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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Zariski's lemma ： ウィキペディア英語版 Zariski's lemma In algebra, Zariski's lemma, introduced by Oscar Zariski, states that if ''K'' is a finitely generated algebra over a field ''k'' and if ''K'' is a field, then ''K'' is a finite field extension of ''k''. An important application of the lemma is a proof of the weak form of Hilbert's nullstellensatz: if ''I'' is a proper ideal of $k(...,\; t\_n)$ (''k'' algebraically closed field), then ''I'' has a zero; i.e., there is a point ''x'' in $k^n$ such that $f(x)\; =\; 0$ for all ''f'' in ''I''.〔Proof: it is enough to consider a maximal ideal $\backslash mathfrak$. Let $A\; =\; k(...,\; t\_n)$ and $\backslash phi:\; A\; \backslash to\; A\; /\; \backslash mathfrak$ be the natural surjection. By the lemma, $A\; /\; \backslash mathfrak\; =\; k$ and then for any $f\; \backslash in\; \backslash mathfrak$, :$f(\backslash phi(t\_1),\; \backslash cdots,\; \backslash phi(t\_n))\; =\; \backslash phi(f(t\_1,\; \backslash cdots,\; t\_n))\; =\; 0$; that is to say, $x\; =\; (\backslash phi(t\_1),\; \backslash cdots,\; \backslash phi(t\_n))$ is a zero of $\backslash mathfrak$.〕 The lemma may also be understood from the following perspective. In general, a ring ''R'' is a Jacobson ring if and only if every finitely generated ''R''-algebra that is a field is finite over ''R''. Thus, the lemma follows from the fact that a field is a Jacobson ring. == Proof == Two direct proofs, one of which is due to Zariski, are given in Atiyah–MacDonald. The lemma is also a consequence of the Noether normalization lemma. Indeed, by the normalization lemma, ''K'' is a finite module over the polynomial ring $k(\backslash ldots\; ,\; x\_d)$ where $x\_1,\; \backslash ldots\; ,\; x\_d$ are algebraically independent over ''k''. But since ''K'' has Krull dimension zero, the polynomial ring must have dimension zero; i.e., $d=0$. For Zariski's original proof, see the original paper.〔http://projecteuclid.org/download/pdf_1/euclid.bams/1183510605〕 In fact, the lemma is a special case of the general formula $\backslash dim\; A\; =\; \backslash operatorname\_k\; A$ for a finitely generated ''k''-algebra ''A'' that is an integral domain, which is also a consequence of the normalization lemma. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Zariski's lemma」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.