attachment":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=574"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=574"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=574"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n \u76ee\u5f55<\/a><\/span>
1. \u5f15\u8a00<\/a><\/span>
2. \u4eff\u5c04\u6982\u578b<\/a><\/span>
\n1. \u5f15\u8a00<\/a><\/span>\n\n\u5728\u672c\u7bc7\u4e2d, \u6211\u4eec\u4e3b\u8981\u5e0c\u671b\u5c06\u7c07\u4e0a\u7684\u8ba4\u8bc6, \u63a8\u5e7f\u5230\u65b0\u6982\u5ff5 “\u6982\u5f62 (scheme)” \u4e0a\u6765. \u7a0d\u5fae\u56de\u5fc6\u4e00\u4e0b: \u6211\u4eec\u9996\u5148\u5728\u4ee3\u6570\u95ed\u57df$K$\u4e0a, \u5b9a\u4e49\u4e86\u4eff\u5c04\u7c07\u7684\u6982\u5ff5, \u5176\u4e0a\u7684\u6b63\u5219\u51fd\u6570 (regular functions) \u8d4b\u4e88\u4e86\u4eff\u5c04\u7c07\u73af\u7a7a\u95f4 (ringed space) \u7684\u7ed3\u6784. \u901a\u8fc7\u6709\u9650\u591a\u4e2a\u4eff\u5c04\u7c07\u7684\u7c98\u63a5, \u6211\u4eec\u63d0\u70bc\u51fa\u4e86\u9884\u7c07 (prevariety) \u7684\u6982\u5ff5, \u6700\u540e\u628a\u5177\u6709\u5206\u79bb\u6027\u7684\u9884\u7c07\u79f0\u4e3a\u7c07 (variety). \u5bf9\u4e8e\u6982\u5f62, \u6211\u4eec\u540c\u6837\u5e0c\u671b\u8fd9\u6837\u505a. \u5728\u672c\u7bc7\u4e2d, \u6211\u4eec\u4f1a\u501f\u52a9\u5bf9\u6982\u5f62\u5404\u79cd\u6982\u5ff5\u7684\u5b9a\u4e49, \u6765\u5de9\u56fa\u5bf9\u7c07\u7684\u8ba4\u8bc6.<\/p>\n
\n $$
\n \\spec(R)\\longleftrightarrow\\{Y\\subseteq X\\text{\u662f\u4e0d\u53ef\u7ea6\u5b50\u7c07}\\}
\n $$
\ne.g.\u82e5$R=K[x]=A(\\mathbb{A}_K^1)$, \u5176\u4e2dground field $K$\u662f\u4ee3\u6570\u95ed\u57df, \u90a3\u4e48$R$\u7684\u4eff\u5c04\u6982\u5f62\u4e3a
\n \\[\\spec(R)=\\{\\langle x-a\\rangle, \\langle 0\\rangle\\}.\\]
\n\u5e0c\u671b\u8868\u8fbe\u7684\u662f, \u6982\u578b\u5148\u7ed9\u51fa\u4e86$R$\u4f5c\u4e3a “\u591a\u9879\u5f0f\u73af”, \u7136\u540e\u7528\u591a\u9879\u5f0f\u73af\u6765\u5b9a\u4e49\u5bf9\u8c61. \u4e3a\u4e86\u5b9a\u4e49$R$\u4e0a\u7684\u5143\u7d20\u5982\u4f55\u5728$X$\u4e0a\u8d4b\u503c, \u6211\u4eec\u9996\u5148\u5b9a\u4e49\u4eff\u5c04\u6982\u5f62$X$\u5728$P$\u5904\u7684\\textbf{\u5269\u4f59\u57df} (residue field), \u4f5c\u4e3a$f\\in R$\u5728$P$\u5904\u7684\u53d6\u503c\u8303\u56f4: \u8bbe$P\\in\\spec(R)$, \u5b9a\u4e49
\n\\[
\n \\boxed{K(P):=\\mathrm{Quot}(R\/P)}
\n\\]
\n\u4e3a\u4eff\u5c04\u6982\u5f62$X$\u5728$P$\u5904\u7684\u5269\u4f59\u57df. \u90a3\u4e48\u5143\u7d20$f\\in R$\u5728$p\\in\\spec(R)$\u5904\u7684\u53d6\u503c\u5e94\u8be5\u843d\u5728$K(P)$\u4e2d. \u4e3a\u6b64, \u6211\u4eec\u5b9a\u4e49
\n\\[
\n f(P):=\\frac{[f]}{1}\\in K(P),
\n\\]
\n\u5176\u4e2d$\\boxed{[f]=f+P\\in R\/P}$. \u8fd9\u6837\u4e00\u6765, \u5982\u679c$f(P)=0$, \u90a3\u4e48$[f]=0\\in R\/P$, \u5373$f\\in P$.
\ne.g. \u8bbe$X$\u662f\u4ee3\u6570\u95ed\u57df$K$\u4e0a\u7684\u4eff\u5c04\u7c07, $R=A(X)$.
\n
\n \\[
\n \\begin{aligned}
\n R\/P & \\rightarrow K \\\\
\n [f] & \\mapsto f(a)
\n \\end{aligned}
\n \\]
\n \u7ed9\u51fa\u4e86\u57df\u540c\u6784.
\n <\/li>
\n \u5bf9\u5e94\u4eff\u5c04\u6982\u5f62\u5728$P$\u5904\u7684\u5269\u4f59\u57df\u662f$Y$\u4e0a\u6240\u6709\u6709\u7406\u51fd\u6570\u6784\u6210\u7684\u96c6\u5408, \u5373
\n \\[
\n K(P)=\\mathrm{Quot}(A(Y))=K(Y)\\]
\n \u5176\u4e2d$K(Y)$\u662f$Y$\u4e0a\u6240\u6709\u6709\u7406\u51fd\u6570\u6784\u6210\u7684\u96c6\u5408. \u6b64\u65f6, $\\forall f\\in R=A(X)$, $f(P)\\in K(P)=K(Y)$\u5e94\u5f53\u662f$Y$\u4e0a\u7684\u6709\u7406\u51fd\u6570. \u7ec6\u8282\u4e0a, $f\\in A(X)$\u7684\u7b49\u4ef7\u7c7b$[f]\\in A(X)\/I_X(Y)=A(Y)$\u662f$f|_Y$, \u90a3\u4e48
\n \\[
\n f(P)=\\frac{[f]}{1}=[f|_Y]\\in K(Y)
\n \\]
\n \u662f$f$\u5bf9\u5e94\u7684\u51fd\u6570\u4f5c\u4e3a$Y$\u4e0a\u7684\u6709\u7406\u51fd\u6570\u7684\u7b49\u4ef7\u7c7b.
\n <\/li><\/ol>
\n<\/div>
\n<\/p>\n","protected":false},"excerpt":{"rendered":"