Hilbert-Poincaré series of parity binomial edge ideals and permanental ideals of complete graph
aa r X i v : . [ m a t h . A C ] M a r HILBERT–POINCAR ´E SERIES OF PARITY BINOMIAL EDGEIDEALS AND PERMANENTAL IDEALS OF COMPLETE GRAPHS
DO TRONG HOANG AND THOMAS KAHLE
Abstract.
We give an explicit formula for the Hilbert–Poincar´e series of the paritybinomial edge ideal of a complete graph K n or equivalently for the ideal generatedby all 2 × × n -matrix. It follows that the depth and Castelnuovo–Mumford regularity of these ideals are independent of n . Introduction
Let R = k [ x , . . . , x n , y , . . . , y n ] be a standard graded polynomial ring in 2 n in-determinates. The parity binomial edge ideal of an undirected simple graph G on[ n ] = { , . . . , n } is I G = ( x i x j − y i y j | { i, j } ∈ E ( G )) ⊂ R, where E ( G ) is the edge set of G . This ideal was defined and studied in [11] in formalsimilarity to the binomial edge ideals of [7] and [13]. If char( k ) = 2, then the linearcoordinate change x i ( x i − y i ) and y i x i + y i turns this ideal into the permanentaledge ideal ( x i y i + x j y j | { i, j } ∈ E ( G )) ⊂ R. We aim to understand homological properties of these ideals and we view suchunderstanding as helpful in the context of complexity theory and the dichotomy ofpermanents and determinants. In linear algebra it is known that determinants canbe evaluated quickly with Gaussian elimination, but permanents are × m × n -matricescase in detail.2 × R as the Lov´asz–Saks–Schrijver ideals of [8]. That paper also containsinformation about radicality and Gr¨obner bases of parity binomial edge ideals. Ba-diane, Burke and Sk¨oldberg proved in [2] that the universal Gr¨obner basis and theGraver basis coincide for parity binomial edge ideals of complete graphs. The caseof bipartite graphs is also special, as then binomial edge ideals and parity binomial Mathematics Subject Classification.
Key words and phrases.
Betti numbers, parity binomial edge ideal, Hilbert–Poincar´e series. dge ideals agree up to a linear coordinate change. A coherent presentation of ourknowledge about these binomial ideals can be found in [6], in particular Chapter 7.In this paper we are concerned with permanental ideals of 2 × n -matrices, but switchto the representation as parity binomial edge ideals of complete graphs, as this seemseasier to analyze. For example, the permanental ideal contains monomials by [12,Lemma 2.1] and these make the combinatorics more opaque [10]. Due to the linearcoordinate change, our computations of homological invariants are valid for both idealsunless char( k ) = 2, in which case the permanental ideal and the determinantal idealagree.The binomial edge ideal of a complete graph, also known as the standard determi-nantal ideal of a generic 2 × n -matrix, is well understood. It has a linear minimalfree resolution independent of n , constructed explicitly by Eagon and Northcott [4].Parity binomial edge ideals of complete graphs do not have a linear resolution andtheir Betti numbers have no obvious explanation. Example 1.1.
The package
BinomialEdgeIdeals in Macaulay2 [5] easily generatesthe following Betti table of I K . The Betti table agrees with the Betti table of apermanental ideal of a generic 2 × n one can observe that the Castelnuovo–Mumford regularity (the index of the last row of the Betti table) of R/ I K n appearsto be independent of n ≥ R/ I K n = 3 (see Section 2 for defini-tions). This was conjecture by the second author and Kr¨usemann [9, Remark 2.15]and is now our Theorem 3.6. Our main results are explicit formulas for the Hilbert–Poincar´e series, the depth, the Castelnuovo–Mumford regularity, and some extremalBetti numbers in the case of a complete graph. The proof of our theorem relies ongood knowledge of the primary decomposition of I K n from [11] and the resulting exactsequences. At the moment it is not clear if the techniques can be generalized to othergraphs or maybe even yield the conjectured upper bound reg( R/ I G ) ≤ n from [9,Remark 2.15]. 2. Basics of (parity) binomial edge ideals
Throughout this paper, let G be a simple (i.e. finite, undirected, loopless and with-out multiple edges) graph on the vertex set V ( G ) = [ n ] := { , . . . , n } . Let E ( G )denote the set of edges of G . Each graded R -module and in particular R/ I G has a inimal graded free resolution0 ← R/ I G ← M j R ( − j ) β ,j ( R/ I G ) ← · · · ← M j R ( − j ) β p,j ( R/ I G ) ← . where R ( − j ) denotes the free R -module obtained by shifting the degrees of R by j .The number β i,j ( R/ I G ) is the ( i, j ) -th graded Betti number of R/ I G . Let H R/ I G bethe Hilbert function of R/ I G . The Hilbert–Poincar´e series of the R -module R/ I G is HP R/ I G ( t ) = X i ≥ H R/ I G ( i ) t i . By [14, Theorem 16.2], this series has a rational expression HP R/ I G ( t ) = P R/ I G ( t )(1 − t ) n . The numerator P R/ I G ( t ) := P pi =0 P p + rj =0 ( − i β i,j ( R/ I G ) t j is the Hilbert–Poincar´e poly-nomial of R/ I G . It encodes different homological invariants of R/ I G of which we areparticulary interested in the Castelnuovo–Mumford regularity reg( R/ I G ) = max { j − i | β i,j ( R/ I G ) = 0 } and the projective dimension of R/ I G :pdim( R/ I G ) = max { i | β i,j ( R/ I G ) = 0 for some j } . In terms of Betti tables, the regularity is the index of the last non-vanishing row,while the projective dimension is the index of the last non-vanishing column of theBetti table. Both are finite for any R -module as R is a regular ring.The Auslander–Buchsbaum formula [6, Theorem 2.15] relates depth and projectivedimension over R as depth( R/ I G ) = 2 n − pdim( R/ I G ).The Castelnuovo–Mumford regularity and depth could also be computed from van-ishing of local cohomology. Using that definition allows to easily deduce some basicproperties of the regularity and depth. For instance, the regularity and depth behavewell in a short exact sequence. The following lemma appears as [14, Corollary 18.7]. Lemma 2.1. If → A → B → C → is a short exact sequence of finitely generatedgraded R -modules with homomorphisms of degree , then P B ( t ) = P A ( t ) + P C ( t ) , and(1) reg( B ) ≤ max { reg( A ) , reg( C ) } ,(2) reg( A ) ≤ max { reg( B ) , reg( C ) + 1 } ,(3) reg( C ) ≤ max { reg( A ) − , reg( B ) } ,(4) depth( B ) ≥ min { depth( A ) , depth( C ) } ,(5) depth( A ) ≥ min { depth( B ) , depth( C ) + 1 } ,(6) depth( C ) ≥ min { depth( A ) − , depth( B ) } . s with any binomial ideal, the saturation at the coordinate hyperplanes plays acentral role. To this end, let g = Q i ∈ [ n ] x i y i and let J G := I G : g ∞ := [ t ≥ I G : g t . By [11, Proposition 2.7], the generators of the saturation J G can be explained usingwalks in G . For our purposes it suffices to know the following generating set whichcan be derived from [11, Section 2]. Proposition 2.2. If G is a non-bipartite connected graph, then J G = ( x i − y i | ≤ i ≤ n ) + ( x i y j − x j y i , x i x j − y i y j | ≤ i < j ≤ n ) . Parity binomial edge ideals of complete graphs
We now consider the parity binomial edge ideal I K n of a complete graph K n on n ≥ ≤ i < j ≤ n , let f ij := x i y j − x j y i and g ij := x i x j − y j y i . The parity binomial edge ideal of the complete graph is I K n = ( g ij | ≤ i < j ≤ n ).We need some further notation. For any I ⊆ [ n ] we denote m I := ( x i , y i | i ∈ I ).Let p + := ( x i + y i | i ∈ [ n ]) and p − := ( x i − y i | i ∈ [ n ]). Denote P ij := ( g ij ) + m [ n ] \{ i,j } .By [11, Theorem 5.9], there is a decomposition of I K n as follows. Proposition 3.1.
For n ≥ , we have I K n = J K n ∩ \ ≤ i For ≤ k ≤ n − , we have I k − ⊆ \ ≤ i By Proposition 2.2, f n , . . . , f ( k − n ∈ J K n . Moreover, for all ( ℓ, n ) = ( i, j ) wehave f ℓn ∈ P ij . Thus( f n , . . . , f ( k − n ) ⊆ \ ≤ i 1, it is clear that x n + y n ∈ P ij and so P ij : ( x n + y n ) = R . By[1, Lemma 4.4], P n − ,n : ( x n + y n ) = P n − ,n . Moreover, by Proposition 2.2, we obtainthat J K n : ( x n + y n ) = p − . This implies that I n − : ( x n + y n ) ⊆ p − ∩ P n − ,n and thusthe conclusion I n − : ( x n + y n ) = p − ∩ P n − ,n .In order to prove the second part, note that p − + P n − ,n = ( x n − + y n − , x n + y n ) + m [ n − . Therefore one reads off depth( R/ ( p − + P n − ,n )) = 2 and reg( R/ ( p − + P n − ,n )) = 0. Itis clear that depth( R/ p − ) = n and reg( R/ p − ) = 0. From the exact sequence0 −→ R/ ( p − ∩ P n − ,n ) −→ R/ p − ⊕ R/P n − ,n −→ R/ ( p − + P n − ,n ) −→ , e obtain, using Lemma 2.1, thatdepth( R/I n − : ( x n + y n )) = depth( R/ ( p − ∩ P n − ,n )) ≥ min { n, , } = 3 , reg( R/I n − : ( x n + y n )) = reg( R/ ( p − ∩ P n − ,n )) ≤ max { , , } = 1 , and furthermore, P R/I n − :( x n + y n ) ( t ) = P R/ p − ( t ) + P R/P n − ,n ( t ) − P R/ ( p − + P n − ,n ) ( t )= (1 − t ) n + (1 − t ) n − (1 + t ) − (1 − t ) n − = (1 − t ) n + 2 t (1 − t ) n − . (cid:3) Lemma 3.5. Let J := ( x n + y n , I n − ) . Then depth( R/J ) ≥ min { n, depth( S/ I K n − ) } , reg( R/J ) ≤ max { , reg( S/ I K n − ) } , and P R/J ( t ) = t (1 − t ) n + (1 − t ) P S/ I Kn − ( t ) , where S = k [ x i , y i | ≤ i ≤ n − .Proof. In order to prove the lemma, we first check two following claims: Claim 1: ( J, x n ) = ( x n , y n , I K n − ) . Since y n = ( x n + y n ) − x n ∈ ( x n , J ) and I K n − ⊆ I n − , we have ( x n , y n , I K n − ) ⊆ ( J, x n ). Conversely, x n + y n , g in , f in ∈ ( x n , y n ) for 1 ≤ i ≤ n − J, x n ) ⊆ ( x n , y n , I K n − ). Claim 2: J : x n = p + . One can compute x n ( x i + y i ) = ( x i x n − y i y n ) + y i ( x n + y n ) ∈ J for 1 ≤ i ≤ n , sothat x n p + ⊆ J which implies that p + ⊆ J : x n . Conversely, for 1 ≤ i < j ≤ n , wehave g ij = x i x j − y i y j = ( x i − y i ) x j + y i ( x j − y j ) = ( x i + y i ) x j − y i ( x j + y j ) ,f ij = x i y j − x j y i = ( x i + y i ) y j − y i ( x j + y j ) = ( x i − y i ) y j − y i ( x j − y j ) . Thus, by Proposition 2.2, J K n ⊆ p + ∩ ( x − y , . . . , x n − − y n − , x n , y n ) and f kn ∈ p + ∩ ( x − y , . . . , x n − − y n − , x n , y n ) for all 1 ≤ k ≤ n − 2. Together with Proposition 3.1, J ⊆ \ ≤ i R/J : x n ) = depth( R/ p + ) = n and reg( R/J : x n ) = reg( R/ p + ) = 0 . From the exact sequence0 −→ R/ ( J : x n )( − −→ R/J −→ R/ ( J, x n ) −→ e obtaindepth( R/J ) ≥ min { n, depth( S/ I K n − ) } and reg( R/J ) ≤ max { , reg( S/ I K n − ) } . Moreover, P R/J ( t ) = tP R/J : x n ( t ) + P R/ ( J,x n ) ( t ) = tP R/ p + ( t ) + P R/ ( x n ,y n , I Kn − ) ( t )= t (1 − t ) n + (1 − t ) P S/ I Kn − ( t ) , as required. (cid:3) Theorem 3.6. The Hilbert–Poincar´e polynomial of R/ I K n is P R/ I Kn ( t ) = 2(1 − t ) n + h − t + ( n + n − 62 ) t + ( n − n + 22 ) t i (1 − t ) n − . In particular, depth( R/ I K n ) ≥ and reg( R/ I K n ) ≤ .Proof. The proof is by induction on n . If n = 3, then a simple calculation (e.g. inMacaulay2) gives the result. Now assume n ≥ 4. For any 1 ≤ k ≤ n − −→ R/ ( I k − : f kn )( − · f kn −−→ R/I k − −→ R/I k −→ . By Lemmas 2.1 and 3.3, depth( R/I k − ) ≥ min { , depth( R/I k ) } , reg( R/I k − ) ≤ max { , reg( R/I k ) } and P R/I k − ( t ) = t (1 − t ) n − (1 + t ) + P R/I k ( t ). This impliesthat depth( R/I ) ≥ min { , depth( R/I n − ) } , reg( R/I ) ≤ max { , reg( R/I n − ) } and P R/I ( t ) = ( n − t (1 − t ) n − (1 + t ) + P R/I n − ( t ) . Now consider the following exact sequence0 −→ R/ ( I n − : ( x n + y n ))( − −→ R/I n − −→ R/ ( x n + y n , I n − ) −→ . Let S := k [ x i , y i | ≤ i ≤ n − R/I n − ) ≥ min { , depth( S/ I K n − ) } , reg( R/I n − ) ≤ max { , reg( S/ I K n − ) } and P R/I n − ( t ) = tP R/I n − : x n + y n ( t ) + P R/ ( x n + y n ,I n − ) ( t )= 2 t (1 − t ) n + 2 t (1 − t ) n − + (1 − t ) P S/ I Kn − ( t ) . The induction hypothesis yields depth( S/ I K n − ) ≥ S/ I K n − ) ≤ 3. There-fore depth( R/I n − ) ≥ R/I n − ) ≤ 3. This is enough to conclude thatdepth( R/ I K n ) ≥ R/ I K n ) ≤ 3. Moreover, P R/ I Kn ( t ) = 2 t (1 − t ) n + h ( n − t + nt i (1 − t ) n − + (1 − t ) P S/ I Kn − ( t ) . = 2 t (1 − t ) n + h ( n − t + nt i (1 − t ) n − + 2(1 − t ) n +1 + h − t + ( n − n − 62 ) t + ( n − n + 62 ) t i (1 − t ) n − = 2(1 − t ) n + h − t + ( n + n − 62 ) t + ( n − n + 22 ) t i (1 − t ) n − , s required. (cid:3) If an ideal has a square-free initial ideal, its extremal Betti numbers agree with thatof the initial ideal by [3]. Although the parity binomial edge ideal of complete graphcannot have a square-free initial ideal (see [11, Remark 3.12]), the bottom right Bettinumber agrees with that of the initial ideal for any term order. Corollary 3.7. β n − , n ( R/ I K n ) = β n − , n ( R/ in < ( I K n )) = n − n + 22 . In particular, reg( R/ I K n ) = reg( R/ in < ( I K n )) = depth( R/ I K n ) = depth( R/ in < ( I K n )) = 3 . Proof. From Theorem 3.6 we obtain β p,p + r ( R/ I K n ) = n − n +22 = 0, where p =pdim( R/ I K n ) and r = reg( R/ I K n ). Thus, p + r = 2 n . Since P R/ I Kn ( t ) = P R/ in < ( I Kn ) ( t ),we getreg( R/ I K n ) = reg( R/ in < ( I K n )) , pdim( R/ I K n ) = pdim( R/ in < ( I K n )) and β p,p + r ( R/ I K n ) = β p,p + r ( R/ in < ( I K n )). On the other hand, r ≤ p ≤ n − r = 3 and p = 2 n − (cid:3) Acknowledgement Do Trong Hoang was supported by the NAFOSTED Vietnam under grant number101.04-2018.307. This paper was done when he visited Department of Mathemat-ics, Otto-von-Guericke Universit¨at Magdeburg with the support of Deutscher Aka-demischer Austauschdienst (DAAD). Thomas Kahle acknowledges support from theDFG (314838170, GRK 2297 MathCoRe). References [1] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley(1969).[2] M. Badiane, I. Burke and E. Sk¨oldberg, The Universal Gr¨obner Basis of a Binomial Edge Ideal ,Electron. J. Combin. (4) (2017), 12 pp.[3] A. Conca and M. Varbaro, Square-free Gr¨obner degenerations , to appear in Inventiones Math-ematicae.[4] J. A. Eagon and D. G. Northcott. Ideals defined by matrices and a certain complex associatedwith them , Proceedings of the Royal Society of London. Series A. (1962) no. 1337, 188–204.[5] D. R. Grayson and M. E. 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