A constructive approach to minimal free resolutions of path ideals of trees

For a rooted tree Γ, we consider path ideals of Γ, which are ideals that are generated by all directed paths of a fixed length in Γ. In this paper, we provide a combinatorial description of the minimal free resolution of these path ideals. In particular, we provide a class of subforests of Γ that are in one-to-one correspondence with the multi-graded Betti numbers of the path ideal as well as providing a method for determining the projective dimension and the Castelnuovo-Mumford regularity of a given path ideal.


Introduction
Monomial ideals have been studied extensively in the literature and many applications of monomial ideals have been explored.In this paper, we consider a specific class of monomial ideals, known as path ideals.Path ideals are a type of squarefree monomial ideals that are generated in a fixed degree.Edge ideals, a specific class of path ideals, were first studied by Conca and De Negri in [4].Given a graph Γ = (V, E) having vertex set V and edge set E, we can form the path ideal of length (t − 1) associated to Γ by considering the ideal generated by the monomials corresponding to all (t − 1) length paths in Γ.If V = {x 1 , . . ., x n }, this ideal is considered in the polynomial ring R := k[x 1 , . . ., x n ], where k is a field.
In [10], Nagel and Reiner showed in Proposition 6.1 that the Betti numbers of edge ideals of arbitrary graphs can be as complicated as desired.In particular, this proposition implies that the Betti numbers of the edge ideal of an arbitrary graph can depend on the choice of the field, k.For the case of path ideals of directed, rooted trees Bouchat, Há, and O'Keefe showed in Theorem 2.7 of [3] that the Betti numbers of a path ideal associated to a rooted tree are independent of the choice of the field, k.In this paper, we will restrict our focus to the study of path ideals of rooted trees.Recall that a tree is a simple, connected graph containing no loops or multi-edges.Then a rooted tree is a tree together with a fixed vertex, called the root.It is natural to consider a rooted tree as a directed graph in which all edges are assigned the direction going away from the root.It should be noted that x i will denote both the vertex in the graph Γ as well as the monomial in the polynomial ring R.
Definition 1.1.Given a directed, rooted tree Γ and t ≥ 2, the path ideal of length (t − 1) of Γ is: Thus, for a given directed, rooted tree Γ there can be more than one path ideal associated to Γ as illustrated in Figure 1.
For a given tree Γ, we can successively remove any leaves that occur at level less than (t − 1) when considering the path ideal I t (Γ), as these vertices cannot contribute to a minimal generator in I t (Γ).A tree that has had successive removal of all leaves ocurring at level less than (t − 1) will be said to be in clean form.
The minimal free resolutions of path ideals of trees have been studied by many authors, including Há and Van Tuyl in [7], Katzman in [8], and Kummini in [9].The basic tools and decompositions that will be used in this paper were introduced by Faridi and Alilooee in [1].The aim of this paper is to give a constructive description of the multi-graded Betti numbers for path ideals of rooted trees.This constructive description of the Betti numbers corresponding to a path ideal of a rooted tree will also provide a method to compute the projective dimension as well as the Castelnuovo-Mumford regularity.

Basic definitions
We begin with the background concepts from commutative algebra.
Let M be a finitely generated graded S-module.Associated to M is a minimal free resolution, which is of the form where the maps δ i are exact and where S(−a) denotes the translation of S obtained by shifting the degree of elements of S by a ∈ (N ∪ {0}) n .The numbers β i,a (M ) are called the multi-graded Betti numbers (or N n -graded Betti numbers) of M , and they correspond to the number of minimal generators of degree a occurring in the i th -syzygy module of M .It should be noted that the graded Betti numbers (or N-graded Betti numbers) of M can be defined as β i,j (M ) := M ) where a = (a 1 , . . ., a n ).
There are two invariants corresponding to the minimal free resolution of M which measure the "size" of the resolution.Definition 2.1.Let M be a finitely generated graded S-module.
1.The projective dimension of M , denoted pd(M ), is the length of the minimal free resolution associated to M .
2. The Castelnuovo-Mumford regularity (or regularity), denoted reg(M ), is We also need the following definitions from simplicial topology.
1.An abstract simplicial complex, ∆, on a vertex set X = {x 1 , . . ., x n } is a collection of subsets of X satisfying: (a) {x i } ∈ ∆ for all i, and (b) The elements of ∆ are called faces of ∆, and the maximal faces (under inclusion) are called facets of ∆.The simplicial complex ∆ with faces F 1 , . . ., F s will be denoted by F 1 , . . ., F s .
2. If ∆ is an abstract simplicial complex, then a face τ ∈ ∆ is called a free face if it is contained in a unique facet of ∆.
3. For any Y ⊆ X , an induced subcollection of ∆ on Y, denoted by ∆ Y , is the simplicial complex whose vertex set is a subset of Y and whose facet set is given by It should be noted that if F 1 , . . ., F s are facets in Definition 2.2(1), then F 1 , . . .F s is a minimal representation of ∆.In particular, the complementary complex ∆ c X , described in the fourth part of Definition 2.2, is heavily utilized within this paper.Definition 2.3.
1. Let ∆ be a simplicial complex with vertex set x 1 , . . ., x n .Then the facet ideal of ∆ is defined as 2. Let I be an ideal in R := k[x 1 , . . ., x n ] minimally generated by square-free monomials m 1 , . . ., m s .
The facet complex ∆(I) associated to I has vertex set {x 1 , . . ., x n } and is defined by where The above constructions provide a one-to-one correspondence between simplicial complexes on the vertex set {x 1 , . . ., x n } and the square-free monomial ideals in R := k[x 1 , . . ., x n ].We illustrate this construction with the following example.
Example 2.4.Consider the path ideal I 3 (Γ) = (x 1 x 2 x 3 , x 2 x 3 x 4 , x 2 x 3 x 5 ) associated to the tree, Γ, pictured below.Then the facet complex associated to Γ with respect to t = 3 is The dimension of the reduced homology group of the complementary complex of a path ideal is instrumental in determining the Betti numbers of the corresponding minimal free resolution.We will use a corollary of Theorem 2.8 of Alilooee and Faridi in [1] to help determine the multi-graded Betti numbers for path ideals of rooted trees.This corollary, found below, appears in [2] as Corollary 2.11.
Corollary 2.5.Let S = k[x 1 , . . ., x n ] be a polynomial ring over a field k, and let I be a pure, squarefree monomial ideal in S. Then the multi-graded Betti numbers of I are given by for i ≥ 1 where Γ is an induced subcollection of ∆(I) with Vert(Γ) = {x i | a i = 1} where a = (a 1 , . . ., a n ).
It should be noted that the Betti numbers β 0,a (I) correspond to the minimal generating set of I. Finally, we will rely upon Lemma 4.1 in [1] of Alilooee and Faridi, which is a tool that allows us to determine the i th reduced homology group of the complementary complex using the (i − 1) st reduced homology group of a smaller complex, which we call the deleted complementary complex, that is, Lemma 2.6 ([Lemma 4.1, Alilooe and Faridi]).Suppose Γ is a tree generated by the paths P 1 , P 2 , . . ., P k and suppose P 1 ∩ (P . It should further be noted that in Corollary 6.1 of [5], it is shown that the independence complex of graph forests are simple-homotopy equivalent to a vertex or to a sphere.Thus by Corollary 2.5, the multi-graded Betti numbers of path ideals of directed, rooted trees correspond via simple-homotopy equivalence to a vertex or to a sphere.We conclude this section with some graph theoretic definitions before stating our three line graph constructions on directed, rooted trees.Definition 2.7.Let Γ be a rooted tree with root x and vertex set Vert(Γ), and let y ∈ Vert(Γ).
1.The level of y, denoted level(y), is the length of the unique path in Γ from x to y.The height of Γ is the maximal level over all vertices in Γ.
2. The parent vertex of the non-root vertex y is the unique vertex z such that yz forms an edge in Γ and level(z) = level(y) − 1.

3.
A subgraph H of Γ is called an induced subgraph if for every pair of vertices x, y ∈ H the following condition holds: if {x, y} is an edge of Γ, then it is also an edge of H.
Now we can define the essential constructions for this paper.Starting with a tree Γ which is composed of generating paths of length (t − 1) in the set {P 1 , P 2 , . . ., P k }, we assume Γ c Vert(Γ) is not contractible, that is, there exists an i ≥ 0 for which dim k H i Γ c Vert(Γ) = 1.We create a new tree Γ such that all edges and vertices of Γ are also contained in Γ as follows.
Remark 2.9.We will also use the terms glue and split as constructions on the empty graph.Consider t ≥ 2. Starting with the empty graph, a split will result in the line graph having exactly t vertices, and a glue from the empty graph will result in the line graph having exactly t + 1 vertices.It should be noted that adding less than t new vertices to the empty graph will result in a graph having no minimal generators for the ideal I t (Γ).
Definition 2.8 along with Lemma 2.6 will allow us to begin with a tree whose complementary complex is homotopic to a sphere in some dimension, adjoin a line graph, and consequently determine the homotopy type of the resulting complementary simplicial complex of the modified tree.In particular, we will show that these three cases, glue, split, and contraction, are the only needed constructions and consequently the new simplicial complex will be homotopic to a sphere in either one or two dimensions higher, or it will be contractible.

Constructions and Betti numbers
In this section, we will explore how a glue, split, and contraction of a directed graph Γ will determine the non-zero Betti number of top dimension in the corresponding minimal free resolution of the path ideal of the newly constructed graph, Γ and consequently will determine the projective dimension.
When we create a new tree, Γ , from Γ by adding n new vertices {y 1 , y 2 , . . ., y n } through a glue, split, or contraction, at most n new generating paths of length (t − 1) will be added to the path ideal of I t (Γ); depending on the level of the vertex y 1 .Let E i be the unique directed path of length (t − 1) which terminates in vertex y i for 1 ≤ i ≤ j where j = min{level(y 1 ) − t + 2, n}.The number j of new generating paths depends on the level of x 0 in the tree Γ and the number of vertices added, as there may not exist a path of length (t − 1) which terminates in vertex y i .Thus Γ is generated by the paths {P 1 , . . ., P k , E 1 , . . ., E j }.
In each of the following cases we consider the deleted complementary complex that was described in Section 2. Since E 1 always contains a leaf of Γ , the conditions of Lemma 2.6 are satisfied and We begin with the case of a glue.
Proposition 3.1.Let Γ be a tree in clean form with respect to t ≥ 2 such that there exists i ≥ 0 for which β i,a (I t (Γ)) = 1, where a = (a 1 , . . ., a m ) and a j = 1 if and only if x j ∈ Γ.Let Γ be a glue of Γ, then where a = (a 1 , . . ., a m+t+1 ) and a j = 1 if and only if x j ∈ Γ .
Before the proof, we illustrate the deletion methods of Lemma 2.6 in the case of a glue with the following example.
Example 3.2.Consider the tree Γ 1 , occurring as a glue of Γ: Then using the deletion method of Lemma 2.6, we set , and E 7 = {x 2 , x 3 , x 5 }.It follows that: After identifying the vertices x 2 and x 3 , the simplicial complex can be visualized as, where the tetrahedron labeled by x 1 x 2 x 3 x 4 x 5 is solid, but the tetrahedron labeled by x 1 x 4 x 5 y 4 is hollow.Thus dim k H 2 ( x 1 x 2 x 3 x 4 x 5 , x 4 x 5 y 4 , x 1 x 5 y 4 , x 1 x 4 y 4 ) = 1, and thus, by Lemma 2.6, we have Proposition 3.3.Let Γ be a tree in clean form with respect to t ≥ 2 such that there exists i ≥ 0 for which β i,a (I t (Γ)) = 1, where a = (a 1 , . . ., a m ) and a j = 1 if and only if x j ∈ Γ.Let Γ be a split of Γ, then where a = (a 1 , . . ., a m ) and a j = 1 if and only if x j ∈ Γ .
As before, we illustrate the deletion methods of Lemma 2.6 in the case of a split with the following example before providing the proof.
Example 3.4.Consider the tree Γ 2 , occuring as a split of Γ: model, any directed graph Γ containing the directed graph Γ as a subgraph, could have been created from Γ through a sequence of glues, splits, and contractions.
By extension and given the conventions of a glue and split to an empty graph, any clean graph can be created from a sequence of glues, splits, and contractions starting with an empty graph for some t ≥ 2. As the empty graph trivially satisfies the conditions on Propositions 3.1, 3.3, and 3.5, we have that all clean, directed graphs can be constructed from a sequence of glues, splits, and contractions.Given a subforest Λ of Γ, let g(Λ) and s(Λ) denote the number of glues and splits, respectively, required to construct Λ from an empty graph.Then we have the following theorem.
Theorem 4.1.Let Γ be a directed, rooted tree, and let t ≥ 2. Then β i,a (I t (Γ)) = 1 precisely when {x j | a j = 0} corresponds to the vertex set of an induced subforest, Λ, of Γ constructed from the empty graph by a sequence of only glues and splits.Moreover, in this case i = 2g(Λ) + s(Λ) − 1.
Proof.Since multigraded Betti numbers correspond to induced subforests of Γ, the result follows immediately from Propositions 3.1, 3.3, and 3.5.
We illustrate Theorem 4.1 with an example.
Example 4.2.Consider the tree Γ depicted below.The graded minimal free resolution of I 3 (Γ) is also provided below and was obtained using Macaulay 2 (see [6]).
x 1 x 2 x 3 x 4 x 5 x 6 x 8 x 7 x 9 x 10 The Betti numbers, β 0,3 (I 3 (Γ)) correspond to all paths in Γ of length 2, which are all induced subforests of Γ formed as a split from the empty graph.The Betti numbers, β 1,j (I 3 (Γ)), correspond to all induced subtrees of Γ formed either by two splits from the empty graph or by one glue from the empty graph.Hence, these Betti numbers correspond to induced subforests of Γ of the following form: Two Splits Two Splits Two Splits One Glue (2 of this type) (1 of this type) (5 of this type) (6 of this type) The Betti numbers, β 2,j (I 3 (Γ)), correspond to all induced subforests of Γ formed either by three splits from the empty graph or by one glue and one split from the empty graph.Lastly, the Betti numbers, β 3,j (I 3 (Γ)), correspond to all induced subforests of Γ formed from either four splits, from one glue and two splits, or from two glues.The only possible induced subforests of these types in Γ are depicted below: One Glue & Two Splits Two Glues Two Glues (1 of this type) (1 of this type) (1 of this type) It should be noted that the vertex sets of the induced subforests correspond to the multi-graded Betti numbers in the minimal free resolution of I 3 (Γ).
Theorem 4.1 also implicitly describes the projective dimension and regularity of the I t (Γ).We have the following corollaries.

Figure 1 .
Figure 1.An example of a graph, Γ, and its associated path ideals
One Glue & One Split One Glue & One Split One Glue & One Split One Glue & One Split