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10.71. Depth

Here is our definition.

Definition 10.71.1. Let $R$ be a ring, and $I \subset R$ an ideal. Let $M$ be a finite $R$-module. The $I$-depth of $M$, denoted $\text{depth}_I(M)$, is defined as follows:

  1. if $IM \not = M$, then $\text{depth}_I(M)$ is the supremum in $\{0, 1, 2, \ldots, \infty\}$ of the lengths of $M$-regular sequences in $I$,
  2. if $IM = M$ we set $\text{depth}_I(M) = \infty$.

If $(R, \mathfrak m)$ is local we call $\text{depth}_{\mathfrak m}(M)$ simply the depth of $M$.

Explanation. By Definition 10.67.1 the empty sequence is not a regular sequence on the zero module, but for practical purposes it turns out to be convenient to set the depth of the $0$ module equal to $+\infty$. Note that if $I = R$, then $\text{depth}_I(M) = \infty$ for all finite $R$-modules $M$. If $I$ is contained in the radical ideal of $R$ (e.g., if $R$ is local and $I \subset \mathfrak m_R$), then $M \not = 0 \Rightarrow IM \not = M$ by Nakayama's lemma. A module $M$ has $I$-depth $0$ if and only if $M$ is nonzero and $I$ does not contain a nonzerodivisor on $M$.

Example 10.67.2 shows depth does not behave well even if the ring is Noetherian, and Example 10.67.3 shows that it does not behave well if the ring is local but non-Noetherian. We will see depth behaves well if the ring is local Noetherian.

Lemma 10.71.2. Let $R$ be a ring, $I \subset R$ an ideal, and $M$ a finite $R$-module. Then $\text{depth}_I(M)$ is equal to the supremum of the lengths of sequences $f_1, \ldots, f_r \in I$ such that $f_i$ is a nonzerodivisor on $M/(f_1, \ldots, f_{i - 1})M$.

Proof. Suppose that $IM = M$. Then Lemma 10.19.1 shows there exists an $f \in I$ such that $f : M \to M$ is $\text{id}_M$. Hence $f, 0, 0, 0, \ldots$ is an infinite sequence of successive nonzerodivisors and we see agreement holds in this case. If $IM \not = M$, then we see that a sequence as in the lemma is an $M$-regular sequence and we conclude that agreement holds as well. $\square$

Lemma 10.71.3. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M$ be a nonzero finite $R$-module. Then $\dim(\text{Supp}(M)) \geq \text{depth}(M)$.

Proof. The proof is by induction on $\dim(\text{Supp}(M))$. If $\dim(\text{Supp}(M)) = 0$, then $\text{Supp}(M) = \{\mathfrak m\}$, whence $\text{Ass}(M) = \{\mathfrak m\}$ (by Lemmas 10.62.2 and 10.62.7), and hence the depth of $M$ is zero for example by Lemma 10.62.18. For the induction step we assume $\dim(\text{Supp}(M)) > 0$. Let $f_1, \ldots, f_d$ be a sequence of elements of $\mathfrak m$ such that $f_i$ is a nonzerodivisor on $M/(f_1, \ldots, f_{i - 1})M$. According to Lemma 10.71.2 it suffices to prove $\dim(\text{Supp}(M)) \geq d$. We may assume $d > 0$ otherwise the lemma holds. By Lemma 10.62.10 we have $\dim(\text{Supp}(M/f_1M)) = \dim(\text{Supp}(M)) - 1$. By induction we conclude $\dim(\text{Supp}(M/f_1M)) \geq d - 1$ as desired. $\square$

Lemma 10.71.4. Let $R$ be a Noetherian ring, $I \subset R$ an ideal, and $M$ a finite nonzero $R$-module such that $IM \not = M$. Then $\text{depth}_I(M) < \infty$.

Proof. Since $M/IM$ is nonzero we can choose $\mathfrak p \in \text{Supp}(M/IM)$ by Lemma 10.39.2. Then $(M/IM)_\mathfrak p \not = 0$ which implies $I \subset \mathfrak p$ and moreover implies $M_\mathfrak p \not = IM_\mathfrak p$ as localization is exact. Let $f_1, \ldots, f_r \in I$ be an $M$-regular sequence. Then $M_\mathfrak p/(f_1, \ldots, f_r)M_\mathfrak p$ is nonzero as $(f_1, \ldots, f_r) \subset I$. As localization is flat we see that the images of $f_1, \ldots, f_r$ form a $M_\mathfrak p$-regular sequence in $I_\mathfrak p$. Since this works for every $M$-regular sequence in $I$ we conclude that $\text{depth}_I(M) \leq \text{depth}_{I_\mathfrak p}(M_\mathfrak p)$. The latter is $\leq \text{depth}(M_\mathfrak p)$ which is $< \infty$ by Lemma 10.71.3. $\square$

Lemma 10.71.5. Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak m$. Let $M$ be a nonzero finite $R$-module. Then $\text{depth}(M)$ is equal to the smallest integer $i$ such that $\mathop{\mathrm{Ext}}\nolimits^i_R(R/\mathfrak m, M)$ is nonzero.

Proof. Let $\delta(M)$ denote the depth of $M$ and let $i(M)$ denote the smallest integer $i$ such that $\mathop{\mathrm{Ext}}\nolimits^i_R(R/\mathfrak m, M)$ is nonzero. We will see in a moment that $i(M) < \infty$. By Lemma 10.62.18 we have $\delta(M) = 0$ if and only if $i(M) = 0$, because $\mathfrak m \in \text{Ass}(M)$ exactly means that $i(M) = 0$. Hence if $\delta(M)$ or $i(M)$ is $> 0$, then we may choose $x \in \mathfrak m$ such that (a) $x$ is a nonzerodivisor on $M$, and (b) $\text{depth}(M/xM) = \delta(M) - 1$. Consider the long exact sequence of Ext-groups associated to the short exact sequence $0 \to M \to M \to M/xM \to 0$ by Lemma 10.70.6: $$ \begin{matrix} 0 \to \mathop{\mathrm{Hom}}\nolimits_R(\kappa, M) \to \mathop{\mathrm{Hom}}\nolimits_R(\kappa, M) \to \mathop{\mathrm{Hom}}\nolimits_R(\kappa, M/xM) \\ \phantom{0~} \to \mathop{\mathrm{Ext}}\nolimits^1_R(\kappa, M) \to \mathop{\mathrm{Ext}}\nolimits^1_R(\kappa, M) \to \mathop{\mathrm{Ext}}\nolimits^1_R(\kappa, M/xM) \to \ldots \end{matrix} $$ Since $x \in \mathfrak m$ all the maps $\mathop{\mathrm{Ext}}\nolimits^i_R(\kappa, M) \to \mathop{\mathrm{Ext}}\nolimits^i_R(\kappa, M)$ are zero, see Lemma 10.70.8. Thus it is clear that $i(M/xM) = i(M) - 1$. Induction on $\delta(M)$ finishes the proof. $\square$

Lemma 10.71.6. Let $R$ be a local Noetherian ring. Let $0 \to N' \to N \to N'' \to 0$ be a short exact sequence of finite $R$-modules.

  1. $\text{depth}(N) \geq \min\{\text{depth}(N'), \text{depth}(N'')\}$
  2. $\text{depth}(N'') \geq \min\{\text{depth}(N), \text{depth}(N') - 1\}$
  3. $\text{depth}(N') \geq \min\{\text{depth}(N), \text{depth}(N'') + 1\}$

Proof. Use the characterization of depth using the Ext groups $\mathop{\mathrm{Ext}}\nolimits^i(\kappa, N)$, see Lemma 10.71.5, and use the long exact cohomology sequence $$ \begin{matrix} 0 \to \mathop{\mathrm{Hom}}\nolimits_R(\kappa, N') \to \mathop{\mathrm{Hom}}\nolimits_R(\kappa, N) \to \mathop{\mathrm{Hom}}\nolimits_R(\kappa, N'') \\ \phantom{0~} \to \mathop{\mathrm{Ext}}\nolimits^1_R(\kappa, N') \to \mathop{\mathrm{Ext}}\nolimits^1_R(\kappa, N) \to \mathop{\mathrm{Ext}}\nolimits^1_R(\kappa, N'') \to \ldots \end{matrix} $$ from Lemma 10.70.6. $\square$

Lemma 10.71.7. Let $R$ be a local Noetherian ring and $M$ a nonzero finite $R$-module.

  1. If $x \in \mathfrak m$ is a nonzerodivisor on $M$, then $\text{depth}(M/xM) = \text{depth}(M) - 1$.
  2. Any $M$-regular sequence $x_1, \ldots, x_r$ can be extended to an $M$-regular sequence of length $\text{depth}(M)$.

Proof. Part (2) is a formal consequence of part (1). Let $x \in R$ be as in (1). By the short exact sequence $0 \to M \to M \to M/xM \to 0$ and Lemma 10.71.6 we see that the depth drops by at most 1. On the other hand, if $x_1, \ldots, x_r \in \mathfrak m$ is a regular sequence for $M/xM$, then $x, x_1, \ldots, x_r$ is a regular sequence for $M$. Hence we see that the depth drops by at least 1. $\square$

Lemma 10.71.8. Let $(R, \mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module. Let $x \in \mathfrak m$, $\mathfrak p \in \text{Ass}(M)$, and $\mathfrak q$ minimal over $\mathfrak p + (x)$. Then $\mathfrak q \in \text{Ass}(M/x^nM)$ for some $n \geq 1$.

Proof. Pick a submodule $N \subset M$ with $N \cong R/\mathfrak p$. By the Artin-Rees lemma (Lemma 10.50.2) we can pick $n > 0$ such that $N \cap x^nM \subset xN$. Let $\overline{N} \subset M/x^nM$ be the image of $N \to M \to M/x^nM$. By Lemma 10.62.3 it suffices to show $\mathfrak q \in \text{Ass}(\overline{N})$. By our choice of $n$ there is a surjection $\overline{N} \to N/xN = R/\mathfrak p + (x)$ and hence $\mathfrak q$ is in the support of $\overline{N}$. Since $\overline{N}$ is annihilated by $x^n$ and $\mathfrak p$ we see that $\mathfrak q$ is minimal among the primes in the support of $\overline{N}$. Thus $\mathfrak q$ is an associated prime of $\overline{N}$ by Lemma 10.62.8. $\square$

Lemma 10.71.9. Let $(R, \mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module. For $\mathfrak p \in \text{Ass}(M)$ we have $\dim(R/\mathfrak p) \geq \text{depth}(M)$.

Proof. If $\mathfrak m \in \text{Ass}(M)$ then there is a nonzero element $x \in M$ which is annihilated by all elements of $\mathfrak m$. Thus $\text{depth}(M) = 0$. In particular the lemma holds in this case.

If $\text{depth}(M) = 1$, then by the first paragraph we find that $\mathfrak m \not \in \text{Ass}(M)$. Hence $\dim(R/\mathfrak p) \geq 1$ for all $\mathfrak p \in \text{Ass}(M)$ and the lemma is true in this case as well.

We will prove the lemma in general by induction on $\text{depth}(M)$ which we may and do assume to be $> 1$. Pick $x \in \mathfrak m$ which is a nonzerodivisor on $M$. Note $x \not \in \mathfrak p$ (Lemma 10.62.9). By Lemma 10.59.12 we have $\dim(R/\mathfrak p + (x)) = \dim(R/\mathfrak p) - 1$. Thus there exists a prime $\mathfrak q$ minimal over $\mathfrak p + (x)$ with $\dim(R/\mathfrak q) = \dim(R/\mathfrak p) - 1$ (small argument omitted; hint: the dimension of a Noetherian local ring $A$ is the maximum of the dimensions of $A/\mathfrak r$ taken over the minimal primes $\mathfrak r$ of $A$). Pick $n$ as in Lemma 10.71.8 so that $\mathfrak q$ is an associated prime of $M/x^nM$. We may apply induction hypothesis to $M/x^nM$ and $\mathfrak q$ because $\text{depth}(M/x^nM) = \text{depth}(M) - 1$ by Lemma 10.71.7. We find $\dim(R/\mathfrak q) \geq \text{depth}(M/x^nM)$ and we win. $\square$

Lemma 10.71.10. Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $R \to S$ be a finite ring map. Let $\mathfrak m_1, \ldots, \mathfrak m_n$ be the maximal ideals of $S$. Let $N$ be a finite $S$-module. Then $$ \min\nolimits_{i = 1, \ldots, n} \text{depth}(N_{\mathfrak m_i}) = \text{depth}(N) $$

Proof. By Lemmas 10.35.20, 10.35.22, and Lemma 10.35.21 the maximal ideals of $S$ are exactly the primes of $S$ lying over $\mathfrak m$ and there are finitely many of them. Hence the statement of the lemma makes sense. We will prove the lemma by induction on $k = \min\nolimits_{i = 1, \ldots, n} \text{depth}(N_{\mathfrak m_i})$. If $k = 0$, then $\text{depth}(N_{\mathfrak m_i}) = 0$ for some $i$. By Lemma 10.71.5 this means $\mathfrak m_i S_{\mathfrak m_i}$ is an associated prime of $N_{\mathfrak m_i}$ and hence $\mathfrak m_i$ is an associated prime of $N$ (Lemma 10.62.16). By Lemma 10.62.13 we see that $\mathfrak m$ is an associated prime of $N$ as an $R$-module. Whence $\text{depth}(N) = 0$. This proves the base case. If $k > 0$, then we see that $\mathfrak m_i \not \in \text{Ass}_S(N)$. Hence $\mathfrak m \not \in \text{Ass}_R(N)$, again by Lemma 10.62.13. Thus we can find $f \in \mathfrak m$ which is not a zerodivisor on $N$, see Lemma 10.62.18. By Lemma 10.71.7 all the depths drop exactly by $1$ when passing from $N$ to $N/fN$ and the induction hypothesis does the rest. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 16950–17252 (see updates for more information).

    \section{Depth}
    \label{section-depth}
    
    \noindent
    Here is our definition.
    
    \begin{definition}
    \label{definition-depth}
    Let $R$ be a ring, and $I \subset R$ an ideal. Let $M$ be a finite $R$-module.
    The {\it $I$-depth} of $M$, denoted $\text{depth}_I(M)$, is defined as follows:
    \begin{enumerate}
    \item if $IM \not = M$, then $\text{depth}_I(M)$ is the supremum in
    $\{0, 1, 2, \ldots, \infty\}$ of the lengths of $M$-regular sequences in $I$,
    \item if $IM = M$ we set $\text{depth}_I(M) = \infty$.
    \end{enumerate}
    If $(R, \mathfrak m)$ is local we call $\text{depth}_{\mathfrak m}(M)$ simply
    the {\it depth} of $M$.
    \end{definition}
    
    \noindent
    Explanation. By Definition \ref{definition-regular-sequence} the empty
    sequence is not a regular sequence on the zero module, but for practical
    purposes it turns out to be convenient to set the depth of the $0$ module
    equal to $+\infty$. Note that if $I = R$, then $\text{depth}_I(M) = \infty$
    for all finite $R$-modules $M$. If $I$ is contained in the radical ideal
    of $R$ (e.g., if $R$ is local and $I \subset \mathfrak m_R$), then
    $M \not = 0 \Rightarrow IM \not = M$ by Nakayama's lemma.
    A module $M$ has $I$-depth $0$ if and only if $M$ is nonzero and $I$ does
    not contain a nonzerodivisor on $M$.
    
    \medskip\noindent
    Example \ref{example-global-regular} shows depth does not
    behave well even if the ring is Noetherian, and Example
    \ref{example-local-regular} shows that it does not
    behave well if the ring is local but non-Noetherian.
    We will see depth behaves well if the ring is local Noetherian.
    
    \begin{lemma}
    \label{lemma-depth-weak-sequence}
    Let $R$ be a ring, $I \subset R$ an ideal, and $M$ a finite $R$-module.
    Then $\text{depth}_I(M)$ is equal to the supremum of the lengths of
    sequences $f_1, \ldots, f_r \in I$ such that $f_i$ is a nonzerodivisor
    on $M/(f_1, \ldots, f_{i - 1})M$.
    \end{lemma}
    
    \begin{proof}
    Suppose that $IM = M$. Then Lemma \ref{lemma-NAK} shows there exists
    an $f \in I$ such that $f : M \to M$ is $\text{id}_M$. Hence
    $f, 0, 0, 0, \ldots$ is an infinite sequence of successive
    nonzerodivisors and we see agreement holds in this case.
    If $IM \not =  M$, then we see that a sequence as in the lemma
    is an $M$-regular sequence and we conclude that agreement holds as well.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-bound-depth}
    Let $(R, \mathfrak m)$ be a Noetherian local ring.
    Let $M$ be a nonzero finite $R$-module.
    Then $\dim(\text{Supp}(M)) \geq \text{depth}(M)$.
    \end{lemma}
    
    \begin{proof}
    The proof is by induction on $\dim(\text{Supp}(M))$.
    If $\dim(\text{Supp}(M)) = 0$, then
    $\text{Supp}(M) = \{\mathfrak m\}$, whence $\text{Ass}(M) = \{\mathfrak m\}$
    (by Lemmas \ref{lemma-ass-support} and \ref{lemma-ass-zero}), and hence
    the depth of $M$ is zero for example by
    Lemma \ref{lemma-ideal-nonzerodivisor}.
    For the induction step we assume $\dim(\text{Supp}(M)) > 0$.
    Let $f_1, \ldots, f_d$ be a sequence of elements of $\mathfrak m$
    such that $f_i$ is a nonzerodivisor on $M/(f_1, \ldots, f_{i - 1})M$.
    According to Lemma \ref{lemma-depth-weak-sequence} it suffices to prove
    $\dim(\text{Supp}(M)) \geq d$. We may assume
    $d > 0$ otherwise the lemma holds. By
    Lemma \ref{lemma-one-equation-module}
    we have $\dim(\text{Supp}(M/f_1M)) = \dim(\text{Supp}(M)) - 1$.
    By induction we conclude $\dim(\text{Supp}(M/f_1M)) \geq d - 1$
    as desired.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-depth-finite-noetherian}
    Let $R$ be a Noetherian ring, $I \subset R$ an ideal, and $M$ a
    finite nonzero $R$-module such that $IM \not = M$. Then
    $\text{depth}_I(M) < \infty$.
    \end{lemma}
    
    \begin{proof}
    Since $M/IM$ is nonzero we can choose $\mathfrak p \in \text{Supp}(M/IM)$
    by Lemma \ref{lemma-support-zero}. Then $(M/IM)_\mathfrak p \not = 0$
    which implies $I \subset \mathfrak p$ and moreover implies
    $M_\mathfrak p \not = IM_\mathfrak p$ as localization is exact.
    Let $f_1, \ldots, f_r \in I$ be an $M$-regular sequence.
    Then $M_\mathfrak p/(f_1, \ldots, f_r)M_\mathfrak p$ is
    nonzero as $(f_1, \ldots, f_r) \subset I$. As localization is
    flat we see that the images of $f_1, \ldots, f_r$ form a
    $M_\mathfrak p$-regular sequence in $I_\mathfrak p$. Since this
    works for every $M$-regular sequence in $I$ we conclude that
    $\text{depth}_I(M) \leq \text{depth}_{I_\mathfrak p}(M_\mathfrak p)$.
    The latter is $\leq \text{depth}(M_\mathfrak p)$ which is
    $< \infty$ by Lemma \ref{lemma-bound-depth}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-depth-ext}
    Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak m$.
    Let $M$ be a nonzero finite $R$-module. Then $\text{depth}(M)$
    is equal to the smallest integer $i$ such that
    $\Ext^i_R(R/\mathfrak m, M)$ is nonzero.
    \end{lemma}
    
    \begin{proof}
    Let $\delta(M)$ denote the depth of $M$ and let $i(M)$ denote
    the smallest integer $i$ such that $\Ext^i_R(R/\mathfrak m, M)$
    is nonzero. We will see in a moment that $i(M) < \infty$.
    By Lemma \ref{lemma-ideal-nonzerodivisor} we have
    $\delta(M) = 0$ if and only if $i(M) = 0$, because
    $\mathfrak m \in \text{Ass}(M)$ exactly means
    that $i(M) = 0$. Hence if $\delta(M)$ or $i(M)$ is $> 0$, then we may
    choose $x \in \mathfrak m$ such that (a) $x$ is a nonzerodivisor
    on $M$, and (b) $\text{depth}(M/xM) = \delta(M) - 1$.
    Consider the long exact sequence
    of Ext-groups associated to the short exact sequence
    $0 \to M \to M \to M/xM \to 0$ by Lemma \ref{lemma-long-exact-seq-ext}:
    $$
    \begin{matrix}
    0
    \to \Hom_R(\kappa, M)
    \to \Hom_R(\kappa, M)
    \to \Hom_R(\kappa, M/xM)
    \\
    \phantom{0\ }
    \to \Ext^1_R(\kappa, M)
    \to \Ext^1_R(\kappa, M)
    \to \Ext^1_R(\kappa, M/xM)
    \to \ldots
    \end{matrix}
    $$
    Since $x \in \mathfrak m$ all the maps $\Ext^i_R(\kappa, M)
    \to \Ext^i_R(\kappa, M)$ are zero, see
    Lemma \ref{lemma-annihilate-ext}.
    Thus it is clear that $i(M/xM) = i(M) - 1$. Induction on
    $\delta(M)$ finishes the proof.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-depth-in-ses}
    Let $R$ be a local Noetherian ring. Let $0 \to N' \to N \to N'' \to 0$
    be a short exact sequence of finite $R$-modules.
    \begin{enumerate}
    \item
    $\text{depth}(N) \geq \min\{\text{depth}(N'), \text{depth}(N'')\}$
    \item
    $\text{depth}(N'') \geq \min\{\text{depth}(N), \text{depth}(N') - 1\}$
    \item
    $\text{depth}(N') \geq \min\{\text{depth}(N), \text{depth}(N'') + 1\}$
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Use the characterization of depth using the Ext groups
    $\Ext^i(\kappa, N)$, see Lemma \ref{lemma-depth-ext},
    and use the long exact cohomology sequence
    $$
    \begin{matrix}
    0
    \to \Hom_R(\kappa, N')
    \to \Hom_R(\kappa, N)
    \to \Hom_R(\kappa, N'')
    \\
    \phantom{0\ }
    \to \Ext^1_R(\kappa, N')
    \to \Ext^1_R(\kappa, N)
    \to \Ext^1_R(\kappa, N'')
    \to \ldots
    \end{matrix}
    $$
    from Lemma \ref{lemma-long-exact-seq-ext}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-depth-drops-by-one}
    Let $R$ be a local Noetherian ring and $M$ a nonzero finite $R$-module.
    \begin{enumerate}
    \item If $x \in \mathfrak m$ is a nonzerodivisor on $M$, then
    $\text{depth}(M/xM) = \text{depth}(M) - 1$.
    \item Any $M$-regular sequence $x_1, \ldots, x_r$ can be extended to an
    $M$-regular sequence of length $\text{depth}(M)$.
    \end{enumerate}
    \end{lemma}
    
    \begin{proof}
    Part (2) is a formal consequence of part (1). Let $x \in R$ be as in (1).
    By the short exact sequence $0 \to M \to M \to M/xM \to 0$
    and Lemma \ref{lemma-depth-in-ses} we see that the depth drops by at most 1.
    On the other hand, if $x_1, \ldots, x_r \in \mathfrak m$
    is a regular sequence for $M/xM$, then $x, x_1, \ldots, x_r$
    is a regular sequence for $M$. Hence we see that the depth drops by
    at least 1.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-inherit-minimal-primes}
    Let $(R, \mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module.
    Let $x \in \mathfrak m$, $\mathfrak p \in \text{Ass}(M)$, and $\mathfrak q$
    minimal over $\mathfrak p + (x)$. Then $\mathfrak q \in \text{Ass}(M/x^nM)$
    for some $n \geq 1$.
    \end{lemma}
    
    \begin{proof}
    Pick a submodule $N \subset M$ with $N \cong R/\mathfrak p$.
    By the Artin-Rees lemma (Lemma \ref{lemma-Artin-Rees})
    we can pick $n > 0$ such that $N \cap x^nM \subset xN$.
    Let $\overline{N} \subset M/x^nM$ be the image of $N \to M \to M/x^nM$.
    By Lemma \ref{lemma-ass} it suffices to show
    $\mathfrak q \in \text{Ass}(\overline{N})$.
    By our choice of $n$ there is a surjection
    $\overline{N} \to N/xN = R/\mathfrak p + (x)$
    and hence $\mathfrak q$ is in the support of $\overline{N}$.
    Since $\overline{N}$ is annihilated by $x^n$ and $\mathfrak p$ we see that
    $\mathfrak q$ is minimal among the primes in the support of $\overline{N}$.
    Thus $\mathfrak q$ is an associated prime of $\overline{N}$ by
    Lemma \ref{lemma-ass-minimal-prime-support}.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-depth-dim-associated-primes}
    Let $(R, \mathfrak m)$ be a local Noetherian ring and $M$ a finite $R$-module.
    For $\mathfrak p \in \text{Ass}(M)$ we have
    $\dim(R/\mathfrak p) \geq \text{depth}(M)$.
    \end{lemma}
    
    \begin{proof}
    If $\mathfrak m \in \text{Ass}(M)$ then there is a nonzero element
    $x \in M$ which is annihilated by all elements of $\mathfrak m$.
    Thus $\text{depth}(M) = 0$. In particular the lemma holds in this case.
    
    \medskip\noindent
    If $\text{depth}(M) = 1$, then by the first paragraph
    we find that $\mathfrak m \not \in \text{Ass}(M)$.
    Hence $\dim(R/\mathfrak p) \geq 1$ for all $\mathfrak p \in \text{Ass}(M)$
    and the lemma is true in this case as well.
    
    \medskip\noindent
    We will prove the lemma in general by induction on $\text{depth}(M)$
    which we may and do assume to be $> 1$. Pick $x \in \mathfrak m$ which
    is a nonzerodivisor on $M$. Note $x \not \in \mathfrak p$
    (Lemma \ref{lemma-ass-zero-divisors}).
    By Lemma \ref{lemma-one-equation} we have
    $\dim(R/\mathfrak p + (x)) = \dim(R/\mathfrak p) - 1$.
    Thus there exists a prime $\mathfrak q$ minimal over $\mathfrak p + (x)$ with
    $\dim(R/\mathfrak q) = \dim(R/\mathfrak p) - 1$ (small argument omitted;
    hint: the dimension of a Noetherian local ring $A$ is the maximum
    of the dimensions of $A/\mathfrak r$ taken over the minimal
    primes $\mathfrak r$ of $A$). Pick $n$ as in
    Lemma \ref{lemma-inherit-minimal-primes} so that
    $\mathfrak q$ is an associated prime of $M/x^nM$.
    We may apply induction hypothesis to $M/x^nM$ and $\mathfrak q$
    because $\text{depth}(M/x^nM) = \text{depth}(M) - 1$ by
    Lemma \ref{lemma-depth-drops-by-one}. We find
    $\dim(R/\mathfrak q) \geq \text{depth}(M/x^nM)$ and we win.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-depth-goes-down-finite}
    Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $R \to S$
    be a finite ring map. Let $\mathfrak m_1, \ldots, \mathfrak m_n$
    be the maximal ideals of $S$. Let $N$ be a finite $S$-module.
    Then
    $$
    \min\nolimits_{i = 1, \ldots, n} \text{depth}(N_{\mathfrak m_i}) =
    \text{depth}(N)
    $$
    \end{lemma}
    
    \begin{proof}
    By Lemmas \ref{lemma-integral-no-inclusion}, \ref{lemma-integral-going-up},
    and Lemma \ref{lemma-finite-finite-fibres} the maximal ideals of
    $S$ are exactly the primes of $S$ lying over $\mathfrak m$ and
    there are finitely many of them. Hence the statement of the lemma
    makes sense. We will prove the lemma by induction on
    $k = \min\nolimits_{i = 1, \ldots, n} \text{depth}(N_{\mathfrak m_i})$.
    If $k = 0$, then $\text{depth}(N_{\mathfrak m_i}) = 0$ for some $i$.
    By Lemma \ref{lemma-depth-ext} this means
    $\mathfrak m_i S_{\mathfrak m_i}$ is an associated prime
    of $N_{\mathfrak m_i}$ and hence $\mathfrak m_i$ is an
    associated prime of $N$ (Lemma \ref{lemma-localize-ass}).
    By Lemma \ref{lemma-ass-functorial-Noetherian} we see that
    $\mathfrak m$ is an associated prime of $N$ as an $R$-module.
    Whence $\text{depth}(N) = 0$. This proves the base case.
    If $k > 0$, then we see that $\mathfrak m_i \not \in \text{Ass}_S(N)$.
    Hence $\mathfrak m \not \in \text{Ass}_R(N)$, again by
    Lemma \ref{lemma-ass-functorial-Noetherian}.
    Thus we can find $f \in \mathfrak m$ which is not a zerodivisor on
    $N$, see Lemma \ref{lemma-ideal-nonzerodivisor}. By
    Lemma \ref{lemma-depth-drops-by-one}
    all the depths drop exactly by $1$ when passing from $N$ to
    $N/fN$ and the induction hypothesis does the rest.
    \end{proof}

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