When is $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T13:09:02Z http://mathoverflow.net/feeds/question/118091 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118091/when-is-lim-n-rightarrow-infty-mathrmdepth-r-mathfrakan-mathrmdep When is $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}$? Mahdi Majidi-Zolbanin 2013-01-04T21:44:26Z 2013-01-04T21:51:03Z <p>Suppose $(R,\mathfrak{m})$ is a noetherian local ring. I am interested in ideals $\mathfrak{a}$ of $R$ for which $$\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}.$$ <strong>Question.</strong> Is there a name for such ideals? Have they been studied? When is this satisfied? Do you know examples other than those mentioned below?</p> <p><strong>Background</strong>. <a href="http://arxiv.org/pdf/math/0409178.pdf" rel="nofollow">This paper</a> provides good background <em>related</em> to this question. In particular, Brodmann proved that $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n$ always exists and in fact the sequence becomes constant for $n\gg0$. Brodmann also proved that $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n\leq\dim R-\ell(\mathfrak{a})$, where $\ell(\mathfrak{a})$ is the <em>analytic spread</em> of $\mathfrak{a}$, i.e., $\ell(\mathfrak{a})=\dim\mathcal{R}(\mathfrak{a})/\mathfrak{m}\mathcal{R}(\mathfrak{a})$, where $$\mathcal{R}(\mathfrak{a})=\bigoplus_n\mathfrak{a}^nt^n$$ is the Rees ring of $\mathfrak{a}$. Eisenbud and Huneke showed that $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n=\dim R-\ell(\mathfrak{a})$, if the associated graded ring $\mathrm{gr}_{\mathfrak{a}}(R)$ is Cohen-Macaulay, which is the case, for instance, when $R$ and $\mathcal{R}(\mathfrak{a})$ are Cohen-Macaulay. Theorem 1.2 in <em>loc. cit.</em> summarizes and re-proves these results of Brodmann and Eisenbud-Huneke. Another related concept is the concept of <em>equimultiple ideal</em>. An ideal $\mathfrak{a}$ of $R$ is called equimultiple, if $\mathrm{height}\:\mathfrak{a}=\ell(\mathfrak{a})$.</p> <p><strong>Examples.</strong> 1) The maximal ideal $\mathfrak{m}$ has this property. 2) Any principal ideal has this property. 3) If $R$ and $R/\mathfrak{a}$ and the Rees ring $\mathcal{R}(\mathfrak{a})$ are Cohen-Macaulay and $\mathfrak{a}$ is an equimultiple ideal, then $\mathfrak{a}$ satisfies the property. </p> <p><em><strong>Proof.</em></strong> If $R$ and $\mathcal{R}(\mathfrak{a})$ (and therefore $\mathrm{gr}_{\mathfrak{a}}(R)$) are Cohen-Macaulay and $\mathfrak{a}$ is an equimultiple ideal then <code>$$\dim R/\mathfrak{a}=\dim R-\mathrm{height}\:\mathfrak{a}=\dim R-\ell(\mathfrak{a})=\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n.$$</code> If $R/\mathfrak{a}$ is also Cohen-Macaulay, then we get an example of what I want. </p> <p><strong>P.S.</strong> The problem with this last example is that it seems too technical. It involves too many conditions and I don't know which ideals are equimultiple.</p>