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Suppose $(R,\mathfrak{m})$ is a noetherian local ring. I am interested in ideals $\mathfrak{a}$ of $R$ for which $$\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}.$$ Question. Is there a name for such ideals? Have they been studied? When is this satisfied? Do you know examples other than those mentioned below?

Background. This paper provides good background related 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 analytic spread 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 loc. cit. summarizes and re-proves these results of Brodmann and Eisenbud-Huneke. Another related concept is the concept of equimultiple ideal. An ideal $\mathfrak{a}$ of $R$ is called equimultiple, if $\mathrm{height}\:\mathfrak{a}=\ell(\mathfrak{a})$.

Examples. 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.

Proof. 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 $$\dim R/\mathfrak{a}=\dim R-\mathrm{height}\:\mathfrak{a}=\dim R-\ell(\mathfrak{a})=\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n.$$ If $R/\mathfrak{a}$ is also Cohen-Macaulay, then we get an example of what I want.

P.S. The problem with this last example is that it seems too technical. It involved involves too many conditions and I don't know which ideals are equimultiple.

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# When is $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}$?

Suppose $(R,\mathfrak{m})$ is a noetherian local ring. I am interested in ideals $\mathfrak{a}$ of $R$ for which $$\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}.$$ Question. Is there a name for such ideals? Have they been studied? When is this satisfied? Do you know examples other than those mentioned below?

Background. This paper provides good background related 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 analytic spread 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 loc. cit. summarizes and re-proves these results of Brodmann and Eisenbud-Huneke. Another related concept is the concept of equimultiple ideal. An ideal $\mathfrak{a}$ of $R$ is called equimultiple, if $\mathrm{height}\:\mathfrak{a}=\ell(\mathfrak{a})$.

Examples. 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.

Proof. 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 $$\dim R/\mathfrak{a}=\dim R-\mathrm{height}\:\mathfrak{a}=\dim R-\ell(\mathfrak{a})=\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n.$$ If $R/\mathfrak{a}$ is also Cohen-Macaulay, then we get an example of what I want.

P.S. The problem with this last example is that it seems too technical. It involved too many conditions I don't know which ideals are equimultiple.