aa r X i v : . [ m a t h . A C ] S e p BETTI NUMBERS OF WEIGHTED ORIENTED GRAPHS
BEATA CASIDAY AND SELVI KARA
Abstract.
Let D be a weighted oriented graph and I ( D ) be its edge ideal. In this paper, we investigatethe Betti numbers of I ( D ) via upper-Koszul simplicial complexes, Betti splittings and the mapping coneconstruction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of severalclasses of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge idealshave a unique extremal Betti number which allows us to compute the regularity and projective dimensionfor the identified classes. Furthermore, we characterize the structure of a weighted oriented graph D on n vertices such that pdim( R/I ( D )) = n where R = k [ x , . . . , x n ]. Introduction An oriented graph is an ordered pair D = ( V ( D ) , E ( D )) with the vertex set V ( D ), the edge set E ( D ) and anunderlying graph G on which each edge is given an orientation. If e = { x, y } is an edge in G and e is orientedfrom x to y in D , we denote the oriented edge by ( x, y ) to reflect the orientation. In contrast to directedgraphs, multiple edges or loops are not allowed in oriented graphs. An oriented graph D is called vertex-weighted oriented (or simply weighted ) if each vertex is assigned a weight by a function w : V ( D ) → N + called a weight function. For simplicity, we set w i = w ( x i ) for each x i ∈ V ( D ) . If the weight value w i ofvertex x i is one, we say x i has a trivial weight in D and we call x i a trivial vertex . Otherwise, we say x i hasa non-trivial weight in D and call x i a non-trivial vertex .Let D be a weighted oriented graph with the vertex set V ( D ) = { x , . . . , x n } and R = k [ x , . . . , x n ] be apolynomial ring over a field k . By identifying vertices of D with variables in R , the edge ideal of a weightedoriented graph D is defined as I ( D ) = ( x i x w j j : ( x i , x j ) ∈ E ( D )) . If all vertices of D have trivial weights, then I ( D ) is the edge ideal of the (undirected, unweighted) underlyinggraph of D . Edge ideal of undirected, unweighted graphs are studied extensively in the literature (see [2]).The minimal generators of I ( D ) display only the weights of target vertices for each edge. If a vertex hasonly outgoing edges from it, we call it a source vertex . Since weights of source vertices do not appear in theminimal generators of I ( D ), we shall assume that w i = 1 if x i is a source vertex.One of the known appearances of edge ideals of weighted oriented graphs is in the algebraic coding theoryliterature (see [7, 15]). In particular, the ideal I = ( x i x w j j : 1 ≤ i < j ≤ n ) where 2 ≤ w ≤ · · · ≤ w n is the initial ideal of the vanishing ideal I ( X ) of a projective nested cartesian set X with respect to thelexicographic order (see [7, Proposition 2.11]). Projective nested cartesian codes are images of degree d evaluation maps on X and these type of evaluation codes are introduced to generalize the classical projectiveReedMuller type codes. Note that the ideal I is the edge ideal of a weighted oriented complete graph on n vertices with the edge set { ( x i , x j ) : 1 ≤ i < j ≤ n } such that x is a source vertex and all other verticesare non-trivial. Algebraic invariants of I is used in detecting “good” projective nested cartesian codes, forinstance, the Castelnuovo-Mumford regularity of R/I , denoted by reg(
R/I ), is a strict upper bound for an“optimal” degree d. Edge ideals of weighted oriented graphs are fairly new objects in the combinatorial commutative algebracommunity and there have been a few papers investigating algebraic properties and invariants of theseobjects (see [5, 11, 12, 17, 18]). There has been an extensive literature on the edge ideals of (unweighted,unoriented) graphs, and one of the reasons for such fruitful outcomes is due to the squarefree nature of edgeideals of graphs. On the other hand, edge ideals of weighted oriented graphs are not squarefree in general, so any of the established squarefree connections such as Hochster’s formula and independence complexes areno longer available in studying our objects of interest. Furthermore, descriptions of edge ideals of weightedoriented graphs differ depending on the orientation and positions of non-trivial weights, making it morecomplicated to obtain general results for this class of ideals.Our general goal is to address these issues in the study of Betti numbers of edge ideals of weighted orientedgraphs. One of the essential elements of the paper is the upper-Koszul simplicial complex. The followingHochster-like formula given in [3, Theorem 2.2] exploits the structure of upper Koszul-simplicial complexesand allows one to compute Betti numbers of I ( D ) in terms of the holes of upper-Koszul simplicial complexes. β i, b ( I ) = dim k e H i − ( K b ( I ); k )where K b ( I ) is the upper-Koszul simplicial complex of a monomial ideal I at multidegree b ∈ N n .Current literature on algebraic invariants of edge ideals of weighted oriented graphs is obtained by focusingon specific classes of graphs with a predetermined orientation or weight assumptions. For instance, in [18],authors investigate the regularity and the projective dimension of edge ideals of weighted cycles and rootedforests with the assumption that all non-source vertices are non-trivial and graphs are naturally oriented(i.e., all edges are oriented in the same direction). A more general case, class of weighted naturally orientedpaths and cycles under any weight distribution is studied in [5]. All the formulas provided in [5, 18] heavilydepend on the orientation and weight distributions. In order to obtain more general results, we investigatethe Betti numbers and provide recursive formulas for these invariants. With this approach, we successfullyrecover several results from [18] and suggest an explanation for the nature of regularity formulas given in [5].If D is a weighted oriented graph on n vertices, then the projective dimension of R/I ( D ) is at most n byHilbert’s Syzygy Theorem. One of the main results of the paper classifies all weighted oriented graphs suchthat the upper bound for the projective dimension is tight (Theorem 4.2). We achieve it by concludingthat edge ideals of such weighted oriented graphs have a unique extremal Betti number and it occurs at themultidegree b = ( w , . . . , w n ) ∈ N n . In addition, we show that β n − , b is the unique extremal Betti numberfor edge ideals of classes of weighted oriented complete graphs and weighted rooted graphs on n verticeswhere b = ( w , . . . , w n ) ∈ N n (Theorem 4.9 and Theorem 5.2). As a result, we provide formulas for theregularity and projective dimension of the edge ideals of those classes of weighted oriented graphs. Otherresults of the paper focuses on providing recursive formulas for the Betti numbers of edge ideals of weightedoriented graphs. In particular, we provide such formulas for edge ideals of • weighted oriented complete graphs with at least one sink vertex by making use of Betti splittings(Theorem 5.4) and • weighted oriented graphs with at least one sink vertex that is also a leaf by employing the mappingcone construction (Theorem 6.1).Our paper is organized as follows. In Section 2, we recall the necessary terminology and results which willbe used in the paper. Section 3, we provide general results by relating algebraic invariants of edge idealsof weighted oriented graphs and their induced weighted oriented subgraphs. We also show that reducingweight of a non-trivial sink vertex (i.e., a vertex with only incoming edges towards it) reduces the regularityby one and keeps projective dimension unchanged (Corollary 3.11). In Section 4, we prove Theorem 4.2 andTheorem 4.9. Section 5 is devoted to weighted oriented complete graphs on n vertices. In particular, weprovide formulas for the regularity and projective dimension when the edges are oriented in the “natural”way, i.e., ( x i , x j ) for any 1 ≤ i < j ≤ n . We also prove Theorem 5.4 in this section. In Section 6, we use themapping cone construction and obtain recursive formulas for Betti numbers of weighted rooted graphs inTheorem 6.1. As an application of Theorem 6.1, we consider weighted oriented paths and make connectionswith the regularity formula given in [5]. Finally, in Section 7, we raise some questions about the behavior ofBetti numbers of edge ideals under weight reduction operations.2. Preliminaries
In this section, we collect the notation and terminology that will be used throughout the paper. et R = k [ x , . . . , x n ] be a polynomial ring over a field k and M be a finitely generated R module. Thenthe minimal free resolution of M over R is of the form0 −→ M j ∈ Z R ( − j ) β p,j ( M ) −→ M j ∈ Z R ( − j ) β p − ,j ( M ) −→ · · · −→ M j ∈ Z R ( − j ) β ,j ( M ) −→ M −→ . The exponents β i,j ( M ) are invariants of the module called the Betti numbers of M and these invariantsencode all the information about the minimal free resolution of a module. In general, it is difficult toexplicitly compute the Betti numbers. A common approach to go around this issue is to investigate coarserinvariants of the module associated to Betti numbers. In this paper, we focus on the Castelnuovo-Mumfordregularity (or simply, regularity), projective dimension and the extremal Betti numbers of M = R/I where I is a homogeneous ideal of R . The Castelnuovo-Mumford regularity and the projective dimension of R/I are defined as reg(
R/I ) = max { j − i : β i,j ( R/I ) = 0 } and pdim( R/I ) = max { i : β i,j ( R/I ) = 0 } . A Betti number β k,l ( R/I ) = 0 is called extremal if β i,j ( R/I ) = 0 for all pairs ( i, j ) = ( k, l ) with i ≥ k and j ≥ l. In other words, extremal Betti numbers occupy the upper left corner of a block of zeroes in the Bettidiagram of
R/I in Macaulay 2. The notion of extremal Betti numbers are introduced in [3] as a refinement ofthe notion of the regularity and one can read off the regularity and projective dimension from the extremalBetti numbers. Particularly,
R/I has the unique extremal Betti number if and only if β p,p + r ( R/I ) = 0 where p = pdim( R/I ) and r = reg( R/I ) . Upper-Koszul Simplicial Complexes.
An important connection in the field of combinatorial com-mutative algebra is the Stanley-Reisner correspondence which allows one to relate a squarefree monomialideal with a simplicial complex (see [10]). As part of the Stanley-Reisner theory, one can compute the Bettinumbers of a squarefree monomial ideal through dimensions of holes of a simplicial complex through theHochster’s formula. When the ideal I is no longer squarefree, there is no Stanley-Reisner complex associ-ated to I and Hochsters formula cannot be applied directly. However, Bayer, Charalambous, and Popescuintroduced the upper Koszul simplicial complex in [3] and provided a Hochster-like formula to compute themultigraded Betti numbers of any monomial ideal. Definition 2.1.
Let I be a monomial ideal in R = k [ x , . . . , x n ] . The upper-Koszul simplicial complex of I at multidegree b ∈ N n is K b ( I ) = { F ⊆ { x , . . . , x n } : x b x F ∈ I } (2.1)where x F = Y x i ∈ F x i . Theorem 2.2. [3] [Theorem 2.2] Given a monomial ideal I in R = k [ x , . . . , x n ] , the multigraded Bettinumbers of I are β i, b ( I ) = dim k e H i − ( K b ( I ); k ) where b ∈ N n . Remark 2.3.
Let I be monomial ideal in R = k [ x , . . . , x n ] and b ∈ N n . If x b is not equal to a least commonmultiple of some of the minimal generators of I, then K b ( I ) is a cone over some subcomplex. Therefore, allnon-zero Betti numbers of I occurs in N n -graded degrees b such that x b is equal to a least common multipleof some minimal generators of I. Betti Splitting.
In the study of Betti numbers of a monomial ideal I , one natural approach is to breakdown the ideal I into smaller pieces and express the Betti numbers of I in terms of the Betti numbers of thesmaller pieces. This strategy was first introduced by Eliahou and Kervaire in [8] for monomial ideals andstudied in more detail by Francisco, H`a, and Van Tuyl in [9]. efinition 2.4. Let
I, J, and K be monomial ideals with generating sets G ( I ) , G ( J ) , and G ( K ) such that G ( I ) is the disjoint union of G ( J ) and G ( K ) . Then I = J + K is called a Betti splitting if β i,j ( R/I ) = β i,j ( R/J ) + β i,j ( R/K ) + β i − ,j ( R/J ∩ K )for all i, j > I = J + K to be a Betti splittingby considering just the generators of I . Theorem 2.5. [9] [Corollary 2.7] Suppose that I = J + K where G ( J ) contains all the generators of I divisible by some variables x i and G ( K ) is a non-empty set containing the remaining generators of I. If J has a linear resolution, then I = J + K is a Betti splitting. General Results
In this section, we recall the notion of inducedness for weighted oriented graphs and introduce a relatednotion called weight reduced form of a weighted oriented graph. Weight reduced form of a weighted orientedgraph D has the same underlying graph as D and the weight of one of the non-trivial vertices in D is reducedby one. We call the process of obtaining a weight reduced form of D a weight reduction process . In the mainresult of this section, we show that the weight reduction of a non-trivial sink vertex reduces the regularityby one but keeps the projective dimension unchanged. Definition 3.1.
Let D be a weighted oriented graph with the underlying graph G . We say D ′ is an inducedweighted oriented subgraph of D if • the underlying graph of D ′ is an induced subgraph of G, • orientation of D ′ is induced from D , i.e., E ( D ′ ) ⊆ E ( D ), and • w x ( D ′ ) = w x ( D ) for all x ∈ V ( D ′ )where w x ( D ) denotes the weight of x in D and w x ( D ′ ) denotes the weight of x in D ′ . We shall use thisnotation throughout the text while considering weight of a vertex in two different weighted oriented graphs.It is well-known that Betti numbers of edge ideals of induced subgraphs for unweighted unoriented graphscan not exceed that of the original graph (see [4, Lemma 4.2]). We provide an analog of this well-knownresult for weighted oriented graphs in the following lemma. Lemma 3.2.
Let D ′ be an induced weighted oriented subgraph of D . Then, for all i, j ≥ , we have β i,j ( I ( D ′ )) ≤ β i,j ( I ( D )) . Proof.
Since the notion of inducedness for weighted oriented graphs is an extension of inducedness forunweighted, unoriented graphs, we can adopt the proof of [4, Lemma 4.2] to obtain the inequality. (cid:3)
Corollary 3.3.
Let D ′ be an induced weighted oriented subgraph of D . Then pdim( R/I ( D ′ )) ≤ pdim( R/I ( D )) and reg( R/I ( D ′ )) ≤ reg( R/I ( D )) . In the study of edge ideals of weighted oriented graphs, we assume that source vertices have trivial weights. Ina similar vein, one can ask whether the same treatment can be applied to sink vertices. In [12], authors assumethat sinks vertices have trivial weights along with source vertices while investigating Cohen-Macaulaynessof edge ideals. In the investigation of algebraic invariants such as regularity, reducing the weight of a non-trivial sink vertex changes the regularity. Thus, one needs to carefully consider the effects of weight reductionprocess on the multigraded Betti numbers of edge ideals and its invariants associated to Betti numbers. Forthis purpose, we introduce a new notion called weight reduced form of a given weighted oriented graph.
Definition 3.4.
Let D be a weighted oriented graph with at least one non-trivial vertex x such that w x ( D ) > . We say D ′ is a weight reduced form of D if D ′ has the same orientation as D , • w y ( D ′ ) = w y ( D ) for all y = x , and • w x ( D ′ ) = w x ( D ) − D ′ is a weightreduced form of D on x .We use the following notation throughout this section. Notation 3.5.
Let D be a weighted oriented graph on n vertices. If D has at least one sink vertex with anon-trivial weight, say x p , such that w p ( D ) = w >
1, then in-neighbors of x p , denoted by N −D ( x p ) , coincidewith all of its neighbors, denoted by N D ( x p ) , where N −D ( x p ) = N D ( x p ) = { x : ( x, x p ) ∈ E ( D ) } 6 = ∅ . We can decompose the edge ideal of D as I ( D ) = I ( D \ x p ) + ( xx wp : x ∈ N D ( x p )) | {z } J where G ( I ( D \ x p )) = G ( I ( D )) \ G ( J ).Let D ′ be the weighted reduced form of D on x p . Then I ( D ′ ) = I ( D \ x p ) + ( xx w − p : x ∈ N D ( x p )) | {z } J ′ . Note that
D \ x p is an induced weighted oriented subgraph of D and D ′ . Remark 3.6.
In the light of Remark 2.3, one can observe that for all i > b ∈ N n , β i, b ( I ( D )) = 0 if b p = 0 , w,β i, b ( I ( D ′ )) = 0 if b p = 0 , w − ,β i, b ( I ( D \ x p )) = 0 if b p = 0because the upper-Koszul complexes of these ideals at the corresponding multidegrees are all cones withapex x p . For the sake of completion, we include the proof below for one of the ideals. Lemma 3.7.
Let b ∈ N n such that b p > . Then K b ( I ( D \ x p )) is a cone with apex x p .Proof. Denote the upper-Koszul complex K b ( I ( D \ x p )) by ∆. It suffices to show that F ∪ { x p } ∈ ∆ for all F ∈ ∆. Let F ∈ ∆ such that x p / ∈ F . Then m := x b · · · x b p p · · · x b n n x F ∈ I ( D \ x p ) . Thus there exists e ∈ G ( I ( D \ x p )) such that m = em ′ for some m ′ ∈ R . Since b p > x p / ∈ F , themonomial m ′ is divisible by x p . Hence mx p = x b · · · x b p p · · · x b n n x F ∪{ x p } ∈ I ( D \ x p )implying that F ∪ { x p } ∈ ∆. Therefore, ∆ is a cone with apex x p . (cid:3) Since only non-zero Betti numbers for the edge ideals of interest can occur at multidegrees b ∈ N n where b p ∈ { , w } for I ( D ) and b p ∈ { , w − } for I ( D ′ ) , we consider these two cases separately by finding relationsbetween the corresponding upper-Koszul simplicial complexes. Lemma 3.8.
Let b ∈ N n such that b p = 0 . Then β i, b ( I ( D )) = β i, b ( I ( D ′ )) = β i, b ( I ( D \ x p )) roof. It suffices to show K b ( I ( D )) = K b ( I ( D ′ )) = K b ( I ( D \ x p ))by Theorem 2.2. It is immediate from the chain of inclusions I ( D \ x p ) ⊆ I ( D ) ⊆ I ( D ′ ) that K b ( I ( D \ x p )) ⊆ K b ( I ( D )) ⊆ K b ( I ( D ′ )) . Let F ∈ K b ( I ( D )). Then m = x b · · · x b p p · · · x b n n x F ∈ I ( D ) . Since b p = 0, none of the generators of J divide m and we must have m ∈ I ( D \ x p ). Thus F ∈ K b ( I ( D \ x p )) , proving the equality K b ( I ( D )) = K b ( I ( D\ x p )) . The remaining equality follows from the same arguments. (cid:3)
Lemma 3.9.
Let b , b ′ ∈ N n such that b p = w , b ′ p = w − and b ′ i = b i for all i = p . Then β i, b ( I ( D )) = β i, b ′ ( I ( D ′ )) Proof.
As in the proof of the previous lemma, it suffices to show K b ( I ( D )) = K b ′ ( I ( D ′ )) by Theorem 2.2.Let F ∈ K b ′ ( I ( D ′ )). By the definition of upper-Koszul simplicial complexes, we have m := x b ′ x F = x b · · · x w − p · · · x b n n x F ∈ I ( D ′ ) . If m ∈ J ′ , then there exists xx w − p ∈ J ′ such that m is divisible by xx w − p . As a result, mx p ∈ J and mx p = x b · · · x wp · · · x b n n x F ∈ J ⊆ I ( D )which implies that F ∈ K b ( I ( D )) . Suppose m / ∈ J ′ . Then m must be contained in I ( D \ x p ), thus mx p ∈ I ( D \ x p ) ⊆ I ( D ) and F ∈ K b ( I ( D )) . It remains to prove the reverse containment. If F ∈ K b ( I ( D )), we have m := x b · · · x wp · · · x b n n x F ∈ I ( D ) . If m ∈ J, then m must be divisible by some xx wp ∈ J which implies that x p / ∈ F . Since w >
1, we have mx p = x b · · · x w − p · · · x b n n x F ∈ J ′ ⊆ I ( D ′ ) . Thus F is a face in K b ′ ( I ( D ′ )) . Suppose m / ∈ J . Then m must be contained in I ( D \ x p ) . Equivalently,there exists a minimal generator e of I ( D \ x p ) such that m = em for a monomial m ∈ R. Since w > m is divisible by x p while e is not. Thus m/x p is still contained in I ( D \ x p ) ⊆ I ( D ′ ) and F ∈ K b ′ ( I ( D ′ )) . Therefore, K b ( I ( D )) = K b ′ ( I ( D ′ )) . (cid:3) Corollary 3.10.
Let D be a weighted oriented graph with a non-trivial sink vertex x p and let D ′ be theweight reduced form of D on x p .(a) The i th total Betti numbers of I ( D ) and I ( D ′ ) are equal for all i ≥ .(b) We have β i,j ( I ( D )) = β i,j − ( I ( D ′ )) − β i,j − ( I ( D \ x p )) + β i,j ( I ( D \ x p )) for all i > , j > . roof. (a) It follows from Lemma 3.8 and Lemma 3.9 that β i ( I ( D )) = X b ∈ N n β i, b ( I ( D )) = X b ∈ N n , b p =0 β i, b ( I ( D )) + X b ∈ N n , b p = w β i, b ( I ( D ))= X b ∈ N n , b p =0 β i, b ( I ( D ′ )) + X b ∈ N n , b p = w − β i, b ( I ( D ′ ))= X b ∈ N n β i, b ( I ( D ′ )) = β i ( I ( D ′ )) . (b) By making use of the equalities of multigraded Betti numbers obtained in Lemma 3.8 and Lemma 3.9,one can express the Betti numbers of I ( D ) in terms of those of I ( D ′ ) and I ( D \ x p ). In particular, we have β i,j ( I ( D )) = X b ∈ N n , | b | = j β i, b ( I ( D )) = X | b | = j b p = w β i, b ( I ( D )) + X | b | = j b p =0 β i, b ( I ( D ))= X | b ′ | = j − b ′ p = w − β i, b ′ ( I ( D ′ )) + X | b | = j b p =0 β i, b ( I ( D \ x p ))= β i,j − ( I ( D ′ )) − β i,j − ( I ( D \ x p )) + β i,j ( I ( D \ x p ))Note that D \ x p is an induced weighted oriented subgraph of D ′ . Thus, by Lemma 3.2, we have β i,j ( I ( D ′ )) ≥ β i,j ( I ( D \ x p )) for all i, j . (cid:3) Corollary 3.11.
Let D be a weighted oriented graph with a non-trivial sink vertex x p and let D ′ be theweight reduced form of D on x p . Then(a) pdim( R/I ( D )) = pdim( R/I ( D ′ )) , (b) reg( R/I ( D )) = reg( R/I ( D ′ )) + 1 . Proof.
Equality of projective dimensions immediately follows from Corollary 3.10 (a). By making use ofCorollary 3.10 (b), we havereg(
R/I ( D )) = max { reg( R/I ( D ′ )) + 1 , reg( R/I ( D \ x p )) } . Since reg(
R/I ( D ′ ) ≥ reg( R/I ( D \ x p ) by Corollary 3.3, we obtain the desired equalityreg( R/I ( D )) = reg( R/I ( D ′ )) + 1 . (cid:3) Remark 3.12.
Let D be a weighted oriented graph with non-trivial weights. In general, values of thenon-trivial weights are not necessarily equal. In the light of the above corollary, one can assign w i = 1 for allnon-trivial sink weights as the base case and gradually obtain the multigraded Betti numbers for arbitraryvalues of the non-trivial sink weights.4. Algebraic Invariants via Upper-Koszul Simplicial Complexes
The main focus of this section is to obtain formulas for the projective dimension and regularity of edgeideals of weighted oriented graphs by exploiting the structure of related upper-Koszul simplicial complexes.Structure of an upper-Koszul simplicial complex heavily rely on the choice of a multidegree. An “optimal”choice for a multidegree can be achieved by encoding the weights of all vertices in the multidegree. Weuse the word optimal to emphasize that this particular multidegree can lead us to a unique extremal Bettinumber which in turn enables us to compute the projective dimension and the regularity.
Theorem 4.1. [13, Hilbert’s Syzygy Theorem]
Every finitely generated graded module M over the ring R = k [ x , . . . , x n ] has a graded free resolution of length ≤ n. Hence pdim( M ) ≤ n . y Hilbert’s Syzygy Theorem, pdim( R/I ) ≤ n for any homogeneous ideal I ⊆ R. It is well-known thatthis bound is tight. A famous example for this instance is the graded maximal ideal m = ( x , . . . , x n ) asthe Koszul complex on the variables x , . . . , x n gives a minimal free resolution of R/ m of length n. In arecent paper [1], the class of monomial ideals with the largest projective dimension are characterized usingdominant sets and divisibility conditions.In the first result of this section, we characterize the structure of all weighted oriented graphs on n verticessuch that projective dimension of their edge ideals attain the largest possible value. Theorem 4.2.
Let D be a weighted oriented graph on the vertices V ( D ) = { x , . . . , x n } . Then pdim( R/I ( D )) = n if and only if there is an edge e = ( x j , x i ) oriented towards x i for each x i ∈ V ( D ) such that x j has anon-trivial weight.Furthermore, pdim( R/I ( D )) = n if and only if β n, b ( R/I ( D )) = 0 where b = ( w , . . . , w n ) ∈ N n . Proof.
Observe that pdim(
R/I ( D )) = n if and only if there exists a multidegree a ∈ N n such that β n, a ( R/I ( D ))is non-zero. It follows from Theorem 2.2 that β n, a ( R/I ( D )) = dim k e H n − ( K a ( I ( D ); k )) = 0which happens only when F = { x , . . . , x n } is a minimal non-face of K a ( I ( D )). Note that each a i ∈ { , , w i } .Otherwise, β n, a ( R/I ( D )) = 0 by Remark 2.3.Let ∆ = K a ( I ( D )) and F i := F \ { x i } for each x i ∈ F . Recall that F is a minimal non-face of ∆ whenever F / ∈ ∆ and each F i ∈ ∆. It follows from the definition of upper-Koszul simplicial complexes that F / ∈ ∆ ⇐⇒ m := x a x F = n Y j =1 x a j − j / ∈ I ( D ) , and F i ∈ ∆ ⇐⇒ m i := x a x F i = x a i i (cid:16) Y j = i x a j − j (cid:17) ∈ I ( D ) for each i ∈ { , . . . , n } . Note that each a i = 0 when m / ∈ I ( D ) and each m i ∈ I ( D ). The monomial m i ∈ I ( D ) if and only if thereexists a minimal generator of I ( D ) associated to an edge oriented towards x i , say e = ( x k , x i ), such that x k x w i i ∈ I ( D ) divides m i for some x k ∈ V ( D ). This implies that a i = w i and a k = w k >
1. Therefore,pdim(
R/I ( D )) = n if and only if, for each x i ∈ V ( D ), there exists an edge e = ( x k , x i ) oriented towards x i such that w k >
1. Notice that there are no source vertices when pdim(
R/I ( D )) = n . The latter statementfollows from the conclusion that a i = w i for each i = 1 , . . . , n whenever β n, a ( R/I ( D )) = 0 . (cid:3) Note that β n, b ( R/I ( D ) is the unique extremal Betti number of R/I ( D ). Using this information, we canfurther deduce the formula for the regularity of R/I ( D ). Corollary 4.3.
Let D be a weighted oriented graph on the vertices V ( D ) = { x , . . . , x n } . Then pdim( R/I ( D )) = n if and only if reg( R/I ( D )) = n X i =1 w i − n. Proof.
Observe that reg(
R/I ( D )) = P ni =1 w i − n if and only if β n, b ( R/I ( D )) = 0 where b = ( w , . . . , w n ) ∈ N n . (cid:3) Remark 4.4.
Let D be a weighted oriented graph on n vertices and let b = ( w , . . . , w n ) ∈ N n be amultidegree corresponding to the least common multiple of all minimal generators of I ( D ). If β p, b ( R/I ( D ))is non-zero where p = pdim( R/I ( D ), then β p, | b | ( R/I ( D )) is the unique extremal Betti number of R/I ( D ).Because β i, a ( R/I ( D )) = 0 for a = ( a , . . . , a n ) ∈ N n such that a i > w i for some i = 1 , . . . , n by Remark 2.3. .1. Weighted Oriented Cycles.
Let C n denote a weighted oriented cycle on n vertices x , . . . , x n . Weshall assume that there exists at least one vertex x i such that w i > x i .Otherwise, C n can be considered as an unweighted, unoriented cycle whose Betti numbers are computed in[14].If C n has at least one sink vertex, using Corollary 3.10 (b), we can express the Betti numbers of I ( C n )recursively in terms of Betti numbers of a weighted oriented path on ( n −
1) vertices and a weight reducedform of C n . In the existence of one sink vertex, without loss of generality, we may assume that x n is a sinkby reordering vertices of C n . Corollary 4.5. If x n is a sink in C n such that w n > , then β i,j ( I ( C n )) = β i,j − ( I ( C ′ n )) − β i,j − ( I ( P n − )) + β i,j ( I ( P n − )) for all i > , j > where C ′ n is a weight reduced form of C n . If C n has no sink vertices (or source vertices), then C n must be endowed with the natural orientation (clockwiseor counter clockwise). If all vertices of a naturally oriented weighted cycle C n have non-trivial weights, itsprojective dimension and regularity can be computed by Theorem 4.2 and we recover one of the main resultsof [18]. Corollary 4.6. [18, Theorem 1.4.]
Let C n be naturally oriented weighted cycle where w i > for all i . Then pdim( R/I ( C n )) = n and reg( R/I ( C n )) = n X i =1 w i − n. Weighted Rooted Graphs.
In this subsection, we consider the possibility of having source verticesand their effects on algebraic invariants. In the existence of several source vertices, one needs more infor-mation on the structure of a weighted oriented graph to be able to compute its algebraic invariants. As anatural starting point, we consider weighted rooted graphs.
Definition 4.7.
A weighted graph D is called rooted if there is a vertex distinguished as the root and thereis a naturally oriented path (i.e., all edges on the path are in the same direction) from the root vertex toany other vertex in D . Orientation of D is determined by naturally oriented paths from the root to othervertices. Note that the only source vertex of a weighted rooted graph is its root. Remark 4.8. If D is a weighted rooted graph, it does not fit in the description of weighted oriented graphswhose edge ideals have the largest projective dimension in Theorem 4.2. Thus pdim( R/I ( D )) ≤ n − . Theorem 4.9.
Let D be a weighted rooted graph on the vertices { x , . . . , x n } with the root vertex x . Suppose w i ≥ for all i = 1 . Then pdim(
R/I ( D )) = n − and reg( R/I ( D )) = n X i =1 w i − n + 1 . Proof.
Let b = (1 , w , . . . , w n ) ∈ N n . It suffices to show that β n − , | b | ( R/I ( D )) is the unique extremal Bettinumber of R/I ( D ) . Consider the upper-Koszul complex K b ( I ( D )) of I ( D ). Let F = { x , . . . , x n } and F i = F \{ x i } for 2 ≤ i ≤ n. Our goal is to show that F is a minimal non-face of K b ( I ( D )). First observe that F / ∈ K b ( I ( D )) . Otherwise,by the definition of upper-Koszul simplicial complexes, we must have x Q ni =2 x w i − i ∈ I ( D ) which is notpossible because each generator of I ( D ) must be divisible by x w i i for some i ∈ { , . . . , n } .If each F i ∈ K b ( I ( D )), then F must be a minimal non-face. Note that m i := Q nj =1 x w j j x · · · ˆ x i · · · x n = x (cid:16) n Y i =1 ,j = i x w j − j (cid:17) x w i i ∈ I ( D ) ecause all non-source vertices have non-trivial weights and m i is divisible by x x w i i ∈ I ( D ) or x j x w i i ∈ I ( D )for some i ∈ { , . . . , n } . It implies that F i ∈ K b ( I ( D )) for each i . Thus F is an ( n − K b ( I ( D )). It follows from Theorem 2.2 that β n − , b ( R/I ( D )) = dim k e H n − ( K b ( I ( D )); k )) = 0 . Therefore, β n − , | b | ( R/I ( D )) is the unique extremal Betti number of R/I ( D ) by Theorem 4.2 and Remark 4.4. (cid:3) As an immediate consequence of Theorem 4.9, we recover several results from [18].
Corollary 4.10. [18, Theorem 1.2. and Theorem 1.3.]
Let D be weighted rooted forest with non-trivialweights. Then pdim( R/I ( D )) = n − and reg( R/I ( D )) = n X i =1 w i − n + 1 . Weighted Oriented Complete Graphs
Let K n denote a weighted oriented complete graph on n vertices { x , . . . , x n } for n > I ( K n ) denoteits edge ideal. Throughout this section, we may assume that there exists at least one vertex x p such that w p >
1. Otherwise, K n is the unweighted, unoriented complete graph and Betti numbers of its edge idealare well-understood from its independence complex due to Hochster’s formula. Definition 5.1.
A weighted oriented complete graph K n is called naturally oriented if the oriented edge setis given by { ( x i , x j ) : 1 ≤ i < j ≤ n } . Then the edge ideal of K n is I ( K n ) = ( x i x w j j : 1 ≤ i < j ≤ n ) . Since x is a source vertex, we set w = 1.In what follows, we provide formulas for the projective dimension and the regularity for the edge ideal ofa naturally oriented weighted complete graph. The key ingredient of the proof is the use of upper-Koszulsimplicial complex of I ( K n ). Theorem 5.2.
Let K n be a naturally oriented weighted complete graph such that w p > for some p ≥ .Then(a) pdim( R/I ( K n )) = n − and(b) reg( R/I ( K n )) = n X i =1 w i − n + 1 . Proof.
Let b = ( w , w , . . . , w n ) ∈ N n . We claim that β n − , b ( R/I ( K n )) is the unique extremal Betti numberof R/I ( K n ). As an immediate consequence of the claim, we obtain the expressions given in the statement ofthe theorem as the values of the projective dimension and the regularity of R/I ( K n ).In order to prove the claim, consider the upper Koszul simplicial complex of I ( K n ) in multidegree b anddenote it by ∆ := K b ( I ( K n )). For the first part of the claim, it suffices to show that the only minimalnon-faces of ∆ are ( n − F = { x , . . . , x n } and F = F \ { x } . Since x b − F = Q ni =2 x w i − i and x b − F = x Q ni =2 x w i − i are notcontained in I ( K n ), neither F nor F is a face in ∆. In addition, let F i,j := F \{ x i , x j } for each 1 ≤ i < j ≤ n .Observe that each F i,j is a face in ∆ because x b x F i,j = (cid:16) Y k = i,j x w k − k (cid:17) x w i i x w j j ∈ I ( K n ) . herefore, the upper Koszul simplicial complex ∆ has at least one minimal non-face of dimension ( n − k e H n − (∆; k )) = 0 . Hence, by Theorem 2.2, β n − , b ( R/I ( K n )) = β n − , b ( I ( K n )) = dim k e H n − (∆; k )) = 0 . Note that K n does not belong to the class of graphs expressed in Theorem 4.2 and it follows that pdim( R/I ( K n )) ≤ n −
1. Thus, β n − , | b | ( R/I ( K n )) is the unique extremal Betti number of R/I ( K n ) by Remark 4.4. (cid:3) Remark 5.3.
Recall that, when w i ≥ x i = x , the edge ideal I ( K n ) is the initial ideal of thevanishing ideal of a projective nested cartesian set. As mentioned in the introduction, reg( R/I ( K n )) is astrict upper bound for the degree of the evaluation map used in creating projective nested cartesian codes. Itwas shown in [7] that degree of the evaluation map must be less than P ni =1 w i − n + 1 for a projective nestedcartesian code to have an “optimal” minimum distance ([7, Theorem 3.8]). However, this upper bound isnot obtained by computing reg( R/I ( K n )) explicitly in [7] and the equality reg( R/I ( K n )) = P ni =1 w i − n + 1is rather concluded from their results (see [15, Proposition 6.3]). In a way, Theorem 5.2 part (b) recoversthis embedded conclusion.Above theorem completes the discussion of regularity and projective dimension of I ( K n ) when K n is naturallyoriented. If the orientation of a weighted complete graph is not known, finding regularity and projectivedimension through upper-Koszul simplicial complexes becomes a more difficult task. In the absence of anexplicit orientation, structure of the upper-Koszul simplicial complex of I ( K n ) is more complex. Thus, oneneeds to employ different techniques than upper-Koszul simplicial complexes.In the following result, we provide a recursive formula for the Betti numbers of I ( K n ) in the existence of asink vertex. This condition can be considered as a local property of K n . Note that if there exists at leastone sink vertex, by relabeling the vertices, we may assume that x n is a sink. Theorem 5.4. If x n is a sink in K n , we have β i,j ( R/I ( K n )) = β i,j ( R/I ( K n − )) + (cid:0) n − i (cid:1) + β i − ,j − w n ( R/I ( K n − )) : j = i + w n β i,j ( R/I ( K n − )) + β i − ,j − w n ( R/I ( K n − )) : j = i + w n for all i > . Proof.
Let J = ( x i x w n n : 1 ≤ i < n ). Then one can decompose I ( K n ) as a disjoint sum of I ( K n ) = I ( K n − )+ J where K n − is a weighted oriented complete graph on ( n −
1) vertices. It is clear that K n − is an inducedweighted oriented subgraph of K n . Note that the minimal free resolution of J is obtained from shifting theminimal free resolution of R/ ( x , . . . , x n − ) by degree w n . Thus J has a linear resolution. It then followsfrom Theorem 2.5 that I ( K n ) = I ( K n − ) + J is a Betti splitting because G ( J ) contains all the generators of I ( K n ) divisible by x n . Then, by Definition 2.4, β i,j ( R/I ( K n )) = β i,j ( R/I ( K n − )) + β i,j ( R/J ) + β i − ,j ( R/I ( K n − ) ∩ J )for i >
1. Our goal is to analyze each term of the above expression.It is immediate from the definition of J that β i,j + w n ( R/J ) = β i,j ( R/ ( x , . . . , x n − )) for i > . Recall thatKoszul complex is a minimal free resolution of the R -module R/ ( x , . . . , x n − ) and the only non-zero Bettinumbers occur when j = i. In particular, for 1 ≤ i ≤ n − , we have β i,i ( R/ ( x , . . . , x n − )) = (cid:18) n − i (cid:19) = β i,i + w ( R/J ) (5.1)as the only non-zero Betti numbers of J .Next, observe that I ( K n − ) ∩ J = ( x w n n ) I ( K n − ) . Then, we can express the Betti numbers of the intersectionin terms of iterated Betti numbers of I ( K n − ). More specifically, for all i > ,β i,j + w ( R/I ( K n − ) ∩ J ) = β i,j ( R/I ( K n − )) . (5.2) herefore, one can obtain the expressions given in the statement of the theorem by using Equation (5.1) andEquation (5.2). (cid:3) Corollary 5.5. If x n is a sink in K n , then(a) pdim( R/I ( K n )) ∈ { n − , n } and(b) reg( R/I ( K n )) = reg( R/I ( K n − )) + ( w n − . Proof.
Let p = pdim( R/I ( K n − )) and r = reg( R/I ( K n − ))). Then β p,p + r ( R/I ( K n − )) = 0 and β i,j ( R/I ( K n − )) =0 for i > p or j > p + r. By using Theorem 5.4, we obtain the following top non-zero Betti numbers of
R/I ( K n ). β p +1 ,p + r + w n ( R/I ( K n )) = 0 (5.3) β n − ,n − w n ( R/I ( K n )) = 0 (5.4)Hence, Equation (5.3) and Equation (5.4) imply thatreg( R/I ( K n )) = max { w n , r + w n − } = r + w n −
1= reg(
R/I ( K n − )) + ( w n − . Similarly, using the top non-zero Betti numbers, we havepdim(
R/I ( K n )) = max { n − , p + 1 } Since p ≤ n − , the projective dimension is either n − n. (cid:3) Remark 5.6.
If the underlying graph of a weighted oriented graph D on n vertices is a star, we call D aweighted oriented star graph. Let x n be the center of D . If x n is a sink vertex, we say D is a weightedoriented star with a center sink. The edge ideal of a weighted oriented star with a center sink x n is given as I ( D ) = ( x i x w n n : 1 ≤ i ≤ n − . As discussed in the proof of Theorem 5.4, the module
R/I ( D ) has a linear resolution and it is obtained byshifting the Koszul complex of R/ ( x , . . . , x n − ) by degree w n . Thenpdim( R/I ( D )) = n − R/I ( D )) = w n . Betti Numbers via Mapping Cone Construction
In this section, we provide a recursive formula for the Betti numbers of edge ideals of weighted orientedgraphs with at least one leaf vertex which is also a sink. We achieve it by employing a technique called the mapping cone construction . This technique is different than Betti splittings while being as powerful.Recall that Betti splitting is a method which allows one to express Betti numbers of an ideal in terms ofsmaller ideals. In a similar vein, mapping cone construction allows one to build a free resolution of an R -module M in terms of R -modules associated to M . In particular, given a short exact sequence0 −→ R/M ′ −→ R/M ′′ −→ R/M −→ M ′ , M ′′ and M are graded R -modules, the mapping cone construction provides a free resolution of M in terms of free resolutions of M ′ and M ′′ . For more details on the mapping cone construction, we referthe reader to [16]. In general, given minimal free resolutions for M ′ and M ′′ , the mapping cone constructiondoes not necessarily give a minimal free resolution of M . However, there are classes of ideals in which themapping cone construction provides a minimal free resolution for particular short exact sequences (see [6]).Let D be a weighted oriented graph with the vertex set V ( D ) = { x , . . . , x n } . A vertex is called a leaf ifthere is only one edge incident to it. In the existence of at least one leaf vertex which is also a sink, one canuse the mapping cone construction to describe Betti numbers of R/I ( D ) recursively. Note that there is norestriction on the overall orientation of D . heorem 6.1. Let D be a weighted oriented graph on the vertices x , . . . , x n with a leaf x n . Suppose x n isa sink vertex. Then the mapping cone construction applied to the short exact sequence −→ RI ( D \ x n ) : ( x n − x w n n ) ( − w n − x n − x wnn −−−−−−→ RI ( D \ x n ) −→ RI ( D ) −→ provides a minimal free resolution of R/I ( D ) . In particular, for any i and j , we have β i,j ( R/I ( D )) = β i,j ( R/I ( D \ x n )) + β i − ,j − w − ( R/I ( D \ x n ) : x n − )) . Proof.
Let D ′ denote the weighted oriented induced subgraph D\ x n of D and let x n − be the unique neighborof x n such that ( x n − , x n ) ∈ E ( D ). Since x w n n does not divide a minimal generator of I ( D ′ ), one has I ( D ′ ) : ( x n − x w n n ) = I ( D ′ ) : x n − . Then, it implies that the exact sequence0 −→ RI ( D ′ ) : ( x n − x w n n )) ( − w n − x n − x wnn −−−−−−→ δ RI ( D ′ ) −→ RI ( D ) −→ RI ( D ′ ):( x n − x wnn ) ( − w n − RI ( D ′ ) RI ( D ) .R/I ( D ′ ) : x n − x n − x wnn x wnn x n − (6.2)Let F : 0 · · · φ −→ F φ −→ F = R φ −→ R/I ( D ′ ) : ( x n − x w n n ) −→ G : 0 · · · ψ −−→ G ψ −−→ G = R ψ −−→ R/I ( D ′ ) −→ R/I ( D ′ ) : ( x n − x w n n ) and R/I ( D ′ ) , respectively. Then the mapping construc-tion applied to Equation (6.1) provides a free resolution of R/I ( D ) given by0 · · · ϕ −→ G ⊕ F ( − w n − ϕ −→ G ⊕ R ( − w n − ϕ −→ R ϕ −→ R/I ( D ) −→ ϕ i ’s are defined by ϕ = [ ψ − δ ] and ϕ i = (cid:20) ψ i ( − i δ i − φ i − (cid:21) for i > δ i : F i ( − w n − −→ G i is induced from the homomorphism δ .It follows from the factorization in Equation (6.2) that the entries of the matrix of δ i are not units. Since F and G are minimal free resolutions, then none of the entries in the matrix representation of ϕ i can beunits. Thus the mapping cone construction applied to Equation (6.1) results with a minimal free resolutionof R/I ( D ). In particular, this implies the following recursive formula for the Betti numbers of R/I ( D ) β i,j ( R/I ( D )) = β i,j ( R/I ( D ′ )) + β i − ,j − w n − ( R/I ( D ′ ) : x n − )for any i, j . (cid:3) Corollary 6.2.
Let D be a weighted oriented graph on the vertices x , . . . , x n such that x n is a leaf and asink vertex. Then(a) reg( R/I ( D )) = max { reg( R/I ( D \ x n )) , reg( R/I ( D \ x n ) : x n − ) + 1 } and(b) pdim( R/I ( D )) = max { pdim( R/I ( D \ x n )) , pdim( R/I ( D \ x n ) : x n − ) + 1 } . .1. Application.
Let P n denote a weighted naturally oriented path on n vertices. If all non-source verticeshave non-trivial weights, regularity and projective dimension formulas follow from Corollary 4.10. If oneallows non-source vertices to have trivial weights, computing the regularity and the projective dimensionbecomes a much more complicated task as these invariants heavily rely on the orientation of the graph andthe positions of non-trivial weights. Providing formulas for the regularity and projective dimension of anyweighted oriented graph is an open problem.In an attempt to address this general problem, weighted naturally oriented paths and cycles are studied in[5]. Indeed, positions of non-trivial weights is quite crucial in computing the regularity (see [5, Theorem5.9]). Particularly, whenever there are consecutive non-trivial weight vertices x i and x i +2 such that x i +1 has a trivial weight, then x i and x i +2 can not “contribute” to the regularity simultaneously. One needs toconsider the contribution of the one or the other and determine the regularity by taking the maximums ofcorresponding contributions (see [5, Notation 5.4, Definition 5.6 and Theorem 5.9]).In what follows, we consider a more general case than that of [5] and provide a recursive formula for theBetti numbers of egde ideal of a weighted oriented path. Furthermore, our recursive formulas can offer anexplanation about the “distance two away condition” of [5, Theorem 5.9]. Corollary 6.3.
Let P n be a weighted oriented path on the vertices x , . . . , x n such that ( x n − , x n − ) , ( x n − , x n ) ∈ E ( P n ) .(a) If x n − is a non-trivial vertex, then reg( R/I ( P n )) = reg( R/I ( P n − )) + w n − . (b) If x n − is a trivial vertex, then reg( R/I ( P n )) = max { reg( R/I ( P n − )) , reg( R/I ( P n − )) + w n } Proof. (a) Suppose w n − > . Then I ( P n − ) : x n − = I ( P ′ n − ) where P ′ n − is a weighted reduced form of P n − on x n − . Thus, it follows from Corollary 6.2 thatreg( R/I ( P n )) = max { reg( R/I ( P n − )) , reg( R/I ( P ′ n − )) + w n } . (6.3)Since x n − is a sink vertex with a non-trivial weight in P ′ n − , we have reg( R/I ( P n − )) = reg( R/I ( P ′ n − ))+1 . By making use of Corollary 3.11 and the fact that w n ≥
1, Equation (6.3) yields to the followingreg(
R/I ( P n )) = reg( R/I ( P ′ n − )) + w n = (reg( R/I ( P n − )) −
1) + w n . (b) Suppose w n − = 1 . Let I ′ := I ( P n − ) : x n − = I ( P n − ) + ( x n − ). Since x n − does not divide anyminimal generator of I ( P n − ) , one can obtain the minimal free resolution of R/I ′ by taking the tensorproduct of minimal free resolutions of R/I ( P n − ) and R/ ( x n − ). Thenreg( R/I ′ ) = reg( R/I ( P n − )) , and the statement follows from Corollary 6.2 (cid:3) Remark 6.4.
Let P n be a weighted naturally oriented path on the vertices x , . . . , x n . If w n > w n − = 1, we can use Corollary 6.3 part (b) to determine the regularity of R/I ( P n ) inductively by takingthe maximum of the following two expressions.reg( R/I ( P n )) = max { reg( R/I ( P n − )) , reg( R/I ( P n − )) + w n } Note that the ideal in the first expression contains x n − in its support. However, vertex x n is not in thesupport of the first ideal and its weight does not contribute to the regularity in the first expression. On theother hand, the second expression contains w n , the weight contribution of x n , and the ideal associated toit does not contain x n − in its support. Thus, Corollary 6.3 part (b) exhibits the behavior of distance twoaway non-trivial weights in the regularity computations. . Questions
Question 7.1.
Let D be a weighted oriented graph and G be its underlying graph on n vertices.(a) Is there any relation between the Betti numbers of R/I ( G ) and R/I ( D ) ?(b) Is pdim( R/I ( G )) ≤ pdim( R/I ( D )) ?(b) Is reg( R/I ( G )) ≤ reg( R/I ( D )) ? Intuition and computational evidence suggests that both questions have positive answers.Our next question is motivated by Corollary 3.10 and Corollary 3.11. In these two corollaries we providea positive answer to the following questions when x i is a non-trivial sink vertex. It is natural to wonderwhether it is true for any non-trivial vertex x i . Question 7.2.
Let D be a weighted oriented graph with a non-trivial weight vertex x i and let D ′ be a weightreduced form of D on x i . (a) When is β i ( R/I ( D )) = β i ( R/I ( D ′ )) for all i ≥ ?(b) Is there any relation between β i,j ( R/I ( D )) and β i,j ( R/I ( D ′ )) ?(c) When is pdim( R/I ( D )) = pdim( R/I ( D ′ )) ?(d) When is reg( R/I ( D )) = reg( R/I ( D ′ )) + 1 ? Example 7.3.
Let I ( D ) = ( x x , x x , x x , x x ). Consider the following chain of weight reductions where D ′ is a weight reduced form of D on x , D ′′ is a weight reduced form of D ′ on x , and D ′′′ is a weight reducedform of D ′′ on x with the corresponding edge ideals given as I ( D ′ ) = ( x x , x x , x x , x x ) I ( D ′′ ) = ( x x , x x , x x , x x ) I ( D ′′′ ) = ( x x , x x , x x , x x ) . Below, we present the Betti tables of I ( D ) , I ( D ′ ) , I ( D ′′ ) , and I ( D ′′′ ), in order. Based on the above Betti tables, equalities in Question 7.2 (a),(c),(d) hold for D and D ′ . However, we havepdim( R/I ( D ′ )) = pdim( R/I ( D ′′ )) + 1 , reg( R/I ( D ′′ )) = reg( R/I ( D ′′′ )) , ndicating that suggested equalities in Question 7.2 are not always valid. Computational experiments suggestthat the desired equalities hold for D and a reduced form of D on x i where w i > References [1] Guillermo Alesandroni. Monomial ideals with large projective dimension.
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Department of Mathematics, Yale University, New Haven, CT 06520-8283
E-mail address : [email protected] Department of Mathematics and Statistics, University of South Alabama, 411 University Blvd North, Mobile, AL36688-0002, USA
E-mail address : [email protected]@southalabama.edu