A combinatorial proof of a formula for Betti numbers of a stacked polytope
aa r X i v : . [ m a t h . C O ] F e b A COMBINATORIAL PROOF OF A FORMULA FOR BETTI NUMBERSOF A STACKED POLYTOPE
SUYOUNG CHOI AND JANG SOO KIM
Abstract.
For a simplicial complex ∆, the graded Betti number β i,j ( k [∆]) of the Stanley-Reisner ring k [∆] over a field k has a combinatorial interpretation due to Hochster. Teraiand Hibi showed that if ∆ is the boundary complex of a d -dimensional stacked polytope with n vertices for d ≥
3, then β k − ,k ( k [∆]) = ( k − ` n − dk ´ . We prove this combinatorially. Introduction A simplicial complex ∆ on a finite set V is a collection of subsets of V satisfying(1) if v ∈ V then { v } ∈ ∆,(2) if F ∈ ∆ and F ′ ⊂ F , then F ′ ∈ ∆.Each element F ∈ ∆ is called a face of ∆. The dimension of F is defined by dim( F ) = | F |− dimension of ∆ is defined by dim(∆) = max { dim( F ) : F ∈ ∆ } . For a subset W ⊂ V ,let ∆ W denote the simplicial complex { F ∩ W : F ∈ ∆ } on W .Let ∆ be a simplicial complex on V . Two elements v, u ∈ V are said to be connected if there is a sequence of vertices v = u , u , . . . , u r = u such that { u i , u i +1 } ∈ ∆ for all i = 0 , , . . . , r −
1. A connected component C of ∆ is a maximal nonempty subset of V suchthat every two elements of C are connected.Let V = { x , x , . . . , x n } and let R be the polynomial ring k [ x , . . . , x n ] over a fixed field k .Then R is a graded ring with the standard grading R = ⊕ i ≥ R i . Let R ( − j ) = ⊕ i ≥ ( R ( − j )) i be the graded module over R with ( R ( − j )) i = R j + i . The Stanley-Reisner ring k [∆] of ∆ over k is defined to be R/I ∆ , where I ∆ is the ideal of R generated by the monomials x i x i · · · x i r such that { x i , x i , . . . , x i r } 6∈ ∆. A finite free resolution of k [∆] is an exact sequence(1) 0 / / F r φ r / / F r − φ r − / / · · · φ / / F φ / / F φ / / k [∆] / / , where F i = ⊕ j ≥ R ( − j ) β i,j and each φ i is degree-preserving. A finite free resolution (1) is minimal if each β i,j is smallest possible. There is a minimal finite free resolution of k [∆]and it is unique up to isomorphism. If (1) is minimal, then the ( i, j ) -th graded Betti number β i,j ( k [∆]) of k [∆] is defined to be β i,j ( k [∆]) = β i,j . Hochster’s theorem says β i,j ( k [∆]) = X W ⊂ V | W | = j dim k e H j − i − (∆ W ; k ) . We refer the reader to [1, 5] for the details of Betti numbers and Hochster’s theorem. Sincedim k e H (∆ W ; k ) is the number of connected components of ∆ W minus 1, we can interpret β i − ,i ( k [∆]) in a purely combinatorial way. Date : October 31, 2018.
Definition 1.1.
Let ∆ be a simplicial complex on a finite nonempty set V . Let k be anonnegative integer. The k -th special graded Betti number b k (∆) of ∆ is defined to be(2) b k (∆) = X W ⊂ V | W | = k (cc(∆ W ) − , where cc(∆ W ) denotes the number of connected components of ∆ W .Note that since there is no connected component in ∆ ∅ = {∅} , we have b (∆) = −
1. If k > | V | , then b k (∆) = 0 because there is nothing in the sum in (2). Thus we have b k (∆) = (cid:26) β k − ,k ( k [∆]) , if k ≥ − , if k = 0 . We refer the reader to [7] for the basic notions of convex polytopes. Let P be a simplicialpolytope with vertex set V . The boundary complex ∆( P ) is the simplicial complex ∆ on V such that F ∈ ∆ for some F ⊂ V if and only if F = V and the convex hull of F is a face of P . Note that if the dimension of P is d , then dim(∆( P )) = d − d -dimensional simplicial polytope P , we can attach a d -dimensional simplex to afacet of P . A stacked polytope is a simplicial polytope obtained in this way starting with a d -dimensional simplex.Let P be a d -dimensional stacked polytope with n vertices. Hibi and Terai [6] showed that β i,j ( k [∆( P )]) = 0 unless i = j − i = j − d +1. Since β i − ,i ( k [∆( P )]) = β n − i − d +1 ,n − i ( k [∆( P )]),it is sufficient to determine β i − ,i ( k [∆( P )]) to find all β i,j ( k [∆( P )]). In the same paper, theyfound the following formula for β k − ,k ( k [∆( P )]):(3) β k − ,k ( k [∆( P )]) = ( k − (cid:18) n − dk (cid:19) . Herzog and Li Marzi [4] gave another proof of (3).The main purpose of this paper is to prove (3) combinatorially. In the meanwhile, we getas corollaries the results of Bruns and Hibi [2] : a formula of b k (∆) if ∆ is a tree (or a cycle)considered as a 1-dimensional simplicial complex.2. Definition of t -connected sum In this section we define a t -connected sum of simplicial complexes, which gives anotherequivalent definition of the boundary complex of a stacked polytope. See [3] for the detailsof connected sums. And then, we extend the definition of t -connected sum to graphs, whichhas less restrictions on the construction. Every graph in this paper is simple.2.1. A t -connected sum of simplicial complexes. Let V and V ′ be finite sets. A relabel-ing is a bijection σ : V → V ′ . If ∆ is a simplicial complex on V , then σ (∆) = { σ ( F ) : F ∈ ∆ } is a simplicial complex on V ′ . Definition 2.1.
Let ∆ and ∆ be simplicial complexes on V and V respectively. Let F ∈ ∆ and F ∈ ∆ be maximal faces with | F | = | F | . Let V ′ be a finite set and σ : V → V ′ a relabeling such that V ∩ V ′ = F and σ ( F ) = F . Then the connected sum ∆ F ,F σ ∆ of ∆ and ∆ with respect to ( F , F , σ ) is the simplicial complex (∆ ∪ σ (∆ )) \ { F } on V ∪ V ′ . If ∆ = ∆ F ,F σ ∆ and | F | = | F | = t , then we say that ∆ is a t -connected sum of∆ and ∆ . COMBINATORIAL PROOF ON BETTI NUMBERS 3 ∆ =2 1 3 ∆ =1 3 4 G (∆ ) =2 1 3 4 G (∆ ) G (∆ ) =2 1 3 4 Figure 1.
The 1-skeleton of a 2-connected sum of ∆ and ∆ is not a 2-connected sum of G (∆ ) and G (∆ ).Note that if ∆ and ∆ are ( d − d −
1, then we can only define a d -connected sum of them.Let ∆ , ∆ , . . . , ∆ n be simplicial complexes. A simplicial complex ∆ is said to be a t -connected sum of ∆ , ∆ , . . . , ∆ n if there is a sequence of simplicial complexes ∆ ′ , ∆ ′ , . . . , ∆ ′ n such that ∆ ′ = ∆ , ∆ ′ i is a t -connected sum of ∆ ′ i − and ∆ i for i = 2 , , . . . , n , and ∆ ′ n = ∆.2.2. A t -connected sum of graphs. Let G be a graph with vertex set V and edge set E .Let W ⊂ V . Then the induced subgraph G | W of G with respect to W is the graph with vertexset W and edge set {{ x, y } ∈ E : x, y ∈ W } . Let b k ( G ) = X W ⊂ V | W | = k (cc( G | W ) − , where cc( G | W ) denotes the number of connected components of G | W .Let ∆ be a simplicial complex on V . The 1 -skeleton G (∆) of ∆ is the graph with vertexset V and edge set E = { F ∈ ∆ : | F | = 2 } . By definition, the connected components of ∆ W and G (∆) | W are identical for all W ⊂ V . Thus b k (∆) = b k ( G (∆)).Now we define a t -connected sum of two graphs. Definition 2.2.
Let G and G be graphs with vertex sets V and V , and edge sets E and E respectively. Let F ⊂ V and F ⊂ V be sets of vertices such that | F | = | F | , and G | F and G | F are complete graphs. Let V ′ be a finite set and σ : V → V ′ a relabeling suchthat V ∩ V ′ = F and σ ( F ) = F . Then the connected sum G F ,F σ G of G and G withrespect to ( F , F , σ ) is the graph with vertex set V ∪ V ′ and edge set E ∪ σ ( E ), where σ ( E ) = {{ σ ( x ) , σ ( y ) } : { x, y } ∈ E } . If G = G F ,F σ G and | F | = | F | = t , then we saythat G is a t -connected sum of G and G .Note that in contrary to the definition of t -connected sum of simplicial complexes, it is notrequired that F and F are maximal, and we do not remove any element in E ∪ σ ( E ). Wedefine a t -connected sum of G , G , . . . , G n as we did for simplicial complexes.It is easy to see that, if | F | = | F | ≥ G (∆ F ,F σ ∆ ) = G (∆ ) F ,F σ G (∆ ). Thuswe get the following proposition. Proposition 2.3.
For t ≥ , if ∆ is a t -connected sum of ∆ , ∆ , . . . , ∆ n , then G (∆) is a t -connected sum of G (∆ ) , G (∆ ) , . . . , G (∆ n ) . Note that Proposition 2.3 is not true if t = 2 as the following example shows. Example 2.4.
Let ∆ = { , , } and ∆ = { , , } be simplicial complexes on V = { , , } and V = { , , } . Here 12 means the set { , } . Let F = F = { , } and let σ SUYOUNG CHOI AND JANG SOO KIM be the identity map from V to itself. Then the edge set of G (∆ F ,F σ ∆ ) is { , , , } ,but the edge set of G (∆ ) F ,F σ G (∆ ) is { , , , , } . See Figure 1.3. Main results
In this section we find a formula of b k ( G ) for a graph G which is a t -connected sum of twographs. To do this let us introduce the following notation. For a graph G with vertex set V ,let c k ( G ) = X W ⊂ V | W | = k cc( G | W ) . Note that c k ( G ) = b k ( G ) + (cid:0) | V | k (cid:1) . Lemma 3.1.
Let G and G be graphs with n and n vertices respectively. Let t be a positiveinteger and let G be a t -connected sum of G and G . Then c k ( G ) = k X i =0 (cid:18) c i ( G ) (cid:18) n − tk − i (cid:19) + c i ( G ) (cid:18) n − tk − i (cid:19)(cid:19) − (cid:18) n + n − tk (cid:19) + (cid:18) n + n − tk (cid:19) . Proof.
Let V (resp. V ) be the vertex set of G (resp. G ). We have G = G F ,F σ G forsome F ⊂ V , F ⊂ V , a vertex set V ′ and a relabeling σ : V → V ′ such that V ∩ V ′ = F , σ ( F ) = F , and G | F and G | F are complete graphs on t vertices.Let A be the set of pairs ( C, W ) such that W ⊂ V ∪ V ′ , | W | = k and C is a connectedcomponent of G | W . Let A = { ( C, W ) ∈ A : C ∩ V = ∅} , A = { ( C, W ) ∈ A : C ∩ V ′ = ∅} . Then c k ( G ) = | A | = | A | + | A |−| A ∩ A | . It is sufficient to show that | A | = P ki =0 c i ( G ) (cid:0) n − tk − i (cid:1) , | A | = P ki =0 c i ( G ) (cid:0) n − tk − i (cid:1) and | A ∩ A | = (cid:0) n + n − tk (cid:1) − (cid:0) n + n − tk (cid:1) .Let B be the set of triples ( C , W , X ) such that W ⊂ V , X ⊂ V ′ \ V , | X | + | W | = k and C is a connected component of G | W . Let φ : A → B be the map defined by φ ( C, W ) = ( C ∩ V , W ∩ V , W \ V ). Then φ has the inverse map defined as follows. Fora triple ( C , W , X ) ∈ B , φ − ( C , W , X ) = ( C, W ), where W = W ∪ X and C is theconnected component of G | W containing C . Thus φ is a bijection and we get | A | = | B | = P ki =0 c i ( G ) (cid:0) n − tk − i (cid:1) . Similarly we get | A | = P ki =0 c i ( G ) (cid:0) n − tk − i (cid:1) .Now let B = { W ⊂ V ∪ V ′ : W ∩ F = ∅} . Let ψ : A ∩ A → B be the map defined by ψ ( C, W ) = W . We have the inverse map ψ − as follows. For W ∈ B , ψ − ( W ) = ( C, W ),where C is the connected component of G | W containing W ∩ F , which is guaranteed to existsince G | F = G | F is a complete graph. Thus ψ is a bijection, and we get | A ∩ A | = | B | = (cid:0) n + n − tk (cid:1) − (cid:0) n + n − tk (cid:1) . (cid:3) Theorem 3.2.
Let G and G be graphs with n and n vertices respectively. Let t be apositive integer and let G be a t -connected sum of G and G . Then b k ( G ) = k X i =0 (cid:18) b i ( G ) (cid:18) n − tk − i (cid:19) + b i ( G ) (cid:18) n − tk − i (cid:19)(cid:19) + (cid:18) n + n − tk (cid:19) . COMBINATORIAL PROOF ON BETTI NUMBERS 5
Proof.
Since c k ( G ) = b k ( G ) + (cid:0) n + n − tk (cid:1) , c i ( G ) = b i ( G ) + (cid:0) n i (cid:1) and c i ( G ) = b i ( G ) + (cid:0) n i (cid:1) ,by Lemma 3.1, it is sufficient to show that2 (cid:18) n + n − tk (cid:19) = k X i =0 (cid:18)(cid:18) n i (cid:19)(cid:18) n − tk − i (cid:19) + (cid:18) n i (cid:19)(cid:18) n − tk − i (cid:19)(cid:19) , which is immediate from the identity P ki =0 (cid:0) ai (cid:1)(cid:0) bk − i (cid:1) = (cid:0) a + bk (cid:1) . (cid:3) Recall that a t -connected sum G of two graphs depends on the choice of vertices of eachgraph and the identification of the chosen vertices. However, Theorem 3.2 says that b k ( G )does not depend on them. Thus we get the following important property of a t -connectedsum of graphs. Corollary 3.3.
Let t be a positive integer and let G be a t -connected sum of graphs G , G , . . . , G n .If H is also a t -connected sum of G , G , . . . , G n , then b k ( G ) = b k ( H ) for all k . Using Proposition 2.3, we get a formula for the special graded Betti number of a t -connectedsum of two simplicial complexes for t ≥ Corollary 3.4.
Let ∆ and ∆ be simplicial complexes on V and V respectively with | V | = n and | V | = n . Let t be a positive integer and let ∆ be a t -connected sum of ∆ and ∆ .If t ≥ then b k (∆) = k X i =0 (cid:18) b i (∆ ) (cid:18) n − tk − i (cid:19) + b i (∆ ) (cid:18) n − tk − i (cid:19)(cid:19) + (cid:18) n + n − tk (cid:19) . For an integer n , let K n denote a complete graph with n vertices.Let G be a graph with vertex set V . If H is a t -connected sum of G and K t +1 then H isa graph obtained from G by adding a new vertex v connected to all vertices in W for some W ⊂ V such that G | W is isomorphic to K t . Thus H is determined by choosing such a subset W ⊂ V . Using this observation, we get the following lemma. Theorem 3.5.
Let t be a positive integer. Let G be a t -connected sum of n K t +1 ’s. Then b k ( G ) = ( k − (cid:18) nk (cid:19) . Proof.
We construct a sequence of graphs H , . . . , H n as follows. Let H be the completegraph with vertex set { v , v , . . . , v t +1 } . For i ≥
2, let H i be the graph obtained from H i − by adding a new vertex v t + i connected to all vertices in { v , v , . . . , v t } . Then H n is a t -connected sum of n K t +1 ’s, and we have b k ( G ) = b k ( H n ) by Corollary 3.3. In H n , the vertex v i is connected to all the other vertices for i ≤ t , and v j and v j ′ are not connected to eachother for all t + 1 ≤ j, j ′ ≤ t + n . Thus b k ( H n ) = ( k − (cid:0) nk (cid:1) . (cid:3) Observe that every tree with n + 1 vertices is a 1-connected sum of n K ’s. Thus we getthe following nontrivial property of trees which was observed by Bruns and Hibi [2]. Corollary 3.6. [2, Example 2.1. (b)]
Let T be a tree with n + 1 vertices. Then b k ( T ) doesnot depend on the specific tree T . We have b k ( T ) = ( k − (cid:18) nk (cid:19) . SUYOUNG CHOI AND JANG SOO KIM
Corollary 3.7. [2, Example 2.1. (c)]
Let G be an n -gon. If k = n then b k ( G ) = 0 ; otherwise, b k ( G ) = n ( k − n − k (cid:18) n − k (cid:19) . Proof.
It is clear for k = n . Assume k < n . Let V = { v , . . . , v n } be the vertex set of G .Then ( n − k ) · b k ( G ) = X W ⊂ V | W | = k (cc( G | W ) − X v ∈ V \ W X v ∈ V X W ⊂ V \{ v }| W | = k (cc( G | W ) − X v ∈ V b k ( G | V \{ v } ) . Since each G | V \{ v } is a tree with n − (cid:3) Remark . Bruns and Hibi [2] obtained Corollary 3.6 and Corollary 3.7 by showing thatif ∆ is a tree (or an n -gon), considered as a 1-dimensional simplicial complex, then k [∆]has a pure resolution. Since k [∆] is Cohen-Macaulay and it has a pure resolution, the Bettinumbers are determined by its type (c.f. [1]).Now we can prove (3). Note that, for d ≥
3, if P is a d -dimensional simplicial polytope and Q is a simplicial polytope obtained from P by attaching a d -dimensional simplex S to a facetof P , then ∆( Q ) is a d -connected sum of ∆( P ) and ∆( S ), and thus the 1-skeleton G (∆( Q ))is a d -connected sum of G (∆( P )) and K d +1 . Hence the 1-skeleton of the boundary complexof a d -dimensional stacked polytope is a d -connected sum of K d +1 ’s. Theorem 3.9.
Let P be a d -dimensional stacked polytope with n vertices. If d ≥ , then b k (∆( P )) = ( k − (cid:18) n − dk (cid:19) . If d = 2 , then b k (∆( P )) = (cid:26) , if k = n , n ( k − n − k (cid:0) n − k (cid:1) , otherwise.Proof. Assume d ≥
3. Then the 1-skeleton G (∆( P )) is a d -connected sum of n − d K d +1 ’s.Thus by Theorem 3.5, we get b k (∆( P )) = b k ( G (∆( P ))) = ( k − (cid:0) n − dk (cid:1) .Now assume d = 2. Then G (∆( P )) is an n -gon. Thus by Corollary 3.7 we are done. (cid:3) References [1] Winfried Bruns and J¨urgen Herzog.
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