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15.94 Formally catenary rings

In this section we prove a theorem of Ratliff [Ratliff] that a Noetherian local ring is universally catenary if and only if it is formally catenary.

Definition 15.94.1. A Noetherian local ring $A$ is formally catenary if for every minimal prime $\mathfrak p \subset A$ the ring $A^\wedge /\mathfrak p A^\wedge $ is equidimensional.

The meaning of the definition will be clear after reading the results.

Lemma 15.94.2. Let $(A, \mathfrak m)$ be a Noetherian local ring which is not formally catenary. Then $A$ is not universally catenary.

Proof. By assumption there exists a minimal prime $\mathfrak p \subset A$ such that $A^\wedge /\mathfrak p A^\wedge $ is not equidimensional. After replacing $A$ by $A/\mathfrak p$ we may assume that $A$ is a domain and that $A^\wedge $ is not equidimensional. Let $\mathfrak q$ be a minimal prime of $A^\wedge $ such that $d = \dim (A^\wedge /\mathfrak q)$ is minimal and hence $0 < d < \dim (A)$. We prove the lemma by induction on $d$.

The case $d = 1$. In this case $\dim (A^\wedge _\mathfrak q) = 0$. Hence $A^\wedge _\mathfrak q$ is Artinian local and we see that for some $n > 0$ the ideal $J = \mathfrak q^ n$ maps to zero in $A^\wedge _\mathfrak q$. It follows that $\mathfrak m$ is the only associated prime of $J/J^2$, whence $\mathfrak m^ m$ annihilates $J/J^2$ for some $m > 0$. Thus we can use Lemma 15.93.4 to find $A \to B$ of finite type such that $B^\wedge \cong A^\wedge /J$. It follows that $\mathfrak m_ B = \sqrt{\mathfrak mB}$ is a maximal ideal with the same residue field as $\mathfrak m$ and $B^\wedge $ is the $\mathfrak m_ B$-adic completion (Algebra, Lemma 10.96.7). Then

\[ \dim (B_{\mathfrak m_ B}) = \dim (B^\wedge ) = 1 = d. \]

Since we have the factorization $A \to B \to A^\wedge /J$ the inverse image of $\mathfrak q/J$ is a prime $\mathfrak q' \subset \mathfrak m_ B$ lying over $(0)$ in $A$. Thus, if $A$ were universally catenary, the dimension formula (Algebra, Lemma 10.112.1) would give

\begin{align*} \dim (B_{\mathfrak m_ B}) & \geq \dim ((B/\mathfrak q')_{\mathfrak m_ B}) \\ & = \dim (A) + \text{trdeg}_ A(B/\mathfrak q') - \text{trdeg}_{\kappa (\mathfrak m)}(\kappa (\mathfrak m_ B)) \\ & = \dim (A) + \text{trdeg}_ A(B/\mathfrak q') \end{align*}

This contradictions finishes the argument in case $d = 1$.

Assume $d > 1$. Let $Z \subset \mathop{\mathrm{Spec}}(A^\wedge )$ be the union of the irreducible components distinct from $V(\mathfrak q)$. Let $\mathfrak r_1, \ldots , \mathfrak r_ m \subset A^\wedge $ be the prime ideals corresponding to irreducible components of $V(\mathfrak q) \cap Z$ of dimension $> 0$. Choose $f \in \mathfrak m$, $f \not\in A \cap \mathfrak r_ j$ using prime avoidance (Algebra, Lemma 10.14.2). Then $\dim (A/fA) = \dim (A) - 1$ and there is some irreducible component of $V(\mathfrak q, f)$ of dimension $d - 1$. Thus $A/fA$ is not formally catenary and the invariant $d$ has decreased. By induction $A/fA$ is not universally catenary, hence $A$ is not universally catenary. $\square$

Lemma 15.94.3. Let $A \to B$ be a flat local ring map of local Noetherian rings. Assume $B$ is catenary and equidimensional. Then

  1. $B/\mathfrak p B$ is equidimensional for all $\mathfrak p \subset A$,

  2. $A$ is catenary and equidimensional.

Proof. Let $\mathfrak p \subset A$ be a prime ideal. Let $\mathfrak q \subset B$ be a prime minimal over $\mathfrak pB$. Then $\mathfrak q \cap A = \mathfrak p$ by going down for $A \to B$ (Algebra, Lemma 10.38.18). Hence $A_\mathfrak p \to B_\mathfrak q$ is a flat local ring map with special fibre of dimension $0$ and hence

\[ \dim (A_\mathfrak p) = \dim (B_\mathfrak q) = \dim (B) - \dim (B/\mathfrak q) \]

(Algebra, Lemma 10.111.7). The second equality because $B$ is equidimensional and catenary. Thus $\dim (B/\mathfrak q)$ is independent of the choice of $\mathfrak q$ and we conclude that $B/\mathfrak p B$ is equidimensional of dimension $\dim (B) - \dim (A_\mathfrak p)$. On the other hand, we have $\dim (B/\mathfrak p B) = \dim (A/\mathfrak p) + \dim (B/\mathfrak m_ A B)$ and $\dim (B) = \dim (A) + \dim (B/\mathfrak m_ A B)$ by flatness (see lemma cited above) and we get

\[ \dim (A_\mathfrak p) = \dim (A) - \dim (A/\mathfrak p) \]

for all $\mathfrak p$ in $A$. Applying this to all minimal primes in $A$ we see that $A$ is equidimensional. If $\mathfrak p \subset \mathfrak p'$ is a strict inclusion with no primes in between, then we may apply the above to the prime $\mathfrak p'/\mathfrak p$ in $A/\mathfrak p$ because $A/\mathfrak p \to B/\mathfrak p B$ is flat and $B/\mathfrak p B$ is equidimensional, to get

\[ 1 = \dim ((A/\mathfrak p)_{\mathfrak p'}) = \dim (A/\mathfrak p) - \dim (A/\mathfrak p') \]

Thus $\mathfrak p \mapsto \dim (A/\mathfrak p)$ is a dimension function and we conclude that $A$ is catenary. $\square$

Lemma 15.94.4. Let $A$ be a formally catenary Noetherian local ring. Then $A$ is universally catenary.

Proof. We may replace $A$ by $A/\mathfrak p$ where $\mathfrak p$ is a minimal prime of $A$, see Algebra, Lemma 10.104.8. Thus we may assume that $A^\wedge $ is equidimensional. It suffices to show that every local ring essentially of finite type over $A$ is catenary (see for example Algebra, Lemma 10.104.6). Hence it suffices to show that $A[x_1, \ldots , x_ n]_\mathfrak m$ is catenary where $\mathfrak m \subset A[x_1, \ldots , x_ n]$ is a maximal ideal lying over $\mathfrak m_ A$, see Algebra, Lemma 10.53.5 (and Algebra, Lemmas 10.104.7 and 10.104.4). Let $\mathfrak m' \subset A^\wedge [x_1, \ldots , x_ n]$ be the unique maximal ideal lying over $\mathfrak m$. Then

\[ A[x_1, \ldots , x_ n]_\mathfrak m \to A^\wedge [x_1, \ldots , x_ n]_{\mathfrak m'} \]

is local and flat (Algebra, Lemma 10.96.2). Hence it suffices to show that the ring on the right hand side is equidimensional and catenary, see Lemma 15.94.3. It is catenary because complete local rings are universally catenary (Algebra, Remark 10.154.9). Pick any minimal prime $\mathfrak q$ of $A^\wedge [x_1, \ldots , x_ n]_{\mathfrak m'}$. Then $\mathfrak q = \mathfrak p A^\wedge [x_1, \ldots , x_ n]_{\mathfrak m'}$ for some minimal prime $\mathfrak p$ of $A^\wedge $ (small detail omitted). Hence

\[ \dim (A^\wedge [x_1, \ldots , x_ n]_{\mathfrak m'}/\mathfrak q) = \dim (A^\wedge /\mathfrak p) + n = \dim (A^\wedge ) + n \]

the first equality by Algebra, Lemma 10.111.7 and the second because $A^\wedge $ is equidimensional. This finishes the proof. $\square$

reference

Lemma 15.94.6. Let $(A, \mathfrak m)$ be a Noetherian local ring with geometrically normal formal fibres. Then

  1. $A^ h$ is universally catenary, and

  2. if $A$ is unibranch (for example normal), then $A$ is universally catenary.

Proof. By Lemma 15.93.8 the number of branches of $A$ and $A^\wedge $ are the same, hence Lemma 15.93.2 applies. Then for any minimal prime $\mathfrak q \subset A^ h$ we see that $A^\wedge /\mathfrak q A^\wedge $ has a unique minimal prime. Thus $A^ h$ is formally catenary (by definition) and hence universally catenary by Proposition 15.94.5. If $A$ is unibranch, then $A^ h$ has a unique minimal prime, hence $A^\wedge $ has a unique minimal prime, hence $A$ is formally catenary and we conclude in the same way. $\square$


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