Multivariate generalized splines and syzygies on graphs
MMULTIVARIATE GENERALIZED SPLINES AND SYZYGIES ON GRAPHS
SELMA ALTINOK AND SAMET SARIO ˘GLAN
Abstract.
Given a graph G whose edges are labeled by ideals of a commutative ring R with identity,a generalized spline is a vertex labeling of G by the elements of R so that the difference of labels onadjacent vertices is an element of the corresponding edge ideal. The set of all generalized splines on agraph G with base ring R has a ring and an R -module structure.In this paper, we focus on the freeness of generalized spline modules over certain graphs with thebase ring R = k [ x , . . . , x d ] where k is a field. We first show the freeness of generalized spline moduleson graphs with no interior edges over k [ x, y ] such as cycles or a disjoint union of cycles with freeedges. Later, we consider graphs that can be decomposed into disjoint cycles without changing theisomorphism class of the syzygy modules. Then we use this decomposition to show that generalizedspline modules are free over k [ x, y ] and later we extend this result to the base ring R = k [ x , . . . , x d ]under some restrictions. Introduction
The concept of spline has been studied in two major approaches: Classical splines and generalizedsplines. Classical splines are collections of polynomials defined on the faces of a polyhedral complexthat agree to a certain degree of smoothness on the intersection of faces. They are useful tools tocontrol the curvature of objects in industry and have many applications related to numerical analysis,geometric design and solutions of partial differential equations. They form a module. For these andother applications, it is useful to study splines as a module.Generalized splines are defined on edge labeled graphs. Given a finite graph G = ( V, E ) and acommutative ring R with identity, an edge labeling function α is a function that labels the edges of G by the ideals of R . The pair ( G, α ) is called an edge labeled graph. A generalized spline on an edgelabeled graph (
G, α ) is a vertex labeling F ∈ R | V | such that for each edge uv ∈ E , the difference f u − f v isan element of the ideal α ( uv ) ∈ R where f v denotes the label on vertex v ∈ V . The set of all generalizedsplines on ( G, α ) with base ring R , denoted by R ( G,α ) , has a ring and an R -module structure. If R isa multivariate polynomial ring, we call such generalized splines as multivariate generalized splines. Weuse two notations for generalized splines: the column matrix notation with entries in order from bottomto top and the vector notation. Consider the edge labeled graph ( G, α ) in Figure 1. v v vx x + x + Figure 1.
Edge labeled graph (
G, α ) a r X i v : . [ m a t h . A C ] F e b SELMA ALTINOK AND SAMET SARIO ˘GLAN
A generalized spline on (
G, α ) can be presented by F = x + 2 x + 1 x + 11 or F = (1 , x + 1 , x + 2 x + 1).The motivation for studying generalized splines is based on algebraic geometry and topology. Studiesin these areas showed that the ring structure of generalized splines corresponds to the equivariantcohomology rings of toric and other algebraic varieties [5, 11, 12]. Goresky and the others [5] studiedthe combinatorial structure of the equivariant cohomology ring corresponding to an algebraic varietyX with a proper torus action. In this work, they defined the edge labeled graph ( G X , α ) correspondsto the algebraic variety X and showed that the equivariant cohomology ring of X agrees with the ringstructure of R ( G X ,α ) where R is a polynomial ring. Thus the algebraic structure of generalized splinesover polynomial rings has importance.Our interest in the paper is to study the freeness of generalized splines as a module. In [8], Rosestudied classical splines and observed that the module of splines on a polyhedral complex ∆ can beviewed as a direct sum of the syzygy module of its dual graph with edges labeled by powers of linearforms. In [9], by using a decomposition of an edge labeled graph G without changing the isomorphismclass of the syzygy module, Rose obtained results related to the freeness of the syzygy module generatedby linear edge labels that meet certain conditions. We extend these ideas to generalized splines onarbitrary graphs which can be decomposed into disjoint cycles and free edges.This paper consists of five sections. In Section 2, we present basic definitions and properties related toprojective modules and dimension. In Section 3, we discuss the rank of a cycle and the relation betweenrank and projective dimension. In Section 4, we first introduce the module M v , then we define the edgedecomposition operations and finally we analyze the effects of these operations on the isomorphism classof the generalized spline module. The main results of this paper are presented in Section 5. In thissection, we first prove the freeness of R ( G,α ) where R = k [ x, y ] and G is a cycle. Then we generalize thisresult to a graph containing only one cycle or having no interior edges. In the case of R = k [ x , . . . , x d ],we first give freeness criteria for generalized spline modules on cycles and then we generalize our resultsto graphs that decompose into disjoint cycles and free edges under some conditions as the final resultof the paper.In the rest of the paper, we refer to multivariate generalized splines as splines.2. Projective Modules and Projective Dimension
Let P be an R -module. P is said to be projective if for every surjective module homomorphism f : A → B and every module homomorphism g : P → B , there exists a module homomorphism h : P → A such that the following diagram commutes, namely f ◦ h = g . P h (cid:127) (cid:127) g (cid:15) (cid:15) A f (cid:47) (cid:47) B (cid:47) (cid:47) R is a principal ideal domain,then every projective R -module is free. Every finitely generated projective module over polynomial ringsis also free by Quillen-Suslin Theorem. Basic properties of projective modules are given below. Proposition 2.1.
Let P be an R -module. Thena) P is projective if and only if all short exact sequences → A → B → P → of R -modules split. ULTIVARIATE GENERALIZED SPLINES AND SYZYGIES ON GRAPHS 3 b) P is projective if and only if it is a direct summand of a free R -module.Proof. See Proposition 6.73 and Theorem 6.76 in [10]. (cid:3)
Proposition 2.2.
Let { P i | i ∈ I } be a family of R -modules. Then P = (cid:76) i ∈ I P i is projective if and onlyif P i is projective for all i ∈ I .Proof. See Proposition 6.75 in [10]. (cid:3)
Let A be an R -module. A projective resolution of A is an exact sequence P • : . . . → P → P → P → A → P n is projective. Every R -module A has a projective resolution. The length of a shortestprojective resolution of A is called the projective dimension of A , denoted by pd A . If A has no finiteprojective resolution, then pd A = ∞ . An R -module A is projective if and only if pd A = 0.The following theorem is given without proof in [7] page 76. Theorem 2.3.
Let → B → P → A → be a short exact sequence of R -modules with P is projective.Then pd A = a > implies pd B = a − .Proof. Let pd A = a > B = b . We show that a = b + 1. Since pd B = b , then there is aprojective resolution of B with length b such as 0 → P b → . . . → P → P → B → A with length b + 1 by setting 0 → P b → . . . → P → P → P → A →
0. Hencewe conclude that a ≤ b + 1.Now assume that a < b + 1. Let 0 → Q b → . . . → Q → Q → B → B with length b . This complex breaks up into short exact sequences 0 → B i +1 → Q i → B i → R -modules with B = B and B b +1 = 0. We can do the same thing to a projective resolution of A withlength a . The resolution 0 → P a → . . . → P → P → A → → A i +1 → P i → A i → R -modules with A = A and A a +1 = 0. Applying Schanuel Lemma to thefollowing short exact sequences 0 → B → P → A → → A → P → A → P ⊕ A ∼ = P ⊕ B . Now applying Schanuel Lemma to the following short exact sequences0 → P ⊕ B → P ⊕ Q → P ⊕ B → → P ⊕ A → P ⊕ P → P ⊕ A → P ⊕ Q ⊕ P (cid:124) (cid:123)(cid:122) (cid:125) Projective ⊕ A ∼ = P ⊕ P ⊕ P (cid:124) (cid:123)(cid:122) (cid:125) Projective ⊕ B .Iterating this argument yields Projective ⊕ A j +1 ∼ = Projective ⊕ B j . Here A a is projective, so thatProjective ⊕ A a is projective. Hence B a − is projective by Theorem 2.2, so the resolution for B ends at B a − or before. Then b ≤ a −
1, which is a contradiction to our assumptation. Hence a = b + 1. (cid:3) An upper bound for the projective dimension of modules over polynomial rings is given by HilbertSyzygy Theorem.
Theorem 2.4. (Hilbert Syzygy Theorem)
Let R = k [ x , . . . , x n ] be the polynomial ring and M be afinitely generated R -module. Then pd M ≤ n . SELMA ALTINOK AND SAMET SARIO ˘GLAN
Proof.
See Corollary 10.167 in [10]. (cid:3)
In order to give a useful property of the projective dimension, we need the concept of regular se-quences. Given a ring R , an R -regular sequence is a finite sequence of elements r , . . . , r k ∈ R such that r i is not a zero divisor of R / (cid:104) r ,...,r i − (cid:105) for each i and (cid:104) r , . . . , r k (cid:105) R (cid:54) = R . Here k is called the length ofthe sequence. Theorem 2.5.
Let R be a commutative ring with identity and I ⊂ R be an ideal generated by an R -regular sequence of length n . Then pd R / I = n .Proof. Let I = (cid:104) x , . . . , x n (cid:105) where x , . . . , x n is an R -regular sequence. We use induction on n . For n = 1, we have the following short exact sequence(1) 0 → R f −→ R π −→ R /
1. Assume that pd R /
1, we fix R ∗ = R /
1. Hence we get pd R ∗ ( R ∗ /
Let ( C n , α ) be an edge labeled cycle with edge labels l , . . . , l n . The rank of C n , denoted by rk C n isthe dimension of the linear span (cid:104) l , . . . , l n (cid:105) . In particular, the rank can be seen as the codimension ofthe intersection of the hyperplanes l i = 0 for all i when all l i are linear.Given an edge labeled cycle ( C n , α ) with edge labels { l , . . . , l n } , fix the ideal I = (cid:104) l , . . . , l n (cid:105) . In thecase all edge labels are homogeneous and linear, the relation between rk C n and pd R / I is given by thefollowing theorem. Theorem 3.1.
Let ( C n , α ) be an edge labeled cycle with all edge labels are linear and homogeneous.Then pd R / I = rk C n . Proof.
See Theorem 4.2 in [8]. (cid:3)
If the edge labels are not linear, then the statement of Theorem 3.1 does not hold. In this case, thereis no even inequality between rk C n and pd R / I . The following example illustrates this fact: Example 3.2.
Consider the following edge labeled 3-cycles ( G , α ) and ( G , α ) with base rings k [ x, y, z ] and k [ x, y, z, t ] respectively. ULTIVARIATE GENERALIZED SPLINES AND SYZYGIES ON GRAPHS 5 v v v x yz + xz y + x y + G v v v G x y xz yt + Figure 2.
Relation between rk C and pd R / I Fix the ideals I = (cid:104) x + yz , x + y , xz + y (cid:105) and I = (cid:104) x , y , xz + yt (cid:105) . It can be computed byMacaulay2 [6] that rk G = 3, pd (cid:0) k [ x,y,z ] / I (cid:1) = 2 and rk G = 3, pd (cid:0) k [ x,y,z,t ] / I (cid:1) = 4.Even though Theorem 3.1 does not hold in general, we have the following theorem for cycles of rankat most two. Theorem 3.3.
Let ( C n , α ) be an edge labeled cycle with rank at most two. Then pd R / I ≤ rk C n .Proof. Let ( C n , α ) be an edge labeled cycle with edge labels { l , . . . , l n } . Fix the ideal I = (cid:104) l , . . . , l n (cid:105) .We suppose that rk C n = 2. Thus I has a generating set with two elements, which may not be minimal.Without loss of generality, say I = (cid:104) l , l (cid:105) .If l , l are coprime, then l , l is an R -regular sequence. In order to see this, assume that l , l arecoprime but not an R -regular sequence. Hence l is a zero divisor of R / (cid:104) l (cid:105) , namely there exists anelement 0 (cid:54) = r ∈ R / (cid:104) l (cid:105) such that rl ∈ (cid:104) l (cid:105) and so l divides rl . Since l , l are coprime l divides r butthis contradics the fact that 0 (cid:54) = r ∈ R / (cid:104) l (cid:105) . Therefore l , l is a regular sequence so that by Theorem 2.5,pd R / I = 2.If a greatest common divisor ( l , l ) (cid:54) = l or l then there exists a nonconstant common factor r ∈ R satisfying l = f r and l = f r , where f , f ∈ R are coprime, so that (cid:104) f , f (cid:105) ∼ = (cid:104) l , l (cid:105) as an R -module isomorphism via the map α : (cid:104) f , f (cid:105) → (cid:104) l , l (cid:105) α ( f ) = rf. Hence pd R / I = pd R / (cid:104) f ,f (cid:105) = 2 as explained above.If we assume that without loss of generality l = rl , where r ∈ R is a nonconstant, then I = (cid:104) l , l (cid:105) = (cid:104) l (cid:105) . Since R is an integral domain, l is an R -regular sequence of length 1 and hence pd R / I = 1 byTheorem 2.5. Thus the inequality holds. In particular, when r is constant or equivalenty rk C n = 1 wehave pd R / I = 1 = rk C n . (cid:3) Decomposition of Edge Labeled Graphs
In this section, we first present the module M v introduced by Tymoczko and the others in [4] to givea characterization of R ( G,α ) . Then we touch on decompositions of an edge labeled graph given by Rosein [9].Let ( G, α ) be an edge labeled connected graph and fix a vertex v ∈ V . Then every spline f ∈ R ( G,α ) can be expressed uniquely as f = r + f v , where is the trivial spline whose all entries are 1, f v ∈ R ( G,α ) with f vv = 0 and r = f v . In order to see this, define f v to be the spline f v = f − r . Hence f vv = f v − r v = r − r = 0. This observation leads to the following theorem. SELMA ALTINOK AND SAMET SARIO ˘GLAN
Theorem 4.1. [4]
Let ( G, α ) be an edge labeled connected graph and v ∈ V . If M v = (cid:104) f : f v = 0 (cid:105) then R ( G,α ) = R ⊕ M v as R -modules. We can relate the module M v to the syzygy module generated by the edge labels as follows: Supposethat G is the dual graph of a hereditary polyhedral complex ∆ and R = R [ x , . . . , x n ]. Let l e be anaffine form that generates the polynomials vanishing on the intersection of faces in ∆ corresponding toeach edge e of G . Define the edge labeling function α as α ( e ) = l e and let B ( G,α ) = (cid:40) ( r , . . . , r | E | ) ∈ R | E | : (cid:88) e ∈ C r e l e = 0 for all cycles C ∈ G (cid:41) . Rose [8] proved that R ( G,α ) ∼ = R ⊕ B ( G,α ) as R -modules. Together with Theorem 4.1, we conclude that M v ∼ = B ( G,α ) as R -modules for a fixed v ∈ V . We use B ( G,α ) instead of M v for the rest of the paper.Given an edge labeled graph ( G, α ), a cycle C in G that does not contain any smaller cycle is calleda minimal cycle. The set of all minimal cycles in G is denoted by B . This set is also called a cycle spacebasis of G , see [3]. The syzygy module B ( G,α ) can be presented as the kernel of the matrix A as follows:The rows of A are indexed by the elements of B and the columns are indexed by the edges of G . Then A ij = , if e j / ∈ C i ± α ( e j ) , if e j ∈ C i . The sign of α ( e j ) depends on the orientation of e j in C i . If G has no cycles, A can be taken to be therow matrix (0 , . . . , e be an edge contained in two or more cycles in G . Our goal is to find conditions for deleting e in one of the cycles, without changing the isomorphism class of B ( G,α ) . Definition 4.2.
Let (
G, α ) be an edge labeled graph and e ∈ E . If e is contained in two or more cycles,we call e an interior edge. If e is contained in only one cycle, we call e an exterior edge, otherwise wecall e a free edge.We want to delete all interior edges from a cycle C without changing the isomorphism class of B ( G,α ) if it is possible. We give the following definition. Definition 4.3.
Let (
G, α ) be an edge labeled graph and e i ∈ E . If e i is an interior edge of a cycle C ⊂ G such that l i ∈ (cid:104) e ∈ C | e is an exterior edge (cid:105) , we say that e i is removable from C . If G (cid:48) isobtained from G by a sequence of removals of interior edges from cycles, and G (cid:48)(cid:48) has the same matrixas G (cid:48) , we say that G decomposes into G (cid:48)(cid:48) . If a cycle C has no interior edges in G (cid:48) , we say that C splitsoff from G .We represent the removal of an edge e i from a single cycle algebraically, by replacing the currentvalue of l i with zero. Notice that when a cycle C splits off, it does so with the removable edges deletedand their endpoints identified. Example 4.4.
Consider the following edge labeled diamond graph ( D , , α ). ULTIVARIATE GENERALIZED SPLINES AND SYZYGIES ON GRAPHS 7 v v v v l x = l x = l y = − l y = − l x y = − − l x y = − − ( ) , G α Figure 3.
Edge labeled D , The edge l is an interior edge. It is removable from the right-side 3-cycle of ( D , , α ) since l = yl − xl .Hence we obtain the following graph G (cid:48) by setting l zero on the right-side 3-cycle. v v v v x x y − y − x y − − ( ) ', ' G α Figure 4.
Removing an interior edgeFinally we split off the right-side 3-cycle as follows: v x v v v x v y − y − x y − − ( ) '', '' G α Figure 5.
Splitting off cyclesThe following theorem shows that the decomposition operations do not affect the isomorphism classof the module B ( G,α ) . Theorem 4.5 and Corollary 4.7 are proved in [9] for classical spline case. However,the proofs are identical for generalized splines. Theorem 4.5. If G decomposes into G (cid:48)(cid:48) , then B ( G,α ) ∼ = B ( G (cid:48)(cid:48) ,α (cid:48)(cid:48) ) as R -modules. SELMA ALTINOK AND SAMET SARIO ˘GLAN
Proof.
See Theorem 4.7 in [9]. (cid:3)
Example 4.6.
Consider the edge labeled graphs given in Figures 3, 4 and 5. Recall that the module B ( G,α ) is represented as the kernel of the following matrix A = (cid:32) x − y − x − y − y x − x − y (cid:33) . Since B ( G (cid:48) ,α (cid:48) ) and B ( G (cid:48)(cid:48) ,α (cid:48)(cid:48) ) are both given by the following matrix: A (cid:48) = (cid:32) x − y − x − y − y x (cid:33) , where the matrix A (cid:48) corresponds to the graph G (cid:48) obtained by removing the edge e from the right-side3-cycle of ( D , , α ) , simply setting the label of e to be 0 in Figure 3, they are isomorphic. Now weconsider a map between B ( G,α ) and B ( G (cid:48) ,α (cid:48) ) defined by φ : B ( G,α ) → B ( G (cid:48) ,α (cid:48) ) ( r , r , r , r , r ) = ( r , r , r + yr , r − xr , r ) . It can be easily observed that φ is an R -module isomorphism so that B ( G,α ) ∼ = B ( G (cid:48)(cid:48) ,α (cid:48)(cid:48) ) . Alternatively, G decomposes into G (cid:48) and finally into two disjoint cycles in G (cid:48)(cid:48) . Hence by Theorem 4.5, we obtain thesame result. Corollary 4.7. If G (cid:48)(cid:48) is a decomposition of G into disjoint cycles C , . . . , C s and p free edges, then B ( G,α ) ∼ = s (cid:77) i =1 B ( C i ,α i ) ⊕ R p . as R -modules.Proof. See Corollary 4.9 in [9]. (cid:3)
Corolloary 4.7 shows that, if G has no interior edges then B ( G,α ) is isomorphic to the direct sum ofthe syzygy modules of cycles in G and the number of free edges copy of R .In case all edge labels of graph G are homogeneous polynomials of the same degree, each isomorphismin Theorem 4.1, Theorem 4.5 and Corollary 4.7 is a graded R -module isomorphism. In this case, wecan talk about the Hilbert series of the syzygy module B ( G,α ) . Proposition 4.8.
Let ( G, α ) be an edge labeled graph such that all edge labels are homogeneous withthe same degree r . Then R ( G,α ) ∼ = R ⊕ B ( G,α ) as graded R -modules with a degree shift in B ( G,α ) of r .Proof. See Theorem 2.2 in [8]. (cid:3)
Proposition 4.9. If G decomposes into disjoint cycles C , . . . , C s and p free edges, then HS ( B ( G,α ) ) = HS ( B ( C ,α ) ) + · · · + HS ( B ( C s ,α s ) ) + p (1 − t ) d where HS ( M ) denotes the Hilbert series of M .Proof. See Corollary 4.9 in [9]. (cid:3)
Example 4.10.
The edge labeled graph (
G, α ) in Figure 6 decomposes into disjoint cycles C and C in Figure 7. ULTIVARIATE GENERALIZED SPLINES AND SYZYGIES ON GRAPHS 9 v v v v x y − − x y − − x x y − y − ( ) , G α Figure 6.
Edge labeled D , v v v v v x y − − l x x y − y − ( ) , C α ( ) , C α Figure 7.
Decomposition of D , Here we have B ( G,α ) = (cid:10) ( y , x , , , , (0 , , x , y , , (1 , − , − , , (cid:11) ,B ( C ,α ) = (cid:10) ( y , x , , (1 , − , (cid:11) ,B ( C ,α ) = (cid:10) ( x , y ) (cid:11) . It follows that the Hilbert series of these syzygy modules are given by HS ( B ( G,α ) ) = 1 + 2 t (1 − t ) , HS ( B ( C ,α ) ) = 1 + t (1 − t ) , HS ( B ( C ,α ) ) = t (1 − t ) . We conclude that HS ( B ( G,α ) ) = HS ( B ( C ,α ) ) + HS ( B ( C ,α ) ), in agreement with Proposition 4.9. Inparticular, HS ( R ( G,α ) ) = t + 2 t (1 − t ) + 1(1 − t ) with a degree shift in B ( G,α ) of 2 by Proposition 4.8.More details on graded generalized spline modules can be found in [2].5. Freeness of the Spline Module R ( G,α ) Let the base ring R = k [ x , . . . , x d ] and ( C n , α ) be an edge labeled cycle with edge labels { l , . . . , l n } .Fix the ideal I = (cid:104) l , . . . , l n (cid:105) . Then there exists a canonical short exact sequence as follows:(2) 0 → I → R → R / I → I = pd R / I − R / I > → B ( C n ,α ) i −→ R n f −→ I → i is the inclusion map and f : R n → I defined by f ( r , . . . , r n ) = n (cid:80) t =1 r t l t . From this short exactsequence, we conclude that pd B ( C n ,α ) = pd I − I >
0. Combining two resultsobtained from (2) and (3), we have(4) pd B ( C n ,α ) = pd R / I − . This observation leads us to the following result.
Proposition 5.1.
Let ( C n , α ) be an edge labeled cycle and the base ring R be the bivariate polynomialring k [ x, y ] . Then R ( C n ,α ) is a free R -module. Proof.
Fix the ideal I = (cid:104) l , . . . l n (cid:105) generated by the edge labels on ( C n , α ). If pd R / I > I > B ( C n ,α ) = pd R / I − ≤ − R / I ≤ B ( C n ,α ) = 0 and therefore B ( C n ,α ) is a projective R -module. Since R ( C n ,α ) ∼ = B ( C n ,α ) ⊕ R , the spline module R ( C n ,α ) is alsoprojective. By Quillen-Suslin Theorem, R ( C n ,α ) is a free R -module.If pd R / I = 0, then R / I is a projective R -module and hence the short exact sequence (2) splits byTheorem 2.1 (a), which means R ∼ = I ⊕ R / I . Here I is also projective R -module by Theorem 2.2 and theshort exact sequence (3) splits so that R n ∼ = B ( C n ,α ) ⊕ I . Therefore B ( C n ,α ) is projective by Theorem 2.2.Since R ( C n ,α ) ∼ = B ( C n ,α ) ⊕ R , the spline module R ( C n ,α ) is also projective. By Quillen-Suslin Theorem, R ( C n ,α ) is a free R -module.If pd I = 0, then I and B ( C n ,α ) are projective R -modules as explained above. Hence R ( C n ,α ) is alsoprojective and so it is a free R -module. (cid:3) We can generalize Proposition 5.1 to graphs that contain only one cycle as follows:
Corollary 5.2.
Let ( G, α ) be an edge labeled graph and R be the bivariate polynomial ring k [ x, y ] . If G contains only one cycle, then R ( G,α ) is a free R -module.Proof. Let C n ⊂ G be the only cycle contained in G with edge labels { l , . . . l n } . Then B ( G,α ) ∼ = B ( C n ,α (cid:48) ) ⊕ R p by Corollary 4.7 where p is the number of the free edges of G . Here B ( C n ,α (cid:48) ) is projectiveby Proposition 5.1 and so is B ( G,α ) . Hence R ( G,α ) is also projective and it is free. (cid:3) Another generalization of Proposition 5.1 can be given as follows:
Corollary 5.3.
Let ( G, α ) be an edge labeled graph with no interior edges and R be the bivariatepolynomial ring k [ x, y ] . Then R ( G,α ) is a free R -module.Proof. If G has no interior edge, then we can assume that G consists of disjoint cycles C , . . . , C s and p free edges. Then B ( G,α ) ∼ = s (cid:76) i =1 B ( C i ,α i ) ⊕ R p by Corollary 4.7. Here B ( C i ,α i ) is projective for all i byProposition 5.1 and hence B ( G,α ) is also projective. Thus R ( G,α ) is free. (cid:3) If G has interior edges, then R ( G,α ) may not be free even the base ring is k [ x, y ]. We consider thefollowing example: Example 5.4.
Let (
G, α ) be the diamond graph as in the Figure 8. v v v v xy y − y y + x y + x − Figure 8.
Edge labeled diamond graphIn this example e is an interior edge which is not removable. Computations on CoCoA 4.7.5 [1] showthat the spline module R ( G,α ) is not free. ULTIVARIATE GENERALIZED SPLINES AND SYZYGIES ON GRAPHS 11
As a result of Corollary 4.7, we obtain the following outcome:
Corollary 5.5.
Let R = k [ x, y ] . If ( G, α ) decomposes into disjoint cycles with p free edges, then R ( G,α ) is a free R -module. Example 5.6.
Consider the diamond graph G in Figure 3, discussed in Example 4.4. This graphdecomposes into two disjoint cycles as in Figure 5 and hence R ( G,α ) is free by Corollary 5.5.Proposition 5.1 holds also for cycles of rank at most two on R = k [ x , . . . , x d ]. In this case, we useTheorem 3.3 to prove the proposition below: Proposition 5.7.
Let ( C n , α ) be an edge labeled cycle and the base ring R be the polynomial ring k [ x , . . . , x d ] . If rk C n ≤ , then R ( C n ,α ) is a free R -module.Proof. Fix the ideal I = (cid:104) l , . . . l n (cid:105) generated by the edge labels on ( C n , α ). Here pd R / I ≤ rk C n ≤ R ( C n ,α ) is free by Proposition 5.1. (cid:3) As a result of Proposition 5.7, Corollaries 5.2, 5.3 and 5.5 can be generalized as follows.
Corollary 5.8.
Let R be the polynomial ring k [ x , . . . , x d ] . (a) If G contains only one cycle C n and rk C n ≤ , then R ( G,α ) is a free R -module. (b) If G decomposes into disjoint cycles C , . . . , C s and p free edges where rk C i ≤ for all i , then R ( G,α ) is a free R -module. The following example shows that the converse of Corollary 5.8 is false.
Example 5.9.
Consider the edge labeled 3-cycle G over R = k [ x, y, z ] in Figure 2. Let I = (cid:104) x + yz , x + y , xz + y (cid:105) be an ideal of R . By using Macaulay2, we obtain pd (cid:0) R / I (cid:1) = 2. It follows thatthe syzygy module is free by (4) and thus R ( G ,α ) is free although rk G = 3. References [1] J. Abbott, A.M. Bigatti, and L. Robbiano,
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Email address : [email protected] Samet Sario˘glan (Corresponding author), Hacettepe University Department of Mathematics, 06800 BeytepeAnkara Turkey.
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