Let $A$ be a commutative noetherian ring of finite Krull dimension.


Is the set
$W = \{p \in Spec(A) \mid A\_p \mbox{ is equidimensional}\}$
a dense open subset of $Spec(A)$? 


(I guess dense is trivial, because it contains all minimal primes).


And if yes, can I conclude that this $W$ is covered by open sets $D(f)$ such that the ring $A\_f$ is equidimensional?
(or maybe a smaller dense open subset contained in $W$?)


If it helps, I can also assume that $A$ is catenary, and even universally catenary.



Equidimensional locus of a noetherian ring
This is true if $A$ is catenary. We will use that a catenary Noetherian *local* ring $A$ has a dimension function $d\_A \colon \operatorname{Spec} A \to \mathbf Z$ [Stacks, Tags [02I8](https://stacks.math.columbia.edu/tag/02I8) and [0ECF](https://stacks.math.columbia.edu/tag/0ECF)]; such a function is well-defined up to adding a constant. Then $A$ is equidimensional if and only if $d\_A(\mathfrak p\_1) = d\_A(\mathfrak p\_2)$ for any two minimal prime ideals $\mathfrak p\_1,\mathfrak p\_2$ of $A$.


However, even for $A$ essentially of finite type over a field, it is not true that $W$ can be covered by standard opens that are spectra of equidimensional rings. I construct one such example below.


It seems likely that there are counterexamples if you drop the catenary hypothesis, but it is a bit hard to make examples. The most useful tool is [Doering–Lequain, Thm. B]; see also [Heinrich] for similar arguments to construct interesting containment graphs of primes in Noetherian rings. (My example below is inspired by one of Heinrich's constructions.)


**Definition.** For a collection $I$ of minimal primes of $A$, write $Z\_I$ for the locus of primes $\mathfrak q$ such that there exist $\mathfrak p\_1, \mathfrak p\_2 \in I$ with $\mathfrak p\_1, \mathfrak p\_2 \subseteq \mathfrak q$ and $d\_{A\_{\mathfrak q}}(\mathfrak p\_1) \neq d\_{A\_{\mathfrak q}}(\mathfrak p\_2)$.


If $I$ is the full set of minimal primes of $A$, then simply write $Z = Z\_I$. Then this is the complement of the equidimensional locus $W$. Note also that
$$Z = \bigcup\_{\substack{\mathfrak p\_1,\mathfrak p\_2\\\text{minimal}}} Z\_{\{\mathfrak p\_1,\mathfrak p\_2\}}.$$
We will also write $Z\_{\mathfrak p\_1,\mathfrak p\_2}$ for $Z\_{\{\mathfrak p\_1,\mathfrak p\_2\}}$.


**Lemma.** *Let $A$ be a catenary Noetherian ring, let $\mathfrak p\_1, \mathfrak p\_2 \subseteq A$ be minimal prime ideals, and let $I = \mathfrak p\_1 + \mathfrak p\_2$. Let $\mathfrak q \subseteq \mathfrak r$ be prime ideals in $V(I)$. Then $\mathfrak q \in Z\_{\mathfrak p\_1,\mathfrak p\_2}$ if and only if $\mathfrak r \in Z\_{\mathfrak p\_1,\mathfrak p\_2}$.*


*Proof.* The restriction of the dimension function $d\_{A\_\mathfrak r}$ to $\operatorname{Spec} A\_{\mathfrak q}$ is a dimension function. Thus, we see that $d\_{A\_{\mathfrak r}}(\mathfrak p\_1) \neq d\_{A\_{\mathfrak r}}(\mathfrak p\_2)$ if and only if $d\_{A\_{\mathfrak q}}(\mathfrak p\_1) \neq d\_{A\_{\mathfrak q}}(\mathfrak p\_2)$. $\square$


**Corollary.** *In the situation of the lemma, $Z\_{\mathfrak p\_1,\mathfrak p\_2}$ is an open and closed subset of $V(I)$.*


*Proof.* Note that $Z\_{\mathfrak p\_1,\mathfrak p\_2} \subseteq V(\mathfrak p\_1) \cap V(\mathfrak p\_2) = V(I)$ by definition of $Z\_{\mathfrak p\_1,\mathfrak p\_2}$. Moreover, the lemma shows that $Z\_{\mathfrak p\_1,\mathfrak p\_2}$ is stable under both generalisation and specialisation of points within $V(I)$. This implies it is open and closed in $V(I)$.


(For example, if $\mathfrak q \in Z\_{\mathfrak p\_1,\mathfrak p\_2}$ and $\mathfrak r \in V(I) \setminus Z\_{\mathfrak p\_1,\mathfrak p\_2}$, then the closed sets $V(\mathfrak q)$ and $V(\mathfrak r)$ have to be disjoint, showing that $D(\mathfrak q)$ is an open neighbourhood of $\mathfrak q$ in $Z\_{\mathfrak p\_1,\mathfrak p\_2}$ since $\mathfrak r \in V(I)\setminus Z\_{\mathfrak p\_1,\mathfrak p\_2}$ was arbitrary. This shows $Z\_{\mathfrak p\_1,\mathfrak p\_2}$ is open in $V(I)$, and similarly the complement $V(I) \setminus Z\_{\mathfrak p\_1,\mathfrak p\_2}$ is open in $V(I)$.) $\square$


**Corollay.** *Let $A$ be a catenary Noetherian ring. Then the equidimensional locus $W \subseteq \operatorname{Spec} A$ is a dense open.*


*Proof.* By definition, the complement $Z = \operatorname{Spec} A \setminus W$ is the finite union of $Z\_{\mathfrak p\_1,\mathfrak p\_2}$. By the corollary above, each $Z\_{\mathfrak p\_1,\mathfrak p\_2}$ is closed in $\operatorname{Spec} A$, hence so is their union. Then $W$ is open, and it is dense because it contains the open loci
$$V\Big(\text{only } \mathfrak p\Big) := D\left(\bigcap\_{\substack{\mathfrak q \subseteq A \text{ minimal}\\\mathfrak q \neq \mathfrak p}} \mathfrak q\right)$$
of primes $\mathfrak r$ such that $\mathfrak p$ is the only minimal prime contained in $\mathfrak r$, for any minimal prime $\mathfrak p \subseteq A$. $\square$




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The open set $\bigcup\_{\mathfrak p} V(\text{only } \mathfrak p) \subseteq W$ is clearly covered by spectra of equidimensional rings, since it a disjoint union of irreducible components (we exactly removed all intersections between irreducible components of $\operatorname{Spec} A$). However, it is not always possible to cover $W$ by spectra of equidimensional rings:


**Example.** Let $C = k[u,v,w,x,y]/(uy)$ be the union of $\mathbf A^4\_{u,v,w,x} = V(y)$ and $\mathbf A^4\_{v,w,x,y} = V(u)$ along $\mathbf A^3\_{v,w,x} = V(u,y)$. Let $B = S^{-1}C$, where $S$ is the complement of $(u,y) \cup (u-1,v,w,x) \cup (w,x,y-1)$. Then $B$ is a semilocal ring with three maximal ideals $(u,y)$, $(u-1,v,w,x)$, and $(w,x,y-1)$ of height $1$, $4$ and $3$ respectively. The irreducible components $V(y)$ and $V(u)$ of $\operatorname{Spec} A$ have dimension $4$ and $3$ respectively, and are glued along the closed point $V(u,y)$ that has codimension $1$ in both components.


Finally, let $A = B\_b$ for some element $b \in B$ vanishing at $(u-1,v,w,x)$ and $(w,x,y-1)$ but not $(u,y)$; for example $b = (u-1)(y-1)$. Then $\operatorname{Spec} A$ has irreducible components $V(y)$ and $V(u)$ of dimension $3$ and $2$ respectively, and they meet in $V(u,y)$ which has codimension $1$ in each component.


Since $A$ is a localisation of the equidimensional ring $C$, all localisations $A\_{\mathfrak p}$ are equidimensional. In particular, $W = \operatorname{Spec} A$. However, any open neighbourhood of $V(u,y)$ meets infinitely many curves through $(u-1,v,w,x)$ in the first copy of $\mathbf A^4$ and infinitely many surfaces containing $(w,x,y-1)$ in the second copy of $\mathbf A^4$. In other words, $U$ contains infinitely many closed points of maximal height in each component of $\operatorname{Spec} A$. Therefore, $U$ has one component $U \cap V(y)$ of dimension $3$ and one component $U \cap V(u)$ of dimension $2$. $\square$


What's going on is the following: we can locally extend the dimension function to a function $d \colon \operatorname{Spec} A \to \mathbf Z$ [Stacks, Tag [02IC](https://stacks.math.columbia.edu/tag/02IC)]; in this example we can do it globally because $A$ is essentially of finite type over $k$. The arguments above then show that on each connected component of $W$ we must have $d(\mathfrak p\_1) = d(\mathfrak p\_2)$ for any minimal primes $\mathfrak p\_1, \mathfrak p\_2$. The dimension is recovered as $\max(d(\mathfrak p)-d(\mathfrak q)\ |\ \mathfrak p \subseteq \mathfrak q)$. However, we do not know what the values of the dimension function at closed points is.


In the example above we constructed two components that meet in a point having the same codimension in each (forcing $d(\mathfrak p\_1) = d(\mathfrak p\_2)$), but the two components have closed points of different heights.




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**References.**


[Doering–Lequain] A. M. d. S. Doering; Y. Lequain, [*The gluing of maximal ideals. Spectrum of a Noetherian ring. Going up and going down in polynomial rings*](http://dx.doi.org/10.2307/1998024), Trans. Am. Math. Soc. **260**, p. 583-593 (1980). [Available online through the AMS](https://www.ams.org/journals/tran/1980-260-02/S0002-9947-1980-0574801-5/S0002-9947-1980-0574801-5.pdf).


[Heinrich] K. Heinrich, [*Some remarks on biequidimensionality of topological spaces and Noetherian schemes*](http://dx.doi.org/10.1216/JCA-2017-9-1-49), J. Commut. Algebra **9**.1, p. 49-63 (2017). arXiv:[1403.5814](https://arxiv.org/abs/1403.5814) 


