Triangulations of polygons and stacked simplicial complexes: separating their Stanley–Reisner ideals

Abstract

A triangulation of a polygon has an associated Stanley–Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals and describe their separated models. More generally, we do this for stacked simplicial complexes, in particular for stacked polytopes.

1 Introduction

Triangulations of polygons constitute a basic yet rich topic going into many directions. The most classical fact about these is perhaps that they are counted by the Catalan numbers, [12, Chap. 23]. Their Stanley–Reisner ideals seem hitherto not to have been systematically studied. Here, we get a full understanding of their algebraic and combinatorial nature. Considerably more generally, we do this for the Stanley–Reisner ideals of stacked simplicial complexes.

Example 1.1

Consider the triangulation of the heptagon in Fig. 1. This may be built up step by step from triangles, by successively attaching the triangles

$$\begin{aligned} 127,\ 257,\ 567,\ 245,\ 234. \end{aligned}$$

Each triangle after the first is attached to a single edge of some earlier triangle. This is a type of shelling called a stacking, and every triangulation of a polygon is a stacking. Moreover, to a triangulation of the polygon we may associate a tree (drawn in red in Fig. 1), showing how the triangles are attached to each other.

Fig. 1

The triangulation of the heptagon in Example 1.1, and its associated tree in red (Color figure online)

This gives our two fundamental notions: That of stacking and the associated (hyper)tree.

Let X be a simplicial complex on a set A, i.e., a family of subsets of A such that if $F \in X$ and $G \subseteq F$, then $G \in X$. Let $F_1, F_2, \dots , F_k$ be an ordering of the facets (the maximal faces) of X. We assume that the $F_i$’s all have the same cardinality. Let $X_p$ be the simplicial complex generated by $F_1, \ldots , F_p$.

The sequence $F_1,\dots ,F_k$ is a stacking of X if each $F_p$ is attached to $X_{p-1}$ along a single codimension-one face of $X_{p-1}$. So we may write $F_p = G_p \cup \{v_p\}$ where $G_p$ is a face of $X_{p-1}$ and $v_p$ is not a vertex of $X_{p-1}$. This is a shelling, but a particularly simple kind of shelling, since each $F_p$ is attached to a single codimension-one face, in contrast to a union of one or more such faces. A simplicial complex that has a stacking as above is called a stacked simplicial complex. Such simplicial complexes have appeared in the literature also as “facet constructible complexes” (see [6, 11]). Although they have been previously studied from the point of view of commutative algebra, to the best of our knowledge it was with a different perspective than in this paper. For instance, in [6] the focus is more on the homological invariants and Cohen–Macaulayness of such complexes.

To a stacked simplicial complex X, we associate a (hyper)tree as in the example above. Let V be an index set for the facets of X. For a codimension-one face G of X which is on at least two facets, let $e_G = \{ v \in V \, | \, F_v \supseteq G \}$. This gives a hypergraph on V whose edges are the sets $e_G$. In fact, this hypergraph is a hypertree T: it is connected, each pair of edges intersects in at most one vertex, and there are no cycles. The hypertree T is an ordinary tree, like in Fig. 1, when each codimension-one face is on at most two facets. Then, X is a triangulated ball. In fact, X may then be realized as a stacked polytope (see [13]), and every stacked polytope is of this kind. For the relationship between stacked simplicial complexes and stacked polytopes, we refer to Sect. 4.5 of [11].

Given an (ordinary) tree T, let V be the vertices of T, and E the edge set of T. Let $C\subseteq E \times V$ be the incidence relation consisting of pairs (e, v) such that v is a vertex on the edge e. Let $k[x_{C}]$ be the polynomial ring in the variables $x_{e,v}$ for $(e,v) \in C$. (Indexing the variables this way was suggested by Lars Hällström.) We associate a squarefree monomial ideal I(T) in the polynomial ring $k[x_{C}]$ as follows. Given a pair of vertices v, w of T, there is a unique path between v and w in the tree T:

Associate with the pair of vertices $\{v,w\}$ the monomial $m_{v,w} = x_{e,v} \cdot x_{f,w}$.

The ideal I(T) is the monomial ideal generated by the $m_{v,w}$ as v and w run through all distinct pairs of vertices of V.

We show that the Stanley–Reisner ring of any triangulation of a polygon is obtained from $k[x_{C}]/I(T)$ by dividing out by a suitable regular sequence of variable differences $x_{e,v^\prime } - x_{f,w^\prime }$. More generally, any Stanley–Reisner ring of a stacked simplicial complex is obtained this way. The rings $k[x_{C}]/I(T)$ for trees T are thus the “initial objects” or “free objects” for Stanley–Reisner rings of stacked simplicial complexes. Formulated otherwise, let I be the Stanley–Reisner ring of a stacked simplicial complex. The separated models of I are one or more of the I(T).

Fig. 2

The tree in Example 1.2 on the left, and on the right the triangulated heptagon that is the complex associated to the ideal obtained after taking the quotient in Example 1.2

Example 1.2

Consider the directed tree in Fig. 2. The ideal I(T) is generated by the ten monomials

$$\begin{aligned} m_{12}&= x_{a,1}x_{a,2} \quad m_{13} = x_{a,1}x_{b,3} \quad m_{14}= x_{a,1}x_{c,4} \quad m_{15} = x_{a,1}x_{d,5}\\ m_{23}&=x_{b,2}x_{b,3} \quad m_{24} = x_{c,2}x_{c,4} \quad m_{25} = x_{c,2}x_{d,5} \quad m_{34} = x_{b,3}x_{c,4}\\ \quad \qquad m_{35}&= x_{b,3}x_{d,5} \qquad m_{45}=x_{d,4}x_{d,5}. \end{aligned}$$

Then, I(T) is the Stanley–Reisner ideal of a stacked simplicial complex of dimension 3 (with facets of cardinality 4) with eight vertices and five facets. Dividing out by the variable difference $x_{a,2} - x_{d,4}$, we get the Stanley–Reisner ring of the triangulation of the heptagon $k[x_{C}]/(I(T) + (x_{a,2} - x_{d,4}))$. Figure 2, on the right, shows the triangulation with our new labelings of the vertices.

The ideals I(T) are introduced in [2] where they are shown to be all possible polarizations of the square of the graded maximal ideal $(x_e)^2_{e \in E}$ in $k[x_e]_{e \in E}$. If $I_X$ is the Stanley–Reisner ring of a stacked simplicial complex, we therefore have processes:

Each of the arrows above preserves the graded Betti numbers. Hence, every $I_X$ has the same graded Betti numbers as a second power of a graded maximal ideal $(x_e)_{e \in E}$.

Let $k[x_{C}]_1 = \langle x_{e,v} \rangle _{(e,v) \in C}$ be the linear subspace of one-dimensional forms in the polynomial ring $k[x_{C}]$. A subspace L of this linear space is a regular linear space if it has a basis consisting of a regular sequence of variable differences for $k[x_{C}]/I(T)$. The quotient ring by the space L of linear forms will still be a polynomial ring divided by a monomial ideal. We show the following.

Theorem

5.16. There is a one-to-one correspondence between regular linear spaces for $k[x_{C}]/I(T)$ and partitions of the vertex set V.

In particular for the partition with one part, the whole of V, the regular sequence consists of all variable differences $x_{e,v} - x_{e,w}$ for $e = \{v,w\} \in E$, and the quotient ring is $k[x_e]_{e \in E}/m^2$, where $m = ( x_e )_{e \in E}$ is the irrelevant maximal ideal.

Theorem

6.2. There is a one-to-one correspondence between regular linear spaces for $k[x_{C}]/I(T)$ giving squarefree quotient rings, and partitions of the vertex set V into sets of independent vertices.

These quotient rings give the Stanley–Reisner rings of stacked simplicial complexes.

In [10], the first author gives a one-to-one correspondence between partitions of the vertex set of a tree T into $(r+1)$ independent sets, and partitions of the edge set of T into r sets. We recall this in the appendix. The above may then be reformulated as:

Theorem

6.3. There is a one-to-one correspondence between regular linear spaces for $k[x_{C}]/I(T)$ giving squarefree quotient rings, and partitions of the edge set E(T). Moreover, the dimension of the simplicial complex associated with this quotient Stanley–Reisner ring is one less than the number of parts in the partition.

Example 1.3

Consider Example 1.2 above. The regular linear space L is the space $L = \langle x_{a,2} - x_{d,4} \rangle$. It corresponds to the partitions of vertices and partitions of edges of the tree in Fig. 3. These partitions are, respectively,

$$\begin{aligned} V = \{1,5 \} \cup \{ 2\} \cup \{3\} \cup \{4\}, \qquad E = \{ a,d \} \cup \{b \} \cup \{c\}. \end{aligned}$$

There are three parts in the edge partition, and so the dimension of the associated simplicial complex is one less, the dimension of the triangulated polygon.

Fig. 3

The tree in Example 1.3. The partition of the edge set is illustrated by the different colors (Color figure online)

Finally, we show the following.

Theorem

8.7. There is a one-to-one correspondence between regular linear spaces for $k[x_{C}]/I(T)$ giving squarefree quotient rings whose associated simplicial complex is a triangulated ball, and partitions of the edge set E(T) into sets of independent edges.

In particular, the last two theorems above give that triangulations of simplicial polygons correspond to partitions of the edges of trees T into three parts, each part being a set of independent edges. In particular, only trees T whose maximal vertex degree is 3 arise in this context (which is easy to see directly like in Example 1.1).

The organization of this article is as follows. In Sect. 2, we recall the notions of separating and joining variables in monomial ideals. We develop basic auxiliary results for doing this. We also recall the notion of separated model. In Sects. 3 and 4, we recall basic notions for simplicial complexes. We define stacked simplicial complexes and hypertrees. We show that the separated models of stacked simplicial complexes are the ideals I(T).

Section 5 is the main technical part and gives the combinatorial description of which linear spaces of variable differences are regular linear spaces for $k[x_{C}]/I(T)$. Section 6 describes the regular linear spaces that give squarefree quotient rings. Section 7 describes the ordering relation between partitions of vertices that corresponds to inclusion of regular linear spaces. Lastly, in Sect. 8 we describe those regular linear spaces where the quotient ring is associated with a triangulation of a ball, or equivalently of a stacked polytope. We also describe the Stanley–Reisner ring of the boundary of these polytopes, which are simplicial spheres.

The appendix recalls the correspondence between partitions of vertices of V(T) into independent sets and partitions of edges of E(T).

2 Separations and joins for Stanley–Reisner ideals

We recall the notion of separation for monomial ideals I. It is a converse to the notion of dividing a quotient ring S/J out by a variable difference which is a nonzero divisor. When $I = I_X$ or $J = J_X$ is a Stanley–Reisner ideal, we describe how the simplicial complex X transforms under these processes.

For a set V denote by $k[x_V]$, the polynomial ring in the variables $x_v$ for $v \in V$. With some abuse of notation, for $R \subseteq V$ let $x_R$ denote the monomial $\prod _{r \in R} x_r$. (This should not cause confusion since in the polynomial ring we always surround $x_V$ with square brackets).

2.1 Separating a variable

The following definition is from [9, Sect. 2].

Definition 2.1

Let $V^\prime {\mathop {\longrightarrow }\limits ^{p}} V$ be a surjection of finite sets with the cardinality of $V^\prime$ one more than that of V. Let $v_1$ and $v_2$ be the two distinct elements of $V^\prime$ which map to a single element v in V. Let I be a monomial ideal in the polynomial ring $k[x_V]$ and J a monomial ideal in $k[x_{V^\prime }]$. We say J is a simple separation of I if the following hold:

• 1. The monomial ideal I is the image of J by the map $k[x_{V^\prime }] \rightarrow k[x_V]$.

• 2. Both the variables $x_{v_1}$ and $x_{v_2}$ occur in some minimal generators of J (usually in distinct generators).

• 3. The variable difference $x_{v_1} - x_{v_2}$ is a nonzero divisor in the quotient ring $k[x_{V^\prime }]/J$.

More generally, if $V^\prime {\mathop {\longrightarrow }\limits ^{p}} V$ is a surjection of finite sets and $I \subseteq k[x_V]$ and $J \subseteq k[x_{V^\prime }]$ are monomial ideals such that J is obtained by a succession of simple separations of I, J is a separation of I. If J has no further separation, we call J a separated model (of I).

Let X be a simplicial complex on the set V. This is a family of subsets of V such that $F \in X$ and $G \subseteq F$ implies $G \subseteq F$. The set of $v \in V$ with $\{v\} \in X$ is the support of X. For $R \subseteq V$, the restriction $X_R$ if the simplicial complex on R consisting of all $F \in X$ such that $F \subseteq R$. Denote by $X_{-R}$ the restriction $X_{R^c}$ where $R^c$ is the complement of R in V. The link $\mathrm {lk}_X R$ is the simplicial complex on $R^c$ consisting of all $F \subseteq R^c$ such that $F \cup R \in X$. If $Y \subseteq X$ are simplicial complexes, denote by $X \setminus Y$ the relative simplicial complex, consisting of those $F \in X$ which are not in Y.

Let $I_X$ be the Stanley–Reisner ideal of X, the monomial ideal in $k[x_V]$ whose generators are the monomials $x_R$ for $R \not \in X$. Suppose we use v to separate $I_X$ to an ideal $I_{X^\prime }$ in the polynomial ring $k[x_{V^\prime }]$. Write the minimal set of monomial generators of $I_X$ as ${\mathcal {M}}_0 \cup {\mathcal {M}}_v$, where ${\mathcal {M}}_0$ consists of those that do not contain $x_v$ and ${\mathcal {M}}_v$ of those of the form $x_v\cdot x_R$. The separated ideal $I_{X^\prime }$ will then have minimal generators ${\mathcal {M}}_0 \cup {\mathcal {M}}_{v,1} \cup {\mathcal {M}}_{v,2}$ (sets of monomials in $k[x_{V^\prime }]$). Here, ${\mathcal {M}}_{v,i}$ consists of those minimal generators that contain $x_{v_i}$. There is a bijection between ${\mathcal {M}}_{v,1} \cup {\mathcal {M}}_{v,2}$ and ${\mathcal {M}}_v$ by sending $x_{v_i}\cdot x_R$ to $x_v \cdot x_R$.

2.2 Criteria for separating a variable

Here is a general description of how $I_X$ can be separated using the variable $x_v$.

Proposition 2.2

We may separate $I_X$ using the variable $x_v$ iff the following holds: there is a partition of the faces into two non-empty parts

$$\begin{aligned} X_{-\{v\}} \setminus \mathrm {lk}_X v = {\mathcal {F}}_1 \sqcup {\mathcal {F}}_2 \end{aligned}$$

where each ${\mathcal {F}}_i$ is closed under taking smaller sets in the sense that if $G \subseteq F$ and $F \in {\mathcal {F}}_i$, then either $G \in {\mathcal {F}}_i$ or $G \in \mathrm {lk}_X v$. The facets of the simplicial complex $X^\prime$ in the separated ideal $I_{X^\prime }$ are then obtained from the facets F of X as follows:

• If $F = G \cup \{v\}$, then $G \cup \{v_1,v_2\}$ is a facet of $X^\prime$.

• If F is in ${\mathcal {F}}_1$, then $F \cup \{v_1\}$ is a facet of $X^\prime$.

• If F is in ${\mathcal {F}}_2$, then $F \cup \{v_2\}$ is a facet of $X^\prime$.

Proof

Assume first that $I_{X^\prime }$ is a separation of $I_X$. If $F \in X_{-\{v\}} \setminus \mathrm {lk}_X v$, then $x_v x_F \in I_X$ (otherwise F would be in $\mathrm {lk}_X v$). Let ${\mathcal {F}}_1$ be the set of those F such that $x_{v_1}x_F$ is in $I_{X^\prime }$ and $x_F$ is not in $I_{X^\prime }$, or equivalently in $I_X$. Similarly define ${\mathcal {F}}_2$ as the set of F such that $x_{v_2}x_F$ is in $I_{X^\prime }$ and $x_F$ is not in $I_{X^\prime }$. Let us show that ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$ are disjoint. If both $x_{v_1}x_F$ and $x_{v_2}x_F$ are in $I_{X^\prime }$, then since $x_{v_1} - x_{v_2}$ is a nonzero divisor for $k[x_{V^\prime }]/I_{X^\prime }$, we have $x_F$ in $I_{X^\prime }$ and so in $I_X$. But then F could not have been in X.

Suppose conversely we have the partition ${\mathcal {F}}_1 \sqcup {\mathcal {F}}_2$. Let the ideal $I_{X^\prime }$ be constructed as just before this Sect. 2.2, so it is generated by ${\mathcal {M}}_0 \cup {\mathcal {M}}_{v,1} \cup {\mathcal {M}}_{v,2}$. Let us show that $x_{v_1} - x_{v_2}$ is a nonzero divisor of $k[x_{V^\prime }]/I_{X^\prime }$. Suppose $(x_{v_1} - x_{v_2})x_F$ is in $I_{X^\prime }$. Then, $x_{v_1} x_F$ and $x_{v_2} x_F$ are both in $I_{X^\prime }$. By construction of $I_{X^\prime }$ we must have $x_v x_F$ in $I_X$ (and note it is not a minimal generator). So F is not in $\mathrm {lk}_X v$. If $x_F$ is not in $I_X$ (the opposite of what we want), F is in $X_{-\{v\}}$. So F is in ${\mathcal F}_1$ or ${\mathcal F}_2$. In the first case $F \cup \{v_1\}$ is a face of $X^\prime$, and so $x_{v_1} x_F$ is not in $I_{X^\prime }$, a contradiction. Similarly F is not in $\{\mathcal F\}_2$. Hence $x_F$ is in $I_X$ and so $x_{v_1} - x_{v_2}$ is not a zero divisor. $\square$

For $v \in V$ and X a simplicial complex, let the neighborhood of v be

$$\begin{aligned} N(v) = \{w \mid \{ v,w \} \in X \} \subseteq V. \end{aligned}$$

Note that N(v) is non-empty iff v is in the support of X, in which case $v \in N(v)$.

Corollary 2.3

Let X be a flag simplicial complex on V, i.e., $I_X$ is generated by quadratic monomials. Let v be in the support of X. Suppose $X_{-N(v)} = X_1 \cup X_2$ where $X_1$ and $X_2$ are simplicial complexes supported on disjoint vertex sets $V_1$ and $V_2$. Then, using $x_v$ the ideal $I_X \subseteq k[x_V]$ may be separated to an ideal $I_{X^\prime } \subseteq k[x_{V^\prime }]$. The facets of $X^\prime$ correspond to the facets F of X as follows:

• If $F = G \cup \{v\}$ contains v, then $G \cup \{v_1,v_2\}$ is a facet of $X^\prime$.

• If F is supported on $V_1$, then $F \cup \{v_1\}$ is a facet of $X^\prime$.

• If F is supported on $V_2$, then $F \cup \{v_2\}$ is a facet of $X^\prime$.

Proof

Let $U = N(v) \setminus \{v\}$. The link $\mathrm {lk}_X v$ is supported on $U \subseteq V$. Let $F \in X_{-\{v\}}\setminus \mathrm {lk}_X v$. Write F as a disjoint union $F_0 \cup F_1$ where $F_1 = F \cap U$. Since X is flag, if $F_0 = \emptyset$, we would have F in $\mathrm {lk}_X v$. So $F_0$ is non-empty and we show it is a subset of either $V_1$ or $V_2$. Let $a,b \in F_0$. Then, a is not in the link $\mathrm {lk}_X v$, so a is either in $V_1$ or $V_2$. Similarly with b. So $\{a,b\}$ is in $X_{-N(v)}$ and so in either $X_1$ or $X_2$. Hence, a, b are in the same set $V_i$. The upshot is that $F_0$ is a subset of either $V_1$ or $V_2$.

We then let ${\mathcal {F}}_1$ be the set of those F such that $F_0$ is a subset of $V_1$ and similarly for ${\mathcal {F}}_2$. These will then be disjoint and closed under taking smaller sets. $\square$

Example 2.4

In Fig. 2, in the introduction one may apply the above corollary to the vertex $v = 5$. This gives the separated ideal I(T) of Example 1.2. The vertex $v=5$ is the only vertex we may use to get a separated ideal. These things may also be seen by Proposition 2.2.

2.3 Criteria for joining variables

We present here basic results on dividing out a Stanley–Reisner ring by variable differences.

Let X be a simplicial complex on a set V, and F a facet of X. Then, for the algebraic set A(X) in the affine space ${{\mathbb {A}}}^V_k$ defined by the Stanley–Reisner ideal $I_X \subseteq k[x_V]$, the facet F corresponds to the linear space A(F) in ${{\mathbb {A}}}_k^V$ where all coordinates $x_v = 0$ for $v \not \in F$, while the $x_v$ for $v \in F$ may take arbitrary values.

For $v_1,v_2 \in V$, let $V_1 = (V \setminus \{v_1,v_2 \}) \cup \{v\}$. The natural map $V \rightarrow V_1$ sending $v_1, v_2 \mapsto v$ gives a surjection of polynomial rings $k[x_V] \rightarrow k[x_{V_1}]$. Let the ideal I be the image of $I_X$. Then, I may or may not be squarefree. If I is squarefree, we say that $x_{v_1} - x_{v_2}$ cuts squarefree. Then, let $I = I_{X_1}$ where $X_1$ is the associated simplicial complex. We then have a commutative diagram of algebraic sets:

Let $e_v$ be the point in affine space ${{\mathbb {A}}}^V_k$ where $x_v$ takes value 1 and the other variables value 0. The map $\phi$ above sends

$$\begin{aligned} \sum _{i \in V_1} a_i e_i \mapsto a_v e_{v_1} + a_v e_{v_2} + \sum _{i \ne v} a_ie_i. \end{aligned}$$

(1)

Lemma 2.5

Let X be a simplicial complex on a set V.

• a. A variable difference $x_{v_1} - x_{v_2}$ where $v_1, v_2 \in V$ is a nonzero divisor for $S/I_X$ iff for each facet F of X, at least one of the variables $v_1$ or $v_2$ is in F.

• b. The ideal I is squarefree iff whenever $F \cup \{v_1\}$ and $F \cup \{v_2\}$ are faces of X, then $F \cup \{v_1,v_2 \}$ is a face of X.

• c. Let $F_1, \ldots , F_r$ be the facets of X. If the difference $x_{v_1} - x_{v_2}$ is a nonzero divisor and cuts squarefree, the facets of $X_1$ are $G_1, \ldots , G_r$ where:

  • If $F_i$ contains exactly one of $v_1$ and $v_2$, then $G_i = F_i \setminus \{v_1, v_2 \}$.

  • If $v_1, v_2$ are both in $F_i$, then $G_i = (F_i \setminus \{v_1,v_2 \}) \cup \{v \}$.

Proof

• a. The associated primes of $I_X$ are the ideals generated by variables $(x_v)_{v \not \in F}$, one such ideal for each facet F. The variable difference is a nonzero divisor iff it is not in any of these ideals. This means that never both $v_1$ and $v_2$ are in such an ideal, or equivalently never both $v_1$ and $v_2$ are outside of a facet F.

• b. The ideal I is squarefree iff there is no minimal generator $x_{v_1}x_{v_2}x_F$ of $I_X$. But having such a generator means having faces $F \cup \{v_1 \}$ and $F \cup \{v_2\}$ but not a face $F \cup \{v_1, v_2 \}$.

• c. This follows by (1) above.

$\square$

When $S/I_X$ is a Cohen-Macaulay ring, a sequence of linear forms $\ell _1, \ldots , \ell _r$ is a regular sequence for $S/I_X$ iff for every facet F, it cuts down A(F) successively by one dimension for every $\ell _k$ [5].

Corollary 2.6

Let X be a Cohen–Macaulay simplicial complex on a set V. Let B be a forest on V, and denote $B_1, \ldots , B_m$ the trees in B and $V_i$ the support of $B_i$ for each i. Then, $\{ x_v-x_w \mid \{v,w \} \text { edge of } B\}$ is a regular sequence for $S/I_X$ iff for each facet F and each $V_i$, at most one of the vertices of $V_i$ is not in F.

Proof

A facet F of X gives the irreducible component A(F) of the algebraic set associated with X. When cutting down A(F) by the sequence of variable differences associated with the edges of $B_i$ we have:

• If some $u \in V_i$ is not in F, the coordinate $x_u = 0$ on points in F. If $\{u,v\}$ is an edge in $B_i$, then $v \in F$, and when dividing out by this variable difference, we will have $x_v = 0$ on the resulting facet. Continuing dividing out by edges in $B_i$, this reduces dimension by $|V_i| - 1$.

• If $V_i \subseteq F$, all coordinates $x_v$ for $v \in V_i$ become equal. This again reduces dimension by $|V_i| - 1$.

Hence, using the edges of the forest B the linear space A(F) is cut down to a linear space whose dimension is $\sum _i (|V_i|-1)$ less than A(F) for each facet F of X. But then the set of variable differences is a regular sequence.

Conversely assume the sequence is regular. If there are $V_i$ and F with $\{ v^\prime ,v^{\prime \prime } \} \subseteq V_i \setminus F$, then the variable differences $x_v - x_w$ associated with edges $\{v,w \}$ in $B_i$ would only give the restrictions $x_v = 0$ for $v \in V_i \cap F$. This only cuts down dimension at most by $|V_i| - 2$, contrary to the sequence being regular. $\square$

3 Stacked simplicial complexes

Let X be a simplicial complex on a set A. In the previous section, we used V for the vertex set of X, but in the sequel we reserve V for the vertex set of the hypertree T associated with the stacked simplicial complex X. In other words, V is an index set for the facets of X.

We show that $I_X$ may be successively separated to an ideal $I_{X^\prime }$, where $X^\prime$ is a stacked simplicial complex of dimension two less than the number of facets |V|. (See Fig. 4 for two examples of such $X^\prime$).

3.1 Stacked simplicial complexes and associated hypertree

A facet F of a simplicial complex X is a leaf if there is a vertex v of F such that F is the only facet containing v. Such a vertex is a free vertex of X. If v is the only free vertex of F, we say F is stacked on $X_{-\{v\}}$. (In general, a free face of a simplicial complex is a face that is not a facet and that lies on exactly one facet, see for instance [16]. The term “leaf” is quite standard in graph theory, and less common in the setting of simplicial complexes, but see for instance [8]).

Definition 3.1

A pure simplicial complex (i.e., where all the facets have the same dimension) is stacked if there is an ordering of its facets $F_0, F_1, \ldots , F_k$ such that if $X_{p-1}$ is the simplicial complex generated by $F_0, \ldots , F_{p-1}$, then $F_p$ is stacked on $X_{p-1}$.

Remark 3.2

This is a special case of shellable simplicial complexes, see [15, Sect. 8.2]. It is not the same as the notion of simplicial complex being a tree as in [8], even if the tree is pure. Rather the notion of stacked simplicial complex is more general. For instance, the triangulation of the heptagon given in Example 1.1, is not a tree in the sense of [8], since removing the triangles 234 and 257 one has no facet which is a leaf.

Remark 3.3

Stacked simplicial complexes are flag complexes. Every minimal nonface is an edge. Equivalently, the Stanley–Reisner ideal is generated by quadratic monomials.

A hypergraph is an ordered pair $H=(V,E)$ where V is a set and E is a collection of subsets of V such that no $e\in E$ is contained in another $e'\in E$. The elements of V are called the vertices of H, and the elements of E are called the edges of H. A hypergraph H is a hypertree if

• (1) Any two edges intersect in either one or zero elements,

• (2) H is connected, i.e., for any two vertices v and w in V there is a sequence $e_1,\dots ,e_m$ of edges of H with $v \in e_1$ and $w \in e_m$ and such that for every $i\in \{1,\dots ,m-1\}$ one has $e_i\cap e_{i+1}\ne \emptyset$, and

• (3) H has no cycle, i.e., no sequence of distinct vertices $v_0, v_1, \ldots , v_n$ save $v_n = v_0$, with $n \ge 3$ such that each pair $\{v_{i-1},v_i\}$ is contained in an edge, but no triple $\{v_{i-1},v_i,v_{i+1}\}$ is contained in an edge.

If $T^\prime$ and T are hypertrees on the same vertex set, $T^\prime$ is a refinement of T if

• (1) Every edge of $T^\prime$ is contained in an edge of T, and

• (2) Every edge of T is a union of edges of $T^\prime$.

Definition 3.4

Let X be a stacked simplicial complex with facets $F_v$ indexed by a set V. We associate a hypertree to X on the vertex set V. For each codimension-one face G, let $e_G = \{ v \in V \mid F_v \supseteq G \}$. The edge set of the hypertree is $E = \{ e_G \mid |e_G| \ge 2 \}$, the set of those $e_G$ containing at least two facets.

The simplicial complex X is a triangulated ball iff its associated hypertree T is an ordinary tree, [4, Theorem 11.4]. It can then be realized as a stacked polytope. Such polytopes are extremal in the following sense: they have the minimal number of faces, given the number of vertices (see [3]).

Observation 3.5

Let X be a stacked simplicial complex which is a cone with p vertices in the cone apex. Thus, X is a join $X_1 * \Delta _p$ where $\Delta _p$ is a simplex on p elements and $X_1$ is not a cone. Then, $X_1$ is also stacked and both X and $X_1$ have the same associated hypertree.

3.2 Separating stacked simplicial complexes

Lemma 3.6

Let X be a stacked simplicial complex of dimension d with hypertree T. If T has $\ge d+3$ vertices, then using the procedure of Corollary 2.3, X may be separated to a simplicial complex $X^\prime$ which is also stacked, and whose hypertree $T^\prime$ is a refinement of T.

Proof

Note that if $d = 0$, then X is a collection of $\ge 3$ vertices and the hypertree T has one edge, the set of all facet indices V. By Corollary 2.3, X may be separated.

Let $d \ge 1$ and $F_0, \dots , F_k$ be a stacking order for X. The facet $F_k$ is stacked on some previous facet $F_p$, with $p<k$. Let v be the vertex of $F_p\setminus F_k$, w the vertex of $F_k \setminus F_p$, and $G = F_p \cap F_k$. Then, $X_{-N(v)}$ contains $\{w \}$ as a component. If there are other components, we may apply Corollary 2.3.

Suppose then $\{w \}$ is the only component. Then, $X_{-\{w\}}$ must be a cone over v. Let Y be the link $\mathrm {lk}_X v$. It is stacked, of dimension $d-1$ and has $\ge d+2$ facets. By induction, we may use Corollary 2.3 and separate Y, using an element $v^\prime$, to $Y^\prime$ whose tree is a refinement of that of Y. Let

$$\begin{aligned} V_1 \cup V_2 = (V \setminus \{v,w \}) \setminus N_Y(v^\prime ) \end{aligned}$$

be the partition given in Corollary 2.3.

• 1. Suppose $N(v^\prime )$ (the neighborhood considered in X) contains w. Then, $X_{-N(v^\prime )}$ has two components, supported on, respectively, $V_1$ and $V_2$, and we may apply Corollary 2.3.

• 2. Suppose $N(v^\prime )$ contains G and not w. Then, $X_{-N(v^\prime )}$ has components $\{w \}$ together with at least one other component and we may again apply Corollary 2.3.

• 3. Suppose $N(v^\prime )$ does not contain w nor G. Then, there is $u \in G$ not in $N(v^\prime )$. So u is in, say $V_1$. Then, $X_{-N(v^\prime )}$ may be written as a disjoint union $X_1 \cup X_2$, with $X_2$ supported on $V_2$ and $X_1$ supported on $V_1 \cup \{w\}$. Again, we may apply Corollary 2.3.

Let us now show that $T^\prime$ is a refinement of T. Let G be a codimension-one face of X contained in two or more facets $F_i$ for $i \in D$, so D is an edge in T. The variable $x_{v^\prime }$ is the variable used in the separation.

• If G contains v, write $G = G^0 \cup \{v^\prime \}$, and then $F_i$ is $F_i^0 \cup \{v^\prime \}$. Then, $G^0 \cup \{ v^\prime _1, v^\prime _2 \}$ is a codimension-one face in facets $F_i^0 \cup \{ v^\prime _1, v^\prime _2 \}$ for $i \in D$. So D is still an edge in $T^\prime$.

• Suppose G does not contain v. Let $D_1 \subseteq D$ index all $F_i$ in ${\mathcal {F}}_1$ containing G and similarly define $D_2$. There might also be a facet $F = G \cup \{v^\prime \}$ containing G, in which case we extend both $D_1$ and $D_2$ with the index of this facet. Then, $D_1$ and $D_2$ are edges of $T^\prime$ and they have at most one vertex in common.

$\square$

Proposition 3.7

Let X be a stacked simplicial complex of dimension d which is not a cone, and let T be the associated hypertree.

• a. T has $\ge d+2$ vertices,

• b. If T has an edge of cardinality $\ge 3$, then T has $\ge d+3$ vertices

• c. If T has $\ge d+3$ vertices, X may be separated to a simplical complex $X^\prime$ whose tree $T^\prime$ is a refinement of T.

• d. If T is an (ordinary) tree with $d+2$ vertices, then X is inseparable and the isomorphism class of X is uniquely determined by T.

Example 3.8

For $d = 2$, Fig. 4 shows the two stacked simplicial complexes of dimension 2 with four facets. The corresponding trees are also drawn in red.

Fig. 4

The two 2-dimensional complexs with four facets, as in Example 3.8, with their corresponding trees in red (Color figure online)

Proof of Proposition 3.7

a,b. Let $F_0, \ldots , F_k$ be a stacking order of facets. Let $X_p$ be the complex generated by $F_0, \ldots , F_p$. Let $C_p = \cap _{i = 0}^p F_i$ and $G_p$ be the codimension-one face of $F_p$ which attaches it to $X_{p-1}$. Then, for $p \ge 1$, $C_p= \cap ^p_{i = 1} G_i$ and $C_{p} = C_{p-1} \cap G_{p}$. Note $G_{p}$ has codimension one in $F_{p-1}$ and $C_{p-1} \subseteq F_{p-1}$. But then $C_p$ has cardinality

$$\begin{aligned} |C_p| = |C_{p-1} \cap G_p| \ge |C_{p-1}| - 1. \end{aligned}$$

Since $|C_0| = d+1$, we get $|C_p| \ge d+1-p$ and so if X is not a cone, $k \ge d+1$. If T has an edge of cardinality $\ge 3$, some $G_p$ equals some $G_r$ for $r < p$. Then, $C_{p-1} \subseteq G_r \subseteq G_p$ and we get $C_{p-1} = C_p$. Thus, $|C_q| \ge d+2-q$ for $q \ge p$, and so if X is not a cone, $k \ge d+2$.

c. This is shown in Lemma 3.6.

d. Let X have associated tree T. Label the vertices of T with $\{0,1, \ldots , d+1\}$.

We assume the labeling is such that the induced subgraph on [0, p] is always a tree for $p = 0, \ldots , d+1$. Then, the corresponding ordering $F_0, F_1, \ldots , F_{d+1}$ of the facets of X is a stacking order.

Let Y be another stacked simplicial complex with tree S isomorphic to T. Transferring the labeling from T, we get a stacking order $G_0, G_1, \ldots , G_{d+1}$ of the facets of Y. Let

$$\begin{aligned} F_{d+1}\setminus F_\ell = \{v \},&\quad&G_{d+1} \setminus G_\ell = \{w \}, \end{aligned}$$

where $\ell <d+1$ is such that $F_{d+1}$ is stacked on $F_\ell$. The following restrictions are cones by part a, since they have $\le d+1$ vertices

$$\begin{aligned} X_{-\{v\}} = X^\prime * \{v^\prime \}, \quad Y_{-\{w\}} = Y^\prime * \{w^\prime \} \end{aligned}$$

and $X^\prime$ and $Y^\prime$ are not cones (since X and Y are not cones). Their trees are obtained from T and S by removing the vertices labeled $d+1$. The $F_i^\prime = \mathrm {lk}_{F_i} v^\prime$ for $i = 0,1,\ldots , d$ form a stacking order for $X^\prime$ and similarly the $G_i^\prime = \mathrm {lk}_{G_i}w^\prime$ form a stacking order for $Y^\prime$.

By induction, there is a bijection between $V \setminus \{v,v^\prime \}$ and $W \setminus \{w, w^\prime \}$ sending the facet $F_i^\prime$ of $X^\prime$ to the facet $G_i^\prime$ of $Y^\prime$. Extend this to a bijection between V and W by $v \mapsto w, \, \, v^\prime \mapsto w^\prime$. Then, the facet $F_i$ is sent to the facet $G_i$ for $i = 0, \ldots , d$.

So consider the facets $F_{d+1}$ and $G_{d+1}$. Let vertex $(d+1)$ of T be attached to vertex $p \le d$. So $F_{d+1}$ is attached by the codimension-one face $F_{d+1} \cap F_p$. But this is $F_{d+1} \setminus \{v\}$ and does not contain $v^\prime$ ($F_{d+1}$ does not contain $v^\prime$ since X is not a cone). So this codimension-one face is $F_p^\prime$. Similarly, $G_{d+1}$ is attached to $G_p^\prime = G_{d+1} \setminus \{w\}$. Since $F_p^\prime$ is sent to $G_p^\prime$, the facet $F_{d+1}$ is sent to $G_{d+1}$. $\square$

4 Trees and the associated separated model

Given a tree T, we define the ideal I(T). These ideals are the separated models of stacked simplicial complexes.

Let T be a tree whose set of vertices is V. Let $E = E(T)$ be its set of edges. The incidence relation $C\subseteq E \times V$ is the set of pairs (e, v) such that $v \in e$. It comes with a natural involution $\tau : C\rightarrow C$ sending $(e,v) \mapsto (e,w)$ where $e = \{v,w\}$.

For $v,w \in V$, denote by vTw the unique path from v to w

and let e, f be the edges incident to, respectively, v, w on this path. For a set A denote by $(A)_2$, the set of subsets $\{a_1,a_2\}$ of cardinality 2. From the directed tree T on V, we get a map

$$\begin{aligned} \Psi : (V)_2\rightarrow & {} (C)_2 \nonumber \\ \{v,w \}\mapsto & {} \{(e,v), (f,w) \}. \end{aligned}$$

For a graph G on V, those vertices that are incident to an edge of G are called the vertices of G. The edges $\Psi (E(G))$ give a graph $\Psi G$ whose vertices are in $C$.

• If $G_1$ and $G_2$ have disjoint vertex sets, the same holds for $\Psi G_1$ and $\Psi G_2$.

• If G is a forest, then $\Psi G$ is a forest, since a cycle in $\Psi G$ must come from a cycle in G.

The following is a basic object in this article.

Definition 4.1

Let $k[x_{C}]$ be the polynomial ring whose variables are indexed by the incidence relation $C$. The tree ideal I(T) in $k[x_{C}]$ associated with the tree T is the edge ideal of $\Psi T$. It is generated by the monomials $m_{v,w} = x_{e,v}x_{f,w}$, one monomial for each pair of distinct vertices v, w in V. The edges e and f are incident to v and w, respectively, on the path vTw.

These tree ideals are introduced in [2, Sect. 5] (but in a slightly less conceptual setting by indexing the variables by $E \times \{0,1\}$). They are shown to be all the possible separated models for the second power $(x_e \, | \, e\in E(T))^2$ of the irrelevant maximal ideal in the polynomial ring $k[x_e]_{ e\in E(T)}$ whose variables are indexed by the edges of T. In particular, the ideals I(T) are Cohen–Macaulay and their graded Betti numbers are precisely those of the graded free resolution of the second power $(x_e \, | \, e \in E(T))^2$ of the graded maximal ideal of $k[x_e]_{ e\in E(T)}$.

The following is given in [2, Sect. 5].

Lemma 4.2

The facets of the simplicial complex associated with the Stanley–Reisner ideal I(T) are

$$\begin{aligned} F_v = \{ (e,w) \in C\mid w \text { vertex on } e \text { closest to } v \}, \end{aligned}$$

one facet for each vertex $v \in V$. The cardinality of these facets is then the number of edges of T.

Corollary 4.3

The ideal I(T) defines the unique non-cone stacked simplicial complex with tree T of dimension $|E|-1$ with $|E|+1$ vertices, given in Proposition 3.7d.

A variation of the map $\Psi$ above is ${\overline{\Psi }}= (\tau )_2 \circ \Psi$ where $(\tau )_2: (C)_2 \rightarrow (C)_2$ is derived from the involution $\tau$ of $C$. Considering the path between v and w

this variation is defined as

$$\begin{aligned} {\overline{\Psi }}: (V)_2\rightarrow & {} (C)_2 \nonumber \\ \{v,w \}\mapsto & {} \{(e,v^\prime ), (f,w^\prime )\}. \end{aligned}$$

We will divide the ring $k[x_{C}]/I(T)$ by the following variable differences:

Definition 4.4

For each pair $\{v,w\}$ in $(V)_2$ let $h_{v,w}$ be the variable difference associated with the edge ${\overline{\Psi }}\{v,w\}$. So

$$\begin{aligned} h_{v,w} = x_{e,v^\prime }- x_{f,w^\prime }. \end{aligned}$$

Note that $h_{w,v} = -h_{v,w}$. Sometimes we write this as $h_e$ where $e = \{v,w\}$ when this sign plays no role.

5 Regular quotients of tree ideals

We describe precisely what sequences of variable differences are regular for $k[x_{C}]/I(T)$. The combinatorial description is in terms of partitions of the vertex set of T, Theorem 5.16.

Definition 5.1

Let T be a tree with vertex set V.

• The sequence of vertices v, u, w is T-aligned if u is on the path in T linking v and w.

• The set $\{v,u,w\}$ is non-aligned for T, if no ordering of them makes a T-aligned sequence.

Example 5.2

Consider the second tree in Fig. 7 in the appendix. The sequence of vertices 1, 4, 8 is T-aligned, and the set $\{1,5,8\}$ is non-aligned for T.

Recall the variable difference $h_{v,w}$ from Definition 4.4. The variables of the polynomial ring $k[x_{C}]$ (see Definition 4.1) are indexed by the incidence relation $C$.

Lemma 5.3

The variable differences in $k[x_{C}]$ which are nonzero divisors for $k[x_{C}]/I(T)$ are those coming from the edges of ${\overline{\Psi }}T$, i.e., the differences $h_{v,w}$.

Proof

This is by Lemma 2.5 and the description in Lemma 4.2 of the facets of the simplicial complex associated with I(T). Given any edge outside of $\mathrm {im}\,{\overline{\Psi }}$, one may find a facet $F_v$ disjoint from this edge. $\square$

The following is the basic obstruction for a sequence of $h_{v,w}$’s to be regular.

Lemma 5.4

Let v, u, w be T-aligned. Then, $h_{v,u}$ and $h_{v,w}$ do not form a regular sequence.

Proof

Let the path vTw be:

We show that $h_{v,w}$ is not $k[x_{C}]/(I(T)+(h_{v,u}))$-regular, by showing that $x_{g,w}$ is in the colon ideal $(I(T)+(h_{v,u})):h_{v,w}$. Indeed

$$\begin{aligned} x_{g,w}h_{v,w}&=x_{g,w}(x_{e,v^\prime }-x_{g,w^\prime })\\&=-x_{g,w}x_{g,w^\prime }+x_{g,w}x_{e,v^\prime }\\&=-x_{g,w}x_{g,w^\prime }+x_{g,w}x_{f,u^\prime }+x_{g,w}h_{v,u} \end{aligned}$$

is an element of $I(T)+(h_{v,u})$. $\square$

The following is straight-forward.

Lemma 5.5

Let $\{ u,v,w\}$ be non-aligned. Then, $h_{v,u} + h_{u,w} = h_{v,w}$.

Definition 5.6

Let T be a tree with vertex set V. Let $U \subseteq V$ and let S be a tree on U (S is a priori unrelated to T). The tree S flows with T if whenever v, u, w are T-aligned vertices with $v,u,w \in U$, then v, u, w are S-aligned.

Example 5.7

The tree T in Fig. 5 has black edges and seven vertices. The trees S are drawn in red. In the first case, $U = \{2,3,4,6\}$. The sequence of vertices 2, 3, 4 is T-aligned but not S-aligned, so S does not flow with T. In the second case, $U = \{2,4,5,7\}$ and 4, 5, 7 is a T-aligned and S-aligned sequence. This tree S flows with T.

Fig. 5

The trees in Example 5.7

Lemma 5.8

Let $U \subseteq V$ and let S and T be trees with vertex sets U and V, respectively. Then, S flows with T iff whenever $\{v,w\}$ is an edge in S, there is no $u \in U\setminus \{v,w\}$ such that v, u, w are T-aligned.

Proof

Let S flow with T, and let $\{v,w\}$ be an S-edge. If there is u such that v, u, w are T-aligned, then v, u, w would be S-aligned, which is not the case since $\{v,w\}$ is an edge in S.

Conversely suppose the condition holds for edges in S. Let v, u, w be vertices in U which are T-aligned, so $\{v,w\}$ is not an edge of S. Suppose the path vSw does not contain u. We argue by induction on the length $\ell _S(v,w)$ of vSw that this is not possible. Since $\ell _S(v,w) \ge 2$ let $r \in U$ on vSw be distinct from v, w (note that $r \ne u$). Then, $\ell _S(v,w) > \ell _S(v,r)$ and $\ell _S(v,w) >\ell _S(r,w)$.

Consider in T a path p from r to a vertex on the path vTw. We may assume only the end vertex of p is on vTw. If p first hits vTw in the path segment vTu, then r, u, w are T-aligned and with the path rSw being such that $\ell _S(r,w) < \ell _S(v,w)$. By induction, this situation is not possible. The case when p first hits vTw in uTw is similar. $\square$

Corollary 5.9

For any $U \subseteq V$, there is a tree S with vertices U flowing with T.

Proof

Let $v \in V$. Consider v as a center from which the tree T branches out. Let $U_0$ be the subset of U consisting of $w \in U$ such that the path vTw contains no other vertex in U than w (in particular, if $v \in U$ then $U_0 = \{v\}$).

Now, define S to be the tree whose edges are:

• Pairs $\{u,w\} \subseteq U$ where (1) v, u, w are T-aligned (we allow $v = u$ if $v \in U$) and (2) the path uTw intersects U only in $\{u,w\}$.

• Give the vertices in $U_0$ a total order. If $u,w \in U_0$ are successive, let $\{u,w\}$ be an edge in S.

The tree S fulfills the criterion of the lemma above, and hence flows with T. $\square$

Definition 5.10

If S is a tree on the vertex set $U \subseteq V$, let L(S) be the linear space with basis the $h_e = h_{v,w}$ where $e = \{v,w\}$ are the edges of S. If L(S) has a basis that is a regular sequence of variable differences for $k[x_{C}]/I(T)$, we say that L(S) is a regular linear space. (Equivalently some basis or any basis of L(S) is a regular sequence).

Lemma 5.11

Let S be a tree with vertex set U, and assume that only the end vertices v and w of the path vTw are contained in U.

• If S flows with T, then $h_{v,w} \in L(S)$.

• If L(S) is a regular linear space, then $h_{v,w} \in L(S)$.

Proof

Let $v = v_0,v_1, \ldots , v_n = w$ be a path in S of length $n \ge 2$. Then, by assumption $v_1, \ldots , v_{n-1}$ are not in vTw. If the $v_i$-incident edges on $v_{i-1}Tv_i$ and $v_iTv_{i+1}$ are always distinct, the paths would splice to give the unique path from v to w. This cannot be the case since this path vTw only has the end vertices in U. Hence, for at least one $v_p, 1 \le p \le n-1$, these two $v_p$-incident edges are equal. We have three possibilities:

• (1) $v_p, v_{p-1},v_{p+1}$ are T-aligned,

• (2) $v_p, v_{p+1},v_{p-1}$ are T-aligned,

• (3) $\{v_{p-1},v_p,v_{p+1}\}$ is non-aligned for T.

For case (1), if S flows with T, this would give that $v_p,v_{p-1},v_{p+1}$ are S-aligned, which is not the case since $v_{p-1},v_p,v_{p+1}$ are S-aligned. Similarly, the second case (2) is excluded. If L(S) is regular the first and second cases are also excluded by Lemma 5.3. Hence, only the last possibility (3) is left.

If S flows with T, then if $v_{p-1}Tv_{p+1}$ contains an element of U, distinct from $v_{p-1}$ and $v_{p+1}$, such an element would be either on $v_{p-1}Tv_p$ or on $v_pTv_{p+1}$, and distinct from $v_p$. But this is not the case by Lemma 5.8 since $v_{p-1},v_p$ and $v_p, v_{p+1}$ are edges in S. Then, we take out the edge $\{ v_{p-1}, v_p\}$ from S and take in the edge $\{v_{p-1},v_{p+1}\}$ to get a new tree $S^\prime$ which still flows with  T by Lemma 5.8. By induction on the length $\ell _{S^\prime }(v,w)$, we have $h_{v,w} \in L(S)$.

If L(S) is a regular linear space, then we again replace S with $S^\prime$. Due to Lemma 5.5, we have $L(S) = L(S^\prime )$ and again we get $h_{v,w} \in L(S)$. $\square$

Proposition 5.12

Let S be a tree on $U \subseteq V$. If L(S) is a regular linear space, then S flows with T.

Proof

Let $\{v,w\}$ be an edge in S, so $h_{v,w} \in S$. Suppose v, u, w are T-aligned vertices in U. If we show this is not possible, then S flows with T by Lemma 5.8. Choose u as close as possible to v, so vTu only contains v and u from U. By Lemma 5.11, $h_{v,u}$ is in L(S). So both $h_{v,w}$ and $h_{v,u}$ are in L(S). By Lemma 5.4, these two elements do not form a regular sequence, contradicting the fact that L(S) is regular. Hence, there can be no u such that v, u, w are T-aligned. So S flows with T. $\square$

Lemma 5.13

For any two trees R and S on U flowing with T, one has $L(R) = L(S)$. Thus, U determines a unique regular linear space, denoted L(U).

Proof

Let $\{v,w\}$ be an R-edge. We show $h_{v,w} \in L(S)$. Since R flows with T, the path vTw does not contain any elements of U save the end vertices. Since S flows with T, Lemma 5.11 gives $h_{v,w} \in L(S)$. $\square$

Lemma 5.14

Let G be a graph on vertex set V (with G a priori unrelated to T).

• a. If $\{h_e\}_{e\in G}$ is a regular sequence for $k[x_{C}]/I(T)$, then G is a forest.

• b. If G is a forest consisting of the trees $S_1, \dots , S_r$, then $\{h_e\}_{e \in G}$ is a regular sequence iff each $\{ h_e\}_{e \in S_i}$ is a regular sequence.

Proof

a. It is enough to show that if G is a cycle O, then $\{h_e\}_{e \in O}$ is not a regular sequence. Denote by L(O) their linear span, and let the cycle be $v_0, v_1, \ldots , v_{n-1}, v_n = v_0$ of length n.

We now use induction on the length n of the cycle to show that L(O) cannot be regular linear space. Not every sequence $v_{i-1}v_iv_{i+1}$ is T-aligned for $i = 1, \ldots n-1$ since $v_0 = v_n$ and T is a tree. Suppose $v_{p-1}v_pv_{p+1}$ is not T-aligned. If say $v_pv_{p-1}v_{p+1}$ are T-aligned, then $h_{v_p,v_{p-1}}$ and $h_{v_p,v_{p+1}}$ do not form a regular sequence, against the assumption. By the same reason, $v_pv_{p+1}v_{p-1}$ are not T-aligned. Hence, $\{v_{p-1},v_p,v_{p+1}\}$ is non-aligned. Then, $h_{v_{p-1},v_{p+1}}$ is $h_{v_{p-1},v_p} + h_{v_p,v_{p+1}}$. Take the edges $\{v_{p-1},v_p\}$ and $\{v_p,v_{p+1}\}$ out from the cycle O and take in the edge $v_{p-1},v_{p+1}$ to make a new cycle $L(O^\prime ) \subseteq L(O)$. By induction, $L(O^\prime )$ is not a regular linear space and so neither is L(O).

b. Suppose each $S_i$ gives a regular sequence. This sequence is determined by the edges of ${\overline{\Psi }}S_i$, and this is a forest. By Corollary 2.6, this is equivalent to each tree in ${\overline{\Psi }}S_i$ giving a regular sequence. But the disjoint union of the trees in the ${\overline{\Psi }}S_i$ are precisely the trees in ${\overline{\Psi }}S$. Hence, Corollary 2.6 gives the result. $\square$

The following is the converse of Proposition 5.12:

Proposition 5.15

Let S be a tree on $U \subseteq V$. If S flows with T, then L(S) is a regular linear space.

Proof

By Lemma 5.13 above, if U is the vertex set of S, we may choose S to be any tree on U that flows with T.

Let $v \in V$. Consider the face

$$\begin{aligned} F_v = \{ (e,w^\prime ) \mid w^\prime \text { vertex on } e \text { closest to } v\}. \end{aligned}$$

Let $U \subseteq V$ and define the tree S flowing with T with vertices in U as in Corollary 5.9. This tree comes with two types of edges:

• Edges $\{u,w\}$ where u and w are two successive elements in the ordering of $U_0$. Then, ${\overline{\Psi }}\{u,w\} = \{(f,u^\prime ), (g,w^\prime )\}$ where $f = \{u,u^\prime \}$ is the edge on uTv going out from u and similarly $g = \{w,w^\prime \}$ the edge on vTw going out form w. These $\{(f,u^\prime ),(g,w^\prime )\}$ give a tree $T_0$ with vertices from the incidence relation $C$ (actually a line graph).

• Edges $\{u^\prime ,w\}$ where $u^\prime$ is the element in U on the path vTw closest to w. If $f = \{w^\prime , w\}$ and $g = \{u^\prime , u \}$ are the edges on $u^\prime T w$, one has ${\overline{\Psi }}\{u^\prime ,w\} = \{(g,u),(f,w^\prime )\}$. Each such w gives a unique $u^\prime$, but one $u^\prime$ may correspond to several g’s and w’s. For each pair $u^\prime ,g$, these edges form a tree $T_{u^\prime ,g}$, a star, with vertices from $C$.

The trees $T_0$ and $T_{u^\prime ,g}$ (with vertices from $C$) are all disjoint. Together the edges of these trees give all variable differences $h_{v,w}$ for $\{v,w \}$ an S-edge. The vertices of $T_0$ are contained in $F_v$. Each $T_{u^\prime ,g}$ has all its vertices save (g, u) contained in $F_v$. By Corollary 2.6, the linear space L(S) is regular. $\square$

Theorem 5.16

There is a one-to-one correspondence between regular linear spaces for $k[x_{C}]/I(T)$ and partitions Q of the vertex set V. If the partition of V is $Q : U_0 \sqcup U_1 \sqcup \cdots \sqcup U_r$, then this regular linear space is

$$\begin{aligned} L(Q) = L(U_0) \oplus \cdots \oplus L(U_r). \end{aligned}$$

Proof

By Lemma 5.13, each $U_i$ determines a unique linear space $L(U_i)$. If $S_i$ is a tree on $U_i$ flowing with T, then $L(S_i) = L(U_i)$. Let $S = \cup _{i=0}^rS_i$. By Lemma 5.14, the edges of ${\overline{\Psi }}S$ give a regular sequence. This regular sequence is a basis for L(Q).

Conversely if L is a regular linear space generated by the regular elements $\{h_e \}_{e \in G}$ for some graph G on V, by Lemma 5.14 the graph G decomposes into a forest and we get a partition of V where each $U_i$ is the vertex set of each tree in the forest. (The vertices v of V not incident to any edge of G give singletons $\{v\}$ in the partition). $\square$

Corollary 5.17

The length of the longest regular sequence of variable differences for $k[x_{C}]/I(T)$ is |E|. Such a sequence corresponds to the trivial partition of V with only one part, the set V itself. The corresponding tree that flows with T is just T itself. Hence, this regular sequence is given by $\{h_e\}_{e \in E}$ and the quotient ring is $k[x_E]/(x_e \, | \, e \in E)^2$.

6 Squarefree quotients

We determine what regular linear spaces give quotient rings of $k[x_{C}]/I(T)$ whose associated ideals are squarefree. These are the Stanley–Reisner rings of stacked simplicial complexes. Let T be a tree with vertices V.

Lemma 6.1

Let $U \subseteq V$ and let S be a tree on U. If S flows with T, then the regular quotient of $k[x_{C}]/I(T)$ by $\{h_e \}_{e \in S}$ is a squarefree monomial ideal iff the vertex set U is an independent vertex set in V for the tree T.

Proof

The following is essential to note: The variables in the quotient ring modulo the sequence $\{h_e\}_{e \in S}$ correspond precisely to the connected components of the graph ${\overline{\Psi }}S$ with vertex set the incidence relation $C$.

If the vertex set U is dependent, say contains end vertices of an edge $e = \{v,w\}$, then we divide out by $x_{e,v} - x_{e,w}$ and the ideal of the quotient ring will contain $x_e^2$ as a generator and so is not squarefree.

Suppose then that U is independent. Let $\{v,w \}$ be a pair of vertices in U. Suppose the associated monomial $x_{e,v}x_{f,w}$ becomes a square after dividing out by the regular sequence. This means that (e, v) and (f, w) are in the same connected component of ${\overline{\Psi }}S$. Let the edge e have vertices $v,v^\prime$ and the edge f vertices $w^\prime , w$. So $v^\prime$ and $w^\prime$ are on the path vTw. Removing the edge e from T, we get a component $T_v$ containing v, and similarly removing f from T we get a component $T_w$ containing w.

Any edge in ${\overline{\Psi }}S$ containing (e, v) is the image of an edge $\{{\overline{v}},v^\prime \}$ in S where ${\overline{v}} \in T_v$. Similarly, we have an edge $\{w^\prime , {\overline{w}} \}$ in S where ${\overline{w}} \in T_w$. But since (e, v) and (f, w) are in the same connected component of ${\overline{\Psi }}S$, there must in S be an edge $\{ {\tilde{v}}, {\tilde{w}} \}$ where ${\tilde{v}}$ is in $T_v$ and ${\tilde{w}}$ is in $T_w$.

Then, either $v^\prime$ or $w^\prime$ from U is in the interior of the path ${\tilde{v}}T {\tilde{w}}$. Since S flows with T, this cannot be the case by Lemma 5.8. $\square$

Theorem 6.2

There is a one-to-one correspondence between regular linear spaces for $k[x_{C}]/I(T)$ giving squarefree quotient rings, and partitions of V into sets of independent vertices.

Proof

Suppose we have a squarefree quotient ring. Each part $U_i$ of the partition gives a regular linear space $L(U_i)$. By Lemma 6.1, $U_i$ is independent. Conversely, if we have a partition of V into independent sets $U_i$, let $S_i$ be a tree on $U_i$ flowing with T. The images ${\overline{\Psi }}S_i$ have disjoint vertex sets as i varies. Lemma 6.1 above shows that the quotient is squarefree. $\square$

Using Theorem A.1, the above may equivalently be formulated as follows:

Theorem 6.3

There is a one-to-one correspondence between regular linear spaces for $k[x_{C}]/I(T)$ giving squarefree quotient rings, and partitions of the edge set E.

If P is a partition of the edge set corresponding to the partition Q into independent vertex sets, write $L(P) = L(Q)$.

Corollary 6.4

The length of the longest regular sequence of variable differences giving a squarefree quotient of $k[x_{C}]/I(T)$ is $|E| - 1$. It corresponds to the unique partition of V into two independent sets of V for the tree T. Thus, the associated regular linear space is also unique.

7 Partial order on partitions

If Q and $Q^\prime$ are partitions of the vertex set V of a tree T, we get the linear spaces L(Q) and $L(Q^\prime )$. What does the inclusion relation on linear spaces correspond to on partitions? Since the linear spaces depend on additional structure coming from the tree T, this is not simply refinement of partitions.

7.1 Partitions of the vertex set

Definition 7.1

Let $U^\prime \subseteq U \subseteq V$. Then, $U^\prime$ is convex in U if for every $v,w \in U^\prime$, all vertices on the path vTw that are contained in U are in $U^\prime$.

Note that such a $U^\prime$ may be convex in some U while not being convex in V.

Lemma 7.2

Let $U^\prime$ and U be subsets of V. If $L(U^\prime )$ is a subspace of L(U), then $U^\prime$ is a convex subset of U, or $U^\prime$ is a singleton (then $L(U^\prime ) = 0$). Conversely if $U^\prime \subseteq U$ is a convex subset, then $L(U^\prime )$ is a subspace of L(U).

Proof

Suppose $L(U^\prime )$ is a nonzero subspace of L(U) and there exists $v \in U^\prime \setminus U$. There is another $w \in U^\prime$ such that $h_{v,w} \in L(U^\prime )$. Consider the path vTw in T:

Then, $h_{v,w} = x_{e,v^\prime } - x_{f,w^\prime }$ is in $L(U^\prime )$. If the edge e occurred in some $h_{a,b}$ generating L(U), since $v \not \in U$, one of a or b would have to be $v^\prime$, and $v^\prime \in U$. But then this $h_{a,b}$ would contain $x_{e,v}$ instead of $x_{e,v^\prime }$. Hence, $U^\prime \subseteq U$.

Let us show that $U'$ is convex in U. Let $v,w \in U^\prime$ be such that vTw contains some $u \in U \setminus U^\prime$. By possibly moving v and w closer to u, and u closer to v, we may assume on vTw that v and w are the only vertices in $U^\prime$, and on vTu that v and u are the only vertices in U. But then $h_{v,w} \in L(U^\prime )$ and $h_{v,u}\in L(U)$ by Lemma 5.11. If $L(U^\prime ) \subseteq L(U)$, this could not be the case by Lemma 5.4, since L(U) is regular. Hence, if we have inclusion, $U^\prime$ must be convex in U.

Conversely if $U^\prime$ is convex in U, then letting $S^\prime$ be a tree on $U^\prime$ flowing with T, by Lemma 5.11, for each edge e in $S^\prime$ we have $h_e \in L(U)$. $\square$

The following is immediate from the above.

Theorem 7.3

Let $Q^\prime$ and Q be partitions of V. Then, $L(Q^\prime ) \subseteq L(Q)$ iff each part $U_i$ of Q is a union of parts of $Q^\prime$ which are convex for $U_i$. Write then $Q^\prime \preceq Q$.

In a partition Q of V, if $U_i$ and $U_j$ are parts such that either $U_i$ or $U_j$ is not convex in $U_i \cup U_j$, we say that $U_i$ and $U_j$ are intertwined.

Corollary 7.4

The maximal partitions for the partial order $\preceq$ are the partitions Q such that any two parts $U_i$ and $U_j$ in the partition are intertwined.

Example 7.5

In the introduction, looking at Fig. 3, the partition of vertices in Example 1.3 is not maximal. We may join

$$\begin{aligned} \{1,5\} \cup \{2\} \cup \{3\} \cup \{4\} \preceq \{1,5\} \cup \{2,3,4\}. \end{aligned}$$

The latter vertex partition is maximal since it is intertwined. Also note that the first partition is not $\preceq \{1,2,3,4,5\}$, since $\{1,5\}$ and $\{2\}$ (as well as $\{4\}$) are intertwined.

The partition $\{1,2,3,4,5\}$ corresponds to the quotient ring $k[x_E]/(x_e)^2_{e \in E}$, which is $k[x_E]$ divided by the square of the maximal graded ideal. Hence, this ring is not a quotient ring of $k[x_{C}]/(I(T) + (x_{a,2} - x_{d,4}))$ of Example 1.2, by a regular linear space. (But it is of course a quotient taking a suitable general linear space).

Corollary 7.6

Let ${\overline{V}}$ be the partition of V into singletons. Then, for any partition Q of V the interval $[{\overline{V}},Q]$ with respect to the partial order $\preceq$ is a Boolean lattice.

Proof

Given a subset U of V, we must show that the lattice of partitions of U into convex parts is a Boolean lattice. Let v be extremal in U in the sense that every other vertex of U is on the same side of v, i.e., there is an edge $e = \{v,w\}$ from v such that the path from v to any other vertex of U starts with the edge e. Let $U^\prime = U \setminus \{ v\}$. By induction, the lattice of partitions of $U^\prime$ into convex subsets is a Boolean lattice B. The partitions Q of U into convex subsets are now of two types: either $\{v\}$ is a singleton class, or v and w are in the same class. This gives that the lattice of partitions of U identifies as $B \times \{0,1\}$ and so is Boolean. $\square$

7.2 Partitions of the edge set

If $D^\prime \subseteq D \subseteq E$ are sets of edges of T, we may as above define the notion of $D^\prime$ being convex in D. As above, we may show:

Proposition 7.7

Let $P^\prime$ and P be partitions of E. Then, $L(P^\prime ) \subseteq L(P)$ iff each part $E_i$ of P is a union of parts of $P^\prime$ which are convex for $E_i$. Write then $P^\prime \preceq P$.

Corollary 7.8

The maximal partitions for the partial order $\preceq$ are the partitions P such that any two parts $E_i$ and $E_j$ in the partition are intertwined.

Example 7.9

In the introduction, looking at Fig. 3, the partition of edges in Example 1.3 is not maximal. We may join

$$\begin{aligned} \{a,d\} \cup \{b\} \cup \{c\} \preceq \{a,d\} \cup \{b,c\}. \end{aligned}$$

In the latter partition, the parts are intertwined and so it is maximal. It corresponds to the vertex partition $\{1,5\} \cup \{3,4\} \cup \{2\}$. This vertex partition is also maximal (but that does not necessarily follow from the edge partition being maximal).

Corollary 7.10

Let ${\overline{E}}$ be the partition of E into singletons. Then, for any partition P of E the interval $[{\overline{E}},P]$ with respect to the partial order $\preceq$ is a Boolean lattice.

8 Hypertree of quotients and triangulated balls

We describe the squarefree quotients of $k[x_{C}]/I(T)$ by regular linear spaces whose associated simplicial complex is a triangulated ball. In particular, we describe when we get triangulations of polygons.

Let

$$\begin{aligned} P : E_1 \sqcup E_2 \sqcup \cdots \sqcup E_r \end{aligned}$$

be a partition of the edge set E of the tree T. We may think of the edges of $E_i$ as a color class. The partition corresponds by Theorem A.1 to a partition $V = U_0 \sqcup U_1 \sqcup \cdots \sqcup U_r$ of the vertex set into independent sets of vertices. Let $S = S_0 \cup S_1 \cup \cdots \cup S_r$ where the $S_i$ are trees on $U_i$ flowing with T. The image ${\overline{\Psi }}S$ is a forest, and each ${\overline{\Psi }}S_j$ is a collection of connected components (trees) of ${\overline{\Psi }}S$. Moreover, $L(P) = \oplus _{i=1}^r L(S_i)$. In the sequel, we also write ${\overline{\Psi }}P$ for ${\overline{\Psi }}S$.

Let us describe the variables in the quotient ring $k[x_{C}]/L(P)$ (this is a polynomial ring). These variables identify as subsets of of the incidence relation $C$. Those subsets which contain more than one element arise as follows. For each class $E_i$, consider maximal sets of edges $E_{ij} \subseteq E_i$ such that for every pair of edges f, g in $E_{ij}$, the only edges in $E_i$ on the unique path from f to g are f and g themselves. For given i, two such maximal $E_{ij}$ and $E_{ij^\prime }$ have at most one edge in common. (In fact, the $E_{ij}$’s form the set of edges in a hypertree on $E_i$). If f, g are edges in a $E_{ij}$ with path

then $x_{f,v^\prime } - x_{g,w^\prime }$ is a variable difference in L(P). It gives a class $[x_{f,v^\prime }]$, a variable in $k[x_{C}]/L(P)$. This gives one variable in $k[x_{C}]/L(P)$ for each set $E_{ij}$.

Example 8.1

In Fig. 6, we have a partition of the edges into three color classes. The four red edges give eight red variables in $k[x_{C}]$. The red edges give two maximal sets $\{a,c\}$ and $\{c,e,g\}$, each of which combines into one variable, giving five red variables in the quotient ring $k[x_{C}]/L(P)$.

Fig. 6

The tree in Example 8.1. The partition of the edge set is illustrated by the different colors (Color figure online)

We now describe the facets of the simplicial complex corresponding to the quotient $k[x_{C}]/(I(T) + L(P))$. For each $v \in V$ and color class $E_i$, let $E_i^v$ be the set of edges f in $E_i$ such that on the path from f to v the only edge in $E_i$ is f itself. Then, $E_i^v$ is a maximal set $E_{ij}$ as above and hence gives a variable $x_{E_i^v}$ in $k[x_{C}]/L(P)$. We have $x_{E_i^v} = x_{E_j^w}$ iff $i = j$ and there is no edge from $E_i$ on the path from v to w. Let

$$\begin{aligned} F_v = \{ E_i^v \mid i = 1, \ldots , r \}. \end{aligned}$$

Example 8.2

Consider Fig. 6. The facet $F_v$ of $k[x_{C}]/(I(T) + L(P))$ is of cardinality 3. Its elements are the three maximal sets

$$\begin{aligned} E^v_\text {red} = \{c,e,g\}, \quad E^v_\text {blue} = \{h,b\}, \quad E^v_\text {green} = \{f \}. \end{aligned}$$

Lemma 8.3

The facets of the simplicial complex associated with $k[x_{C}]/(I(T)+L(P))$ are the $F_v$’s, for $v \in V$. In particular, the cardinality of each facet is the number of classes in the partition P.

Proof

This follows by repeated use of Lemma 2.5. $\square$

Lemma 8.4

Let $e = \{v,w\}$ be an edge in T, in the class $E_k$. Then, $E_i^v = E_i^w$ for $i \ne k$. The facets $F_v$ and $F_w$ have a codimension-one face in common. It is the set

$$\begin{aligned} F_e = \{ E_i^v (= E_i^w) \mid i \ne k \}. \end{aligned}$$

Proof

This is clear. $\square$

Lemma 8.5

The facets $F_v$ and $F_w$ have a codimension-one face G in common if and only if the path from v to w has all edges of the same color. Then, for all edges e on this path, the $F_e$ are equal, and this is G. In particular, G is common to all facets $F_u$ for u on this path.

Proof

Suppose the edges on the path are all of the same color red. Let the path be $v = u_0, u_1, \ldots , u_m = w$ with $e_i$ the edge $\{u_{i-1},u_i\}$. Then, for each edge $e_i$

$$\begin{aligned} F_{u_{i-1}} = F_{e_i} \cup \{ {(e_i,u_{i-1})}\}, \quad F_{u_i} = F_{e_i} \cup \{ {(e_i,u_i)}\} \end{aligned}$$

for suitable $F_i$. Since $e_i$ and $e_{i+1}$ are successive red edges, we divide out by the variable difference $x_{e_i,u_i} - x_{e_{i+1},u_i}$ and so $(e_i,u_i)$ identifies with $(e_{i+1},u_i)$. We also have

$$\begin{aligned} F_{u_{i}} = F_{e_{i+1}} \cup \{ {(e_{i+1},u_i)}\}, \quad F_{u_{i+1}} = F_{e_{i+1}} \cup \{ {(e_{i+1},u_{i+1})}\}. \end{aligned}$$

We must then have $F_{e_i} = F_{e_{i+1}}$. Hence, all these $F_{e_i}$ are equal.

Suppose the edges on the path are not of the same color. Suppose going from v to w, there is first a sequence of red edges, the first one being $e = \{ v = u_{0},u_1 \}$ and then eventually a blue edge $f = \{u_i, u_{i+1}\}$.

• The facet $F_v$ contains $(e,u_0)$ of color red. The facet $F_w$ also contains a (class) of a red edge. If this red edge was e, it would have to be $(e,u_1)$. Hence, $(e,u_0)$ is in $F_v$ but not in $F_w$.

• Similarly, the blue $(f,u_i)$ is in $F_v$, and by a similar argument as above, $(f,u_i)$ is not in $F_w$.

• The upshot is that $F_v \setminus F_w$ contains at least two elements, and so $F_v$ and $F_w$ do not intersect in codimension one.

$\square$

Recall that a set of edges in the tree T is independent if no two edges in the set are adjacent. The quotient of $k[x_{C}]/I(T)$ by L(P) is a stacked simplicial complex. It is again a quotient of the polynomial ring $k[x_{C}]/L(P)$. Each part $E_i$ of E is a subforest of T. Let $T_{ij}$ be the trees of this subforest and $V_{ij} \subseteq V$ the support of $T_{ij}$. Let $T^\prime$ be the hypertree whose edges are the sets $V_{ij}$. In particular, note that if P is a partition whose parts $E_i$ consist of independent edges, then each $T_{ij}$ is simply an edge, and so $T^\prime = T$.

Proposition 8.6

Let P be a partition of the edge set of T. The quotient of $k[x_{C}]/I(T)$ by L(P) corresponds to a stacked simplicial complex X whose associated hypertree is $T^\prime$.

Proof

Consider then the tree $T_{ij}$. Let v, u, w be three vertices in $V_{ij}$. If they are T-aligned for some ordering, the facets $F_v, F_u, F_w$ of X have a codimension-one face in common by Lemma 8.5. Suppose $\{u,v,w\}$ are non-aligned. Consider the path from v to w and let e be its last edge. Then, e is also the last edge on the path from u to w. Write $F_w = F \cup \{(e,w)\}$. By the argument of Lemma 8.5, all x on these paths have $F_x$ containing F. We readily get that F is a codimension-one face of every $F_x$ for $x \in V_{ij}$. Thus, each $V_{ij}$ form an edge in the hypertree $T^\prime$ associated with the simplicial complex X. $\square$

Theorem 8.7

There is a one-to-one correspondene between:

• Regular linear spaces giving squarefree quotients of $k[x_{C}]/I(T)$ corresponding to triangulated balls, and

• Partitions P of the edge set E of T into sets of independent edges.

The codimension-one faces of this triangulation which are on two facets are precisely the faces $F_e$ of Lemma 8.4. Let B(T, P) be the ideal generated by the $x_{F_e}= \prod _{a \in F_e} x_a$ for $e \in E(T)$. Then, $I(T) + B(T,P)$ is the Stanley–Reisner ideal in $k[x_{C}]/L(P)$ defining the boundary of this triangulated ball, a triangulated sphere.

Proof

When the edges are partitioned into independent sets, the hypertree $T^\prime$ is an ordinary tree T. And when a stacked simplicial complex gives an ordinary tree T, it is a triangulated ball and may be realized as a stacked polytope.

The only faces on a stacked simplical complex not on the boundary are the codimension one faces which are on at least two faces. This gives the statement about the Stanley–Reisner ideal of the boundary. $\square$

Remark 8.8

In [7], the first author et al. give the construction of large classes of triangulated balls, defined by letterplace ideals of posets. The ideal defining the boundary of triangulated balls is given in a similar way there.

In particular, triangulations of simplicial polygons correspond to partitions of trees T into three parts, each part being a set of independent edges. Thus, only trees T whose maximal vertex degree is 3 arise in this context.

Corollary 8.9

The length of the longest regular sequence of variable differences giving a squarefree quotient of $k[x_{C}]/I(T)$ that corresponds to a triangulated ball is $|E|- \Delta$, where $\Delta$ is the maximal degree of a vertex of T.

Proof

This is because the minimal number of parts in a partition of E(T) into independent edges, the edge chromatic number of the tree T, is the maximal degree of a vertex in T, [1]. $\square$

Appendix A. Partitions of the vertices and edges of a tree

We recall the basic result on trees from [10] on the correspondence between partitions of edges and partitions of vertices into independent sets. Let T be a tree with vertex set V and edge set E. We consider partitions of the vertices

$$\begin{aligned} V = V_1 \sqcup V_2 \sqcup \cdots \sqcup V_r \end{aligned}$$

(2)

into disjoint sets such that each $V_i$ is an independent set of vertices. (This is almost the same as a coloring of vertices, but not quite: The symmetric group $S_r$ acts on colorings by permuting the color labels of the $V_i$. So such a partition is an orbit for the actions of $S_r$. The class of such orbits, or equivalently of partitions (2) are also called non-equivalent vertex colorings, see [14]).

We also consider partitions of the edges

$$\begin{aligned} E = E_1 \sqcup E_2 \sqcup \cdots \sqcup E_s. \end{aligned}$$

Here, we have no independence requirements. Any partition is good.

Now, we make a correspondence as follows. Given such a partition of V, make a partition of E as follows: If v and w are vertices consider the unique path in T linking v and w. Let f, respectively, g, be the edge incident to v, respectively, w, on this path. If (i) v and w are in the same part $V_i$ of V and (ii) no other vertex on this path is in the part $V_i$, then put f and g into the same part of E, and write $f \sim ^{\!\! \prime }_E g$. The partition of edges is the equivalence relation $\sim _E$ generated by $\sim ^{\!\! \prime }_E$, i.e., the smallest equivalence relation on E containing $\sim ^{\!\! \prime }_E$. Note that in general $\sim ^{\!\! \prime }_E$ alone would not be reflexive nor transitive.

Conversely, given a partition of the edge set E, make a partition of V as follows: Let v and w be distinct vertices, and consider again the path from v to w. If (i) the edges f and g are distinct, (ii) f and g are in the same part $E_j$, and (iii) no other edge on this path is in the part $E_j$, then put v and w in the same part of V, and write $v \sim ^{\!\! \prime }_V w$. The partition of vertices is the equivalence relation $\sim _V$ generated by $\sim ^{\!\! \prime }_V$.

Fig. 7

The trees in Example A.3. The partitions of the vertex set and the edge set are illustrated by the different colors (Color figure online)

Theorem A.1

([10]) Let T be a tree with vertex set V and edge set E. The above gives a one-to-one correspondence between partitions of the vertices V into $r+1$ independent sets, and partitions of the edges E into r sets.

Example A.2

Any tree has a unique partition of the vertices into two independent sets (two colors modulo $S_2$). This corresponds to the partition of the edges into one part (one color).

Example A.3

In Fig. 7, we partition the edges into red and black color classes. The vertices are then partitioned into three sets, each consisting of independent vertices. The partition of the vertex set of the first tree is

$$\begin{aligned} \{1,3,5\}\cup \{2,6\}\cup \{4\}, \end{aligned}$$

and that of the second tree is

$$\begin{aligned} \{1,3,5,8,10\}\cup \{2,4,7\}\cup \{6,9\}. \end{aligned}$$