uniform bound of the number of generators of prime ideals - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:00:47Z http://mathoverflow.net/feeds/question/112244 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112244/uniform-bound-of-the-number-of-generators-of-prime-ideals uniform bound of the number of generators of prime ideals Pham Hung Quy 2012-11-13T02:40:22Z 2013-03-06T10:25:08Z <p>These questions are inspired from the well known fact (by Sally et. al.) as follows:</p> <p><strong>Theorem 1.</strong> Let $(R, \mathfrak{m})$ be a Noetherian local ring of dimension one. Then the minimal number of generators of ideals of $R$ is bounded above by a constant i.e. there exists a positive integer $C$ such that $\ell (I/\mathfrak{m}I) \leq C$ for all ideal $I$. </p> <p>In higher dimension, it is easy to see Theorem 1 is not true.</p> <p><strong>Question 2.</strong> Let $(R, \mathfrak{m})$ be a Noetherian local domain of dimension two. Does the exist a positive integer $C$ such that $\ell (\mathfrak{p}/\mathfrak{mp}) \leq C$ for all prime ideal $\mathfrak{p}$ of $R$? </p> <p><strong>Question 3.</strong> Let $(R, \mathfrak{m})$ be a Noetherian local domain of dimension two. Does the exist a positive integer $C$ such that $\ell (\mathfrak{p}R_{\mathfrak{p}}/\mathfrak{p}^2R_{\mathfrak{p}}) \leq C$ for all prime ideal $\mathfrak{p}$ of $R$?</p> http://mathoverflow.net/questions/112244/uniform-bound-of-the-number-of-generators-of-prime-ideals/112328#112328 Answer by Qing Liu for uniform bound of the number of generators of prime ideals Qing Liu 2012-11-14T00:08:54Z 2012-11-14T00:08:54Z <p>The answer to Question 3 is yes if $R$ is excellent (it is enough that the normalization of $R$ is finite over $R$). Indeed the normal locus of $\mathrm{Spec}(R)$ is then open, so there are only finitely many $\mathfrak p$ of height $1$ with non-normal $R_{\mathfrak p}$. The other prime ideals are either $0, \mathfrak m$ or height $1$ with normal (hence regular) $R_{\mathfrak p}$.</p> http://mathoverflow.net/questions/112244/uniform-bound-of-the-number-of-generators-of-prime-ideals/112340#112340 Answer by Pham Hung Quy for uniform bound of the number of generators of prime ideals Pham Hung Quy 2012-11-14T03:29:18Z 2013-03-06T10:25:08Z <p>Here is answer for question 3 in arbitrary dimension with a certain restriction on $R$.</p> <p>We assume $R$ is a image of a regular local ring $(S, \mathfrak{n})$ (we always have this assumption by passing the completion). $R = S/ I$ for some ideal $I$ of $S$. Recall that the embedded dimension of $R$ is $\mu(\mathfrak{m}) = \ell (\mathfrak{m}/\mathfrak{m}^2)$, we denote it by $C$. It is not difficult to see that we can assume that the embedded dimension of $S$ is equal to $\mu(\mathfrak{m})$. So $S$ is a regular local ring of dimension $C$. </p> <p>Now let $\mathfrak{p}$ be a prime ideal of $R$. Let $\mathfrak{q} \in Spec (S)$ such that $\mathfrak{p} = \mathfrak{q}/I$. It is well known that $S_{\mathfrak{q}}$ is also a regular local ring of (embedded) dimension $\leq C$. Moreover $R_{\mathfrak{p}} = S_{\mathfrak{q}}/I S_{\mathfrak{q}}$. So the embedded dimension of $R_{\mathfrak{p}}$ is less than or equal to the embedded dimension of $S_{\mathfrak{q}}$. Thus $\mu (\mathfrak{p}R_{\mathfrak{p}}) = \ell (\mathfrak{p}R_{\mathfrak{p}}/ \mathfrak{p}^2R_{\mathfrak{p}}) \leq C$.</p> <p>Edit:</p> <p>I have just read from Sally's book (page 52) the following result:</p> <p>Theorem: Let $(R, \mathfrak{m})$ be a NOetherian local ring. Then $\dim R \leq 2$ iff there is a uniform bound of the number of generators of all ideals which do not have $\mathfrak{m}$ as a associated prime.</p> <p>Thus, Question 3 has an affirmative answer. </p> http://mathoverflow.net/questions/112244/uniform-bound-of-the-number-of-generators-of-prime-ideals/112384#112384 Answer by Karl Schwede for uniform bound of the number of generators of prime ideals Karl Schwede 2012-11-14T14:59:08Z 2012-11-14T14:59:08Z <p>The following paper seems to indicate that there is no such bound in question 2. </p> <ul> <li>T. T. Moh, <em>On the unboundedness of generators of prime ideals in powerseries rings of three variables</em>. J. Math. Soc. Japan Volume 26, Number 4 (1974), 722-734. <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.jmsj/1240435038" rel="nofollow">http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.jmsj/1240435038</a></li> </ul> <p>He seems to construct a sequence of prime ideals $P_n$ in $k[[x,y,z]]$ with $n$ generators.</p> <p>On the other hand, in a positive result, in this paper:</p> <ul> <li>M. Boratyński, D. Eisenbud, D. Rees, <em>On the number of generators of ideals in local Cohen-Macaulay rings.</em> J. Algebra 57 (1979), no. 1, 77–81. <a href="http://www.sciencedirect.com/science/article/pii/0021869379902096" rel="nofollow">http://www.sciencedirect.com/science/article/pii/0021869379902096</a></li> </ul> <p>There they show some bounds for 2 -dimensional Cohen-Macaulay rings.</p>