Linear Strands of Initial Ideals of Determinantal Facet Ideals
aa r X i v : . [ m a t h . A C ] J a n LINEAR STRANDS OF INITIAL IDEALS OF DETERMINANTALFACET IDEALS
AYAH ALMOUSA AND KELLER VANDEBOGERT
Abstract.
A determinantal facet ideal (DFI) is an ideal J ∆ generated by maximalminors of a generic matrix parametrized by an associated simplicial complex ∆. Inthis paper, we construct an explicit linear strand for the initial ideal with respectto any diagonal term order < of an arbitrary DFI. In particular, we show that if ∆has no , then the Betti numbers of the linear strand of J ∆ and its initialideal coincide. We apply this result to prove a conjecture of Ene, Herzog, and Hibion Betti numbers of closed binomial edge ideals in the case that the associated graphhas at most 2 maximal cliques. More generally, we show that the linear strand of theinitial ideal (with respect to < ) of any DFI is supported on a polyhedral cell complexobtained as an induced subcomplex of the complex of boxes , introduced by Nagel andReiner. Introduction
Let R denote the coordinate ring of a generic n × m matrix M , over some field k (with n ≤ m ). The ideal of maximal minors I n ( M ) possesses many surprising anddesirable properties; for example, a result of Sturmfels and Zelevinsky [16] shows thatthe natural generating set consisting of all maximal minors forms of M a reducedGr¨obner basis of I n ( M ), for any term order < . Boocher goes one step further in [3] andshows that the graded Betti numbers of I n ( M ) and any of its initial ideals must alsoagree. As a consequence, the minimal free resolution of any initial ideal of the ideal ofmaximal minors can be obtained by simply setting some of the entries in the matrixrepresentation of the standard Eagon-Northcott differentials equal to 0.One direction for generalizing ideals of maximal minors is to imagine that the column-sets of minors appearing in the generating set are parametrized by some simplicialcomplex ∆. Such ideals are called determinantal facet ideals (DFIs), and have beenstudied in multiple contexts by a wide variety of authors (see [6], [8], [17]). In the casethat ∆ has dimension 1 (that is, ∆ is a graph), DFIs are better known as binomialedge ideals , and are even more well behaved than arbitrary DFIs (see [14], [7], and [9]).DFIs themselves have also been generalized in a few different directions - Mohammadiand Rauh [12] have allowed for the minors to be parametrized by hypergraphs, and in[2], DFIs for nonmaximal minors are studied.In [5], Ene, Herzog, and Hibi conjecture that Betti numbers for a closed binomialedge ideal agree with that of its initial ideal with respect to any diagonal term order. Date : January 20, 2021.
Key words and phrases. determinantal facet ideal, binomial edge ideal, initial ideals, linear strand,free resolutions.AA was partially supported by the NSF GRFP under Grant No. DGE-1650441.
This conjecture is known to be true in the case of Cohen-Macaulay binomial edge ideals,but has remained elusive in generality. In this paper, we give further evidence and aproof of this conjecture in the 2-clique case using techniques related to the computationof linear strands.Given a homogeneous minimal complex of initial degree d , the linear strand can beobtained by restricting the i th differential to basis elements of degree i + d , for all i ≥ generalized Eagon-Northcott complex. In this paper, we provea similar result showing that the linear of of the initial ideal of certain classes of DFIsis obtained as a generalized sparse
Eagon-Northcott complex (see Definition 4.3).More precisely, in this paper we study linear strands of initial ideals of DFIs withrespect to a diagonal term order. We first construct an explicit resolution of in < I n ( M )as a sparse Eagon-Northcott complex (see 3.6). Using this, one can restrict to anappropriate subcomplex parametrized by the clique complex of the simplicial complexassociated to the DFI J ∆ . This subcomplex is not acyclic in general; however, if oneimposes sufficient conditions on ∆, the homology of this subcomplex will vanish in“small” degrees. This implies that the Betti numbers of certain DFIs and their initialideals with respect to a diagonal term order coincide on the linear strand (see Theorem4.13).As an application, we prove the previously mentioned conjecture of Ene, Herzog, andHibi in the case that the associated graph G has at most 2 maximal cliques. Moreover,we pose more generally the conjecture that for any lcm-closed DFI J ∆ , the Betti num-bers of J ∆ and its initial ideal with respect to any diagonal term order coincide (seeConjecture 4.16). Finally, we consider the linear strands of initial ideals (with respectto any diagonal term order) of arbitrary DFIs. In this case, we find that the linearstrand in general is always supported on a polyhedral cell complex. This implies thatthe Betti numbers of a general linear strand can be obtained by looking at certaininduced subcomplexes of the so-called complex of boxes of Nagel and Reiner (see [13]).We conclude with examples of this construction and remarks on further applications.The paper is organized as follows. In Section 2, we introduce the notation andterminology that will be in play for the rest of the paper. This includes the definitionof a DFI (see 2.3), linear strands, and the Eagon-Northcott complex. We include theaforementioned result of Boocher on the Betti numbers of initial ideals of ideals ofmaximal minors, and use this to deduce that the multigraded Betti numbers of anyinitial ideal of the ideal of maximal minors are either 0 or 1. In Section 3 we introducea sparse Eagon-Northcott complex to be used for resolving the initial ideal of the idealof maximal minors with respect to any diagonal term order. As corollaries, we obtainminimal free resolutions of the ideal of squarefree monomials of a given degree and thebox polarization for powers of the graded maximal ideal (and hence by specialization,any power of the graded maximal ideal).In Section 4, we consider subcomplexes of the sparse Eagon-Northcott complex withbasis elements parametrized by a simplicial complex ∆. This subcomplex in general is not acyclic, but it turns out that combinatorial properties of ∆ will allow us to deduce
INEAR STRANDS OF INITIAL IDEALS OF DETERMINANTAL FACET IDEALS 3 exactly when homology is nontrivial in linear degrees. This combinatorial condition isencoded in the existence of so-called i - nonfaces , a generalization of minimal nonfaces(see Definition 4.5). We use this to show that the Betti numbers along the linear strandof a DFI J ∆ and its initial ideal with respect to a diagonal term order coincide if andonly if the clique complex associated to ∆ has no 1-nonfaces of cardinality n + 1. Weapply this result to give a proof of the conjecture of Ene, Herzog, and Hibi for graphshaving at most 2 maximal cliques (see Corollary 4.21).In Section 5, we consider linear strands supported on cellular complexes. In particular,we show that the linear strand of the initial ideal (with respect to a diagonal term order)of any DFI is supported on a polyhedral cellular complex which can be obtained as theinduced subcomplex of the complex of boxes introduced by Nagel and Reiner in [13] (seeTheorem 5.11). This implies that the multigraded Betti numbers of the linear strandof any such ideal are 0 or 1 and moreover allows us to count the Betti numbers in thelinear strand as the f -vector of the associated polyhedral cell complex.2. Background
In this section we introduce some necessary background to be used for the rest ofthe paper. To start, we discuss determinantal facet ideals and establish some notationrelated to these ideals and the simplicial complexes that parametrize them. We thengive the definition of the linear strand of a minimal homogeneous complex, which willbe used extensively in later sections. Finally, we conclude with some results on idealsgenerated by maximal minors, including the definition of the classical
Eagon-Northcott complex. We conclude with an observation stating that the multigraded Betti numbersof the minimal free resolution of the initial ideal of the ideal of maximal minors withrespect to any term order has multigraded Betti numbers that are either 0 or 1.
Notation 2.1.
Let S = k [ x ij | ≤ i ≤ n, ≤ j ≤ m ] be a polynomial ring overan arbitrary field k . Let M be an n × m matrix of variables in S where n ≤ m . Forindices a = { a , . . . , a r } and b = { b , . . . , b r } such that 1 ≤ a < . . . < a r ≤ n and1 ≤ b < · · · < b r ≤ m , set[ a | b ] = [ a , . . . , a r | b , . . . , b r ] = det x a ,b · · · x a ,b r ... . . . ... x a r ,b · · · x a r ,b r where [ a | b ] = 0 if r > n . When r = n , use the simplified notation [ a ] = [1 , . . . , n | a ].The ideal generated by the r -minors of M is denoted I r ( M ). Definition 2.2.
For a simplicial complex ∆ and an integer i , the i -th skeleton ∆ ( i ) of∆ is the subcomplex of ∆ whose faces are those faces of ∆ with dimension at most i . Let S denote the set of simplices Γ with vertices in [ m ] with dim(Γ) ≥ n − ( n − ⊂ ∆.Let Γ , . . . , Γ c be maximal elements in S with respect to inclusion, and let ∆ i :=Γ ( n − i . Each Γ i is called a maximal clique , and any induced subcomplex of Γ i is a clique . The simplicial complex ∆ clique whose facets are the maximal cliques of ∆ is AYAH ALMOUSA AND KELLER VANDEBOGERT called the clique complex associated to ∆. The decomposition ∆ = ∆ ∪ · · · ∪ ∆ c iscalled the maximal clique decomposition of ∆. Definition 2.3.
Let ∆ be a pure ( n − m ]. Let S = k [ x ij | ≤ i ≤ n, ≤ j ≤ m ] be a polynomial ring over an arbitraryfield k . Let M be an n × m matrix of variables in S . The determinantal facet ideal (or DFI ) J ∆ ⊆ S associated to ∆ is the ideal generated by determinants of the form [ a ]where a supports an ( n − a ] correspond to thevertices of some facet of ∆. Notation 2.4.
Let ∆ be a pure ( n − m ] with maximal clique decomposition ∆ = ∆ ∪ · · · ∪ ∆ c . The notation J ∆ i willbe used to denote the DFI associated to the simplicial complex whose facets come fromall a ∈ ∆ i with | a | = r . Definition 2.5.
Let ∆ be a pure ( n − m verticeswith maximal clique decomposition ∆ = S ci =1 ∆ i . The DFI J ∆ is lcm-closed if thefollowing condition holds:(*) For all [ a ] ∈ J ∆ i , [ a ′ ] ∈ J ∆ j with ( a k , b k ) = ( a ′ k , b ′ k ) for some 1 ≤ i ≤ r and [ a ] , [ a ′ ] / ∈ J ∆ i ∩ ∆ j , there exists [ c ] ∈ J ∆ i ∩ ∆ j such that in([ c ]) divideslcm (cid:0) in([ a ]) , in([ a ′ ]) (cid:1) .In [2], it is shown that the standard minimal generating set of an lcm-closed DFI formsa reduced Gr¨obner basis; conjecturally, we believe that Definition 2.5 is equivalent tobeing a Gr¨obner basis for DFIs. The following definition introduces the main theme ofthe current paper. Definition 2.6.
Let F • be a minimal graded R -free complex with F having initialdegree d . Then the linear strand of F • , denoted F lin • , is the complex obtained byrestricting d Fi to ( F i ) d + i for each i ≥ Remark . Observe that the minimality assumption in Definition 2.6 ensures thatthe linear strand is well defined. Choosing bases, the linear strand can be obtained byrestricting to the columns where only linear entries occur in the matrix representationof each differential.The following result, due to Boocher, shows that with respect to any term order < ,the ideal in < I n ( M ) specializes to the ideal of all squarefree monomials of degree n in m variables. Theorem 2.8 ([3, Proof of Theorem 3.1]) . For any term order < , the sequence ofvariable differences { x − x , . . . , x − x n } ∪ · · · ∪ { x m − x m , . . . , x m − x nm } forms a regular sequence on R/ in < I n ( M ) . In particular, β ij ( R/I n ( M )) = β ij ( R/ in < I n ( M )) for all i, j. Definition 2.9 (Eagon-Northcott complex) . Let φ : F → G be a homomorphism offree modules of ranks n and m , respectively, with n ≥ m . Let c φ be the image of φ INEAR STRANDS OF INITIAL IDEALS OF DETERMINANTAL FACET IDEALS 5 under the isomorphism Hom R ( F, G ) ∼ = −→ F ∗ ⊗ G . The Eagon-Northcott complex is thecomplex0 → D m − n ( G ∗ ) ⊗ m ^ F → D m − n − ( G ∗ ) ⊗ m − ^ F → · · · → G ∗ ⊗ n +1 ^ F → n ^ F → n ^ G with differentials in homological degree ≥ c φ ∈ F ∗ ⊗ G , and the map V g F → V g G is V g φ . Notation 2.10.
Let E • denote the Eagon-Northcott complex of Definition 2.9. If F has basis f , . . . , f m and G has basis g , . . . , g n , then define g ∗ ( α ) ⊗ f I := g ∗ ( α )1 · · · g ∗ ( α n ) n ⊗ f i ∧ · · · ∧ f i n + ℓ , where α = ( α , . . . , α n ) and I = ( i < · · · < i n + ℓ ). Observe that E • inherits a Z n × Z m -grading by settingmdeg( g ∗ ( α ) ⊗ f I ) = (1 + α ǫ + · · · + α n ǫ n , ǫ i + · · · ǫ i n + ℓ ) , where ǫ k denotes the appropriately sized vector with 1 in the i th spot and 0 elsewhere,and 1 denotes a length n vector of 1s. Corollary 2.11.
Let F • denote a multigraded resolution of in < I n ( M ) . Then for everymultidegree α , β α ( R/ in < I n ( M )) ≤ . Proof.
By Theorem 2.8, a minimal free resolution of R/ in < I n ( M ) may be obtained bysetting some of the entries in the matrix representation of the differentials of Definition2.9 equal to 0. With respect to the Z n × Z m -grading of the Eagon-Northcott complex E • , one has β α ( R/ in < I n ( M )) ≤ . Since any Z nm -graded minimal free resolution is also Z n × Z m -graded, the result follows. (cid:3) Sparse Eagon-Northcott Complexes
In this section, we construct an explicit example of a sparse
Eagon-Northcott complex.The most complicated part of the construction ends up being the definition of thedifferentials; as it turns out, ayclicity will follow immediately from Theorem 2.8. As aconsequence, we deduce that certain specializations of this complex also yield minimalfree resolutions of the ideal generated by all squarefree monomials of a given degree andpowers of the graded maximal ideal in a polynomial ring. We begin this section withthe following setup:
Setup 3.1.
Let R = k [ x ij | ≤ i ≤ n, ≤ j ≤ m ] and M = ( x ij ) ≤ i ≤ n, ≤ j ≤ m denote ageneric n × m matrix, where n ≤ m . View M as a homomorphism M : F → G of freemodules F and G of rank m and n , respectively.Let f i , i = 1 , . . . , m , g j , j = 1 , . . . , n denote the standard bases with respect to which M has the above matrix representation. Let < denote any diagonal term order on R and in < I n ( M ) the initial ideal with respect to < of the ideal of maximal minors of M . AYAH ALMOUSA AND KELLER VANDEBOGERT
Notation 3.2.
Let α = ( α , . . . , α n ). Define α ≤ i := ( α , . . . , α i ) , where α ≤ i = ∅ if i ≤ α ≤ i = α if i ≥ n . Definition 3.3.
Let α = ( α , . . . , α n ) with | α | = ℓ and I = ( i < · · · < i n + ℓ ). Definethe indexing set I < ( α, I ) := { ( i, I i + j ) | i ∈ { k | α k > } , | α ≤ i − | ≤ j ≤ | α ≤ i |} Example 3.4.
One easily computes: I < ((1 , , , (1 , , , , , { (1 , , (1 , , (2 , , (2 , , (3 , , (3 , }I < ((1 , , , (1 , , , , , { (1 , , (1 , , (3 , , (3 , , (3 , }I < ((2 , , (1 , , , , { (1 , , (1 , , (1 , , (2 , , (2 , } Remark . Each basis element of the Eagon-Northcott complex also has a Z nm -gradingand can be viewed as a monomial in the ideal in( J ∆ ): g ∗ ( α ) ⊗ f σ ↔ ( x σ . . . x σ α ) · ( x σ α . . . x σ α α ) · · · ( x nσ n + i − αn − . . . x nσ n + i − ) =: m α,σ . Observe then that I ( α, σ ) chooses precisely those indices for which m α,σ /x rs ∈ in < I n ( M )for all ( r, s ) ∈ I ( α, σ ).Using the indexing set of Definition 3.3, we can define what will end up being acomplex as follows: Definition 3.6.
Adopt notation and hypotheses as in Setup 3.1. Let E ′• denote thesequence of module homomorphisms with E ′ ℓ = (V n G if ℓ = 0 D ℓ ( G ∗ ) ⊗ V n + ℓ F otherwise , and first differential d ′ : V n F → V n G sending f I in < ( M ( f I )). For ℓ ≥ d ′ ℓ : D ℓ − ( G ∗ ) ⊗ V n + ℓ − F → D ℓ − ( G ∗ ) ⊗ V n + ℓ − F is the map d ℓ ( g ∗ ( α ) ⊗ f I ) = X { i | α i > } X ( i,I j ) ∈I < ( α,I ) ( − j +1 x iI j g ∗ ( α − ǫ i ) ⊗ f I \ I j . Proposition 3.7.
The sequence of homomorphisms E ′• of Definition 3.6 forms a com-plex.Proof. Observe first that the map d ′ : V n F → V n G sends f I x I · · · x nI n g [ n ] . We first verify that d ′ ◦ d ′ = 0. Let g ∗ k ⊗ f I ∈ G ∗ ⊗ V n +1 F ; then: d ′ ◦ d ′ ( g ∗ k ⊗ f I ) = d ′ (( − k +1 x kI k f I \ I k + ( − k +2 x kI k +1 f I \ I k +1 )= ( − k +1 x kI k ( x I · · · d x kI k x kI k +1 · · · x nI n ) g [ n ] + ( − k +2 x kI k +1 ( x I · · · x kI k \ x kI k +1 · · · x nI n ) g [ n ] = 0 . INEAR STRANDS OF INITIAL IDEALS OF DETERMINANTAL FACET IDEALS 7
Assume now that ℓ ≥
1; the fact that d ′ ℓ +1 ◦ d ′ ℓ +2 = 0 is a nearly identical computationto that of the standard Eagon-Northcott differential, where one uses the fact that I < ( α − ǫ i , I \ I j ) = ( I < ( α, I ) \{ ( i, I j ) } if α i > I < ( α, I ) \{ ( i ′ , I j ′ ) | i ′ = i } if α i = 1 . (cid:3) Definition 3.8.
Adopt notation and hypotheses as in Setup 3.1. Then the sparseEagon-Northcott complex (with respect to < ) is the complex of Definition 3.6 Remark . The complex of Definition 3.6 is called sparse because of the relationshipof the differentials to that of the classical Eagon-Northcott complex. Namely, thedifferentials are obtained by simply setting some of the entries equal to 0 in the matrixrepresentation.
Corollary 3.10.
Adopt notation and hypotheses as in Setup 3.1. Then the sparseEagon-Northcott complex E ′• is a minimal free resolution of the ideal in < ( I n ( M )) .Proof. It is clear that the image of each basis element of E ′ i forms a linearly independentsubset of ker( d i − ) for each i ≥
2. Using Theorem 2.8, this image must also be agenerating set. (cid:3)
The following results are quick corollaries of Corollary 3.10. For the definition of thepolarization used in Corollary 3.12, see [1].
Corollary 3.11.
Adopt notation and hypotheses as in Setup 3.1. Let E ′• denote thesparse Eagon-Northcott complex with respect to < . Then E ′• ⊗ R/σ is a minimal freeresolution of the ideal of all squarefree monomials of degree n in m variables, where σ = { x − x , x − x , . . . , x − x n } ∪ { x − x , . . . , x − x n } ∪ . . . ∪{ x m − x m , . . . , x m − x nm } Proof.
This is immediate by Theorem 2.8. (cid:3)
Corollary 3.12.
Adopt notation and hypotheses as in Setup 3.1. Let E ′• denote thesparse Eagon-Northcott complex with respect to < . Then, under the relabelling x ij x j − i +1 ,i ,E ′• is a minimal free resolution of the box polarization of ( x , . . . , x m − n +1 ) n .In particular, with the above relabelling, E ′• ⊗ R/σ is a minimal free resolution of ( x , . . . , x m − n +1 ) n , where σ = { x − x , x − x , . . . , x − x n } ∪ { x − x , . . . , x − x n } ∪ . . . ∪{ x m − n +1 , − x m − n +1 , , . . . , x m − n +1 , − x m − n +1 ,n } AYAH ALMOUSA AND KELLER VANDEBOGERT Linear Strand of Certain DFIs
In this section, we construct an explicit linear strand for the initial ideal of certainclasses of DFIs. This linear strand is built as a generalized sparse Eagon-Northcottcomplex, where the generators in each homological degree are parametrized by facesof the clique complex associated to the simplicial complex ∆ parametrizing the DFI.We then define the notion of an i -nonface of a simplicial complex (see Definition 4.5);it turns out that the nonexistence of 1-nonfaces will guarantee that homology of theassociated generalized sparse Eagon-Northcott complex is trivial in appropriate degrees.In particular, we deduce that so-called lcm-closed DFIs have the property that β i,i + n ( R/J ∆ ) = β i,i + n ( R/ in < J ∆ ) for all i ≥ , for any diagonal term order < . To conclude this section, we verify a conjecture of Ene,Herzog, and Hibi on the Betti numbers of binomial edge ideals when the associatedgraph has at most 2 maximal cliques.To start off, we recall some results on linear strands of general R -modules due toHerzog, Kiana, and Madani. Theorem 4.1 ([8], Theorem 1.1) . Let R be a standard graded polynomial ring over afield k . Let G • be a finite linear complex of free R -modules with initial degree n . Thenthe following are equivalent:(1) The complex G • is the linear strand of a finitely generated R -module with initialdegree n .(2) The homology H i ( G • ) i + n + j = 0 for all i > and j = 0 , . Proposition 4.2 ([8], Corollary 1.2) . Let R be a standard graded polynomial ring overa field k . Let G • be a finite linear complex of free R -modules with initial degree n suchthat H i ( G • ) i + n + j = 0 for all i > , j = 0 , .Let N be a finitely generated R -module with minimal graded free resolution F • . As-sume that there exist isomorphisms making the following diagram commute: G ∼ (cid:15) (cid:15) / / G ∼ (cid:15) (cid:15) F lin / / F lin . Then G • ∼ = F lin • . Notice that the following definition is simply a subcomplex of the sparse Eagon-Northcott complex as in Definition 3.6.
Definition 4.3.
Adopt notation and hypotheses as in Setup 3.1. Let ∆ denote asimplicial complex on the vertex set [ m ]. Define C < (∆ , M ) := V n G . For i ≥
1, let C
Let C < • (∆ , M ) denote the complex induced by the differential d ′ : n ^ F → n ^ G sending f I in < ( M ( f I )) and for ℓ ≥ d ′ ℓ : D ℓ − ( G ∗ ) ⊗ n + ℓ − ^ F → D ℓ − ( G ∗ ) ⊗ n + ℓ − ^ F is the sparse Eagon-Northcott differential d ℓ ( g ∗ ( α ) ⊗ f I ) = X { i | α i > } X ( i,I j ) ∈I < ( α,I ) ( − j +1 x iI j g ∗ ( α − ǫ i ) ⊗ f I \ I j . Definition 4.4.
Adopt notation and hypotheses as in Setup 3.1. Let ∆ denote asimplicial complex on the vertex set [ m ]. The complex of Definition 4.3 will be calledthe generalized sparse Eagon-Northcott complex.Notice that by the definition of a simplicial complex, the above complex is indeedwell-defined. The following definition introduces a slight generalization of the notion ofa minimal nonface . Definition 4.5.
Let ∆ be a simplicial complex. Then an i -nonface σ = { σ < · · · < σ ℓ } is an element σ / ∈ ∆ such that for some j ≥ σ \ σ j + k ∈ ∆ for all k = 0 , . . . , i . Example 4.6.
Consider the following graph G on vertices { , , , } :3 2 ⑧⑧⑧⑧⑧⑧⑧⑧ . ❄❄❄❄❄❄❄ Observe that the associated clique complex has facets { , , } and { , , } , and nominimal nonfaces. However, { , , , } is a 1-nonface of the clique complex, since { , , } and { , , } are both facets.If we instead consider the graph 3 2 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) , ❃❃❃❃❃❃❃❃ then the clique complex has facets { , , } and { , , } . The set { , , , } is not a1-nonface. Likewise, there are no 1-nonfaces of cardinality 3. In the graph3 2 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) , the associated clique complex has facets { , , } and { , , } , and has no 1-nonfacesof cardinality 4. However, { , , } is a 1-nonface of cardinality 3 since { , } and { , } are vertices of G . Remark . Notice that a minimal nonface σ is a dim σ -nonface in the above definition.Moreover, any i -nonface is a k -nonface for all k ≤ i .In the proofs of the results in the remainder of this section, notice that we havechosen to augment our complexes with the ring R . This means that we are resolvingthe quotient ring R/I as opposed to the module I ; this has the effect of shifting theindexing in the statements of Theorem 4.1 and Proposition 4.2. Lemma 4.8.
Adopt notation and hypotheses as in Setup 3.1. If the simplicial complex ∆ has no -nonfaces of cardinality ≥ n + 1 , then the complex C < • (∆ , M ) is the linearstrand of a finitely generated graded R -module with initial degree n .Proof. Employ Theorem 4.1. To avoid trivialities, assume n ≤ dim(∆) + 1. Observefirst that H i ( C < • (∆ , M )) i + n − = 0 for all i ≥ H i ( C < • (∆ , M )) i + n = 0 for all i ≥ ⇒ there exists a 1-nonface of cardinality n + i, for all i ≥ . For convenience, use the notation C < • ( M ) =: E ′• , where C < • ( M ) is as in Definition 4.3.Assume H i ( C < • (∆ , M )) i + n = 0. Let z ∈ C } X ( ℓ,σ j ) ∈I < ( α,σ ) ( − j +1 x ℓσ j g ∗ ( α − ǫ ℓ ) ⊗ f σ \ σ j . Since z = 0, ( ℓ, σ j ) ∈ I < ( α, σ ) for some ℓ , j . By definition of I < ( α, σ ), this means( ℓ, σ k ) ∈ I < ( α, σ ) for all | α ≤ ℓ − | ≤ j ≤ | α ≤ ℓ | . This translates to the fact that σ is an α ℓ -nonface of cardinality n + i . Since α ℓ ≥
1, the result follows. (cid:3)
Remark . The proof of Lemma 4.8 allows one to construct explicit examples ofnonzero homology on the complex C < • (∆ , M ). Let ∆ clique be the simplicial complexassociated to the first graph of Example 4.6. Then the element z = x f , , − x f , , is a cycle which is not a boundary in C < • (∆ clique , M ). Lemma 4.10.
Adopt notation and hypotheses as in Setup 3.1. Then the following areequivalent:(1) H ( C < • (∆ , M )) n +1 = 0 ,(2) ∆ has no -nonfaces of cardinality n + 1 .Proof. The implication (2) = ⇒ (1) is Lemma 4.8. Conversely, assume that σ / ∈ ∆is a 1-nonface of cardinality n + 1. By definition, there exists some j such that σ \ σj and σ \ σ j +1 ∈ ∆. This means that z = ( − j +1 ( x jσ j f σ \ σ j − x jσ j +1 f σ \ σ j +1 ) is a cycle in INEAR STRANDS OF INITIAL IDEALS OF DETERMINANTAL FACET IDEALS 11 C < (∆ , M ) that is not a boundary, since z = d ( g ∗ j ⊗ f σ ), and g ∗ j ⊗ f σ / ∈ C (∆ , M ) byconstruction. (cid:3) Recall that the standard Eagon-Northcott complex inherits a Z n × Z m -grading, asdescribed in Notation 2.10. Since the sparse Eagon-Northcott complexes of Definition4.3 are obtained by simply setting certain entries in the differentials equal to 0, thesemaps will remain multigraded in an identical manner. We tacitly use this multigradingfor the remainder of this section. Theorem 4.11.
Adopt notation and hypotheses as in Setup 3.1. Assume that ∆ is an ( n − -pure simplicial complex such that ∆ clique has no -nonfaces of cardinality n + 1 .Let F • denote the minimal graded free resolution of in < ( J ∆ ) ; then F lin • ∼ = C < • (∆ clique , M ) . Proof.
Let Z lin := (ker d ) n +1 , where d is the first differential of the complex C < • (∆ clique , M ).By construction, C < (∆ clique , M ) is generated in degree n + 1 and hence induces a homo-geneous map ∂ : C < (∆ clique , M ) → Z lin . Let 0 = z ∈ Z lin be an element of multidegree ( ǫ s +1 , ǫ i + · · · + ǫ i n +1 ) (where 1 denotes theappropriately sized vector of all 1’s). Set τ := { i < · · · < i n +1 } ; by multihomogeneity,there are constants λ k ∈ k such that z = n +1 X k =1 λ k x si k f τ \ i k . Since z is a cycle of C < ( M ) (where C < • ( M ) := E ′• is as in Definition 4.3), there exists y ∈ C < ( M ) such that d ( y ) = z . By multihomogeneity, y = λg s ⊗ f τ for some con-stant λ , whence z = λ ( − s +1 ( x sσ s f σ \ σ s − x sσ s +1 f σ \ σ s +1 ). This implies that σ ∈ ∆ clique ,since otherwise ∆ clique would have a 1-nonface of cardinality n + 1, contradicting ourassumptions. Thus Z lin is generated by the set { r s ( σ ) := ( − s +1 ( x sσ s f σ \ σ s − x sσ s +1 f σ \ σ s +1 ) | ≤ s ≤ n, σ ∈ ∆ clique , | σ | = n + 1 } . Moreover, since mdeg( r s ( σ )) = mdeg( r s ′ ( σ ′ ) for s = s ′ or σ = σ ′ , the above is a basis.Finally, d ( g ∗ s ⊗ f σ ) = r s ( σ ), whence the induced map ∂ is an isomorphism of vectorspaces. (cid:3) Remark . Let ∆ be an ( n − clique has nominimal nonfaces in cardinality ≥ n + 1, since any minimal nonface is in particular a1-nonface. This means that ∆ clique satisfies the hypotheses of Theorem 3 . Theorem 4.13.
Adopt notation and hypotheses as in Setup 3.1. Assume that ∆ is an ( n − -pure simplicial complex with no -nonfaces of cardinality n + 1 . Then for all i ≥ , β i,n + i ( J ∆ ) = β i,n + i (in < ( J ∆ )) . Proof.
Notice that the linear strand of J ∆ is C • (∆ clique , M ) where C • is the general-ized Eagon-Northcott complex of [8]. Then, C • and C < • have the same underlying freemodules, so the result follows. (cid:3) Proposition 4.14.
Let ∆ be pure ( n − -dimensional simplicial complex on m vertices,and let J ∆ be its associated n -determinantal facet ideal. If J ∆ is lcm-closed, then thereare no -nonfaces in ∆ clique .Proof. It suffices to show that there are no 1-nonfaces of cardinality n +1 in ∆ clique . Sup-pose, seeking contradiction, that f = { f < · · · < f n +1 } is a 1-nonface of cardinality n +1in ∆ clique . By definition, there exists some f i such that a = { f , . . . , b f i , f i +1 , . . . , f n +1 } and b = { f , . . . , f i , d f i +1 , . . . , f n +1 } are facets of ∆. Let ∆ a and ∆ b be maximal cliquesof ∆ containing a and b , respectively, with nontrivial intersection. By assumption,∆ a = ∆ b . Because J ∆ is lcm-closed, there exists some facet c ∈ ∆ a ∩ ∆ b such that in[ c ]divides lcm(in[ a ] , in[ b ]) = x f x f . . . x if i x if i +1 x i +1 f i +2 . . . x n,f n +1 . The only possible facets c of ∆ satisfying this property are a and b themselves, so theymust be in the same maximal clique of ∆ and f is indeed a face in ∆ clique , giving thedesired contradiction. (cid:3) Corollary 4.15.
Adopt notation and hypotheses as in Setup 3.1. Assume that J ∆ isan lcm-closed DFI. Then for all i , β i,n + i ( J ∆ ) = β i,n + i (in < ( J ∆ )) . Using Corollary 4.15 as initial evidence, we pose the following:
Conjecture 4.16.
Adopt notation and hypotheses as in Setup 3.1. Assume that J ∆ isan lcm-closed DFI. Then β ij ( R/J ∆ ) = β ij ( R/ in < J ∆ ) for all i, j. Remark . It is important to notice that the condition on 1-nonfaces is not sufficientfor the minimal generators to form a Gr¨obner basis, and is hence more general thanthe property of being lcm-closed. For example, let ∆ be the simplicial complex withfacets 1 , ,
3, 1 , ,
5, and 1 , ,
7. One can verify directly that there are no 1-nonfaces ofcardinality 4, but calculations in Macaulay2 show that the minimal generating set ofthe associated determinantal facet ideal does not form a Gr¨obner basis.To conclude this section, we gather some necessary facts to prove (in a special case)a conjecture of Ene, Herzog, and Hibi. The following result, due to Conca and Varbaro,shows that ideals with squarefree initial ideals (with respect to some term order) exhibithomological behavior similar to that of the associated generic initial with respect torevlex.
Theorem 4.18 ([4, Follows from Theorem 1.3]) . Let I be a homogeneous ideal in astandard graded polynomial ring R with term order < . If in < ( I ) is squarefree, then theextremal Betti numbers of R/I and R/ in < ( I ) coincide. In particular, reg( R/I ) = reg( R/ in < ( I )) and pd R ( R/I ) = pd R ( R/ in < ( I )) . INEAR STRANDS OF INITIAL IDEALS OF DETERMINANTAL FACET IDEALS 13
Theorem 4.19 ([15, Theorem 1.1]) . Let I be a graded ideal and let < be any termorder. Then the graded Betti numbers β i,j ( R/I ) can be obtained from the graded Bettinumbers β i,j ( R/ in < ( I )) by a sequence of consecutive cancellations. Theorem 4.20 ([10, Corollary of Theorem 2.3]) . Let G be a closed graph with at most maximal cliques. Then reg( R/J G ) ≤ . Finally, we arrive at the last result of this section.
Corollary 4.21.
Let G be a closed graph with at most maximal cliques. Let J G denotethe associated binomial edge ideal J G and let < denote any diagonal term order. Thenfor all i, j , β i,j ( R/J G ) = β i,j ( R/ in < ( J G )) . Proof.
Since G is a closed graph, J G is lcm-closed and hence the associated clique com-plex of G has no 1-nonfaces (by Proposition 4.14). It is well known that every binomialedge ideal has squarefree Gr¨obner basis with respect to < (see [7]); in particular, The-orem 4.18 conbimed with Theorem 4.20 shows that reg( R/ in < J G ) ≤
2. Theorem 4.19asserts that the Betti numbers of
R/J G are obtained by those of R/ in < J G by a sequenceof consecutive cancellations. However, Theorem 4.13 implies that no cancellations arepossible. (cid:3) Linear Strands Supported on Cellular Complexes
In this section, we introduce the notion of a linear strand supported on a polyhe-dral cell complex (Definition 5.6), generalizing the well-studied phenomenon of cellularresolutions. We show that the first linear strand of the initial ideal of any determinan-tal facet ideal with respect to a diagonal term order is supported on a polyhedral cellcomplex. In particular, this cell complex is an induced subcomplex of the complex ofboxes introduced by Nagel and Reiner (see Construction 5.8), which supports a minimallinear free resolution of squarefree strongly stable and strongly stable ideals [13].We begin by recalling some basic notions from the theory of cellular resolutions. Fora more detailed exposition, see [11, Chapter 4].
Definition 5.1. A polyhedral cell complex P is a finite collection of convex polytopes(called cells or faces of P ) in some Euclidean space, satisfying the following two prop-erties: • if H is a polytope in P , then every face of H also lies in P , and • if H i , H j are both in P , then H i ∩ H j is a face of both H i and H j .Denote by V ( P ) the set of vertices (or 0-dimensional cells) of P . If X ⊆ V ( P ), the induced subcomplex of P on X is the subcomplex { F ∈ P | V ( F ) ⊆ X } . The f -vectorof a d -dimensional polyhedral cell complex P is the vector ( f , f , . . . , f d ), where f i isthe number of i -dimensional cells of P . Construction 5.2.
Set S = k [ x , . . . , x n ] to be a polynomial ring over a field k . Let P be an oriented polyhedral complex and let ( α H ) H ∈P ∈ Z n be a labeling of the cellsof P such that α H = lcm { x α G | G ⊂ H } . The labeled complex ( P , α ) gives rise to an algebraic complex of free Z n -graded S -modules in the following way. Let ( C • , ∂ • ) be the cellular chain complex for P . For twocells G, H ∈ P with dim H = dim G + 1 denote by ǫ ( H, G ) ∈ { , ± } the coefficient of G in the cellular boundary of H . Define the free modules F i := M H ∈P dim H = i +1 S ( − α H ) . The differentials d i : F i → F i − are given by d ( e H ) := X dim G =dim H − ǫ ( H, G ) x α H − α G e G . One can verify this defines an algebraic complex F P . For β ∈ Z n , denote by P ≤ β the subcomplex of P given by all cells H ∈ P with α H ≤ β coordinatewise. Let I = h x α v | v ∈ P a vertex i . Lemma 5.3 ([11, Proposition 4.5]) . Adopt notation and hypotheses of Construction 5.2.Let F P be the algebraic complex obtained from the labeled polyhedral complex ( P , α ) . Iffor every β ∈ Z n the subcomplex P ≤ β is acyclic over k , then F P resolves the quotient S/I . Furthermore, the resolution is minimal if α H = α G for any two faces G ⊂ H with dim H = dim G + 1 . Definition 5.4.
Adopt notation and hypotheses of Construction 5.2. The complex F P is a cellular resolution if it meets the criteria of Lemma 5.3, and the polyhedral complex P supports the resolution.We extend the notion of cellular resolution to study multigraded linear strands sup-ported on a polyhedral cell complex. The following lemma follows naturally from The-orem 4.1. Lemma 5.5.
Adopt notation and hypotheses of Construction 5.2 and assume the com-plex F P is d -linear, i.e., all the generators of I have degree d and all higher syzygy mapsare given by linear forms. Then F P is the first linear strand of S/I if, for any α ∈ Z n with | α | = d + k and k > , ˜ H i ( P ≤ α ) = 0 for i = k and k − . Definition 5.6.
Adopt notation and hypotheses of Construction 5.2. The complex F P is a cellular linear strand if it satisfies the criteria of Lemma 5.5, and the polyhedralcomplex P supports the linear strand of S/I . Setup 5.7.
Let R = k [ x ij | ≤ i ≤ n, ≤ j ≤ m ] and M = ( x ij ) ≤ i ≤ n, ≤ j ≤ m denote ageneric n × m matrix, where n ≤ m . Denote by I n ( M ) the ideal of maximal minors of M . Let < denote any diagonal term order on R , and for any ideal I ⊂ R , denote byin( I ) = in < ( I ) the initial ideal of I with respect to < .Partition the variables of R into n subsets ˇ X i = { x i , . . . , x im } . Set K = { a | a is an n subset of [ m ] } , so the elements of K are in one-to-one correspondence withthe generators of in < ( I n ( M )) via a = { a < · · · < a n } ←→ x a = x a · · · x na n .The following construction by Nagel and Reiner can be defined more generally forsquarefree strongly stable and strongly stable ideals, but for our purposes we onlyconsider the case when the ideal in question is in( I n ( M )). INEAR STRANDS OF INITIAL IDEALS OF DETERMINANTAL FACET IDEALS 15
Construction 5.8. (see [13]) Adopt notation and hypotheses of Setup 5.7. Call asubset of K which is a Cartesian product X × · · · × X n for subsets X j ⊆ ˇ X j a box inside K , and define the complex of boxes of K to be the polyhedral subcomplex of theproduct of simplices 2 ˇ X × · · · × ˇ X n having faces indexed by the boxes inside K . Theorem 5.9 ([13, Theorem 3.12]) . Adopt notation and hypotheses of Construction5.8. Labelling a vertex a in the complex of boxes by the monomial x a gives a minimallinear cellular resolution of R/ in( I n ( M )) .Hence β i,j ( R/ in( I n ( M ))) = 1 where i = P k | X k | − n , j = X ⊔ · · · ⊔ X n for every box X × · · · × X n inside K , and all other Betti numbers vanish. Notation 5.10.
Let ∆ be a pure ( n − m vertices,and let P denote the complex of boxes supporting a minimal linear cellular resolution of R/ in( I n ( M )). Denote by P (∆) the induced polyhedral subcomplex of P on the vertexset labeled by { x a | a a facet of ∆ } . Theorem 5.11.
Adopt notation and hypotheses of Setup 5.7. Let ∆ be a pure ( n − -dimensional simplicial complex on m vertices. Then the linear strand of R/ in( J ∆ ) issupported on P (∆) .Proof. First, observe that although in( J ∆ ) may have generators in higher degree, itslinear strand is completely determined by syzygies among the generators of degree n .Let ℓ = f n − (∆), the number of facets of ∆, and proceed by downward inductionon ℓ . The base case ℓ = (cid:0) mn (cid:1) corresponds to the case where J ∆ = I n ( M ) and is clear.Fix ℓ ≥ ℓ generators, P (∆) supports the first linearstrand of in( J ∆ ). Let ∆ ′ ⊂ ∆ be the subcomplex with a single facet a removed. Then P (∆ ′ ) ⊂ P (∆) is the induced subcomplex P (∆) \ v a , where v a is the vertex labeled bythe generator x a of in( J ∆ ).By Lemma 5.5, it suffices to check that for any multidegree α with | α | = n + k ,˜ H k ( P (∆ ′ ) ≤ α ) = ˜ H k − ( P (∆ ′ ) ≤ α ) = 0. Observe that any face of dimension k in P (∆ ′ ) ≤ α has multidegree β such that | β | = | α | = n + k . Therefore, if dim P (∆ ′ ) ≤ α = k , itis the unique box in P with multidegree α by Theorem 5.9, which is contractible. Ifdim P (∆ ′ ) ≤ α < k , then H k ( P (∆ ′ ) ≤ α ) = 0 trivially.By the inductive hypothesis, ˜ H k − ( P (∆ ′ ) ≤ α ) = 0 as long as x a does not divide x α , sosuppose it does. Let C be a cycle of dimension k − P (∆ ′ ) ≤ α . Since ˜ H k − ( P (∆) ≤ α ) =0 by the inductive hypothesis, there is some boundary B of dimension k in P (∆) ≤ α ofdegree α containing the vertex v a . By Theorem 5.9, there is a unique box in P withmultidegree α , so this must be B . But then ∂ ( B ) will be a linear combination of itscodimension 1 faces, including those containing v a , so it cannot be C . (cid:3) As a consequence of Theorem 5.11, we obtain the following means of computing theBetti numbers in the first linear strand of in( J ∆ ). Corollary 5.12.
Adopt notation and hypotheses of Setup 5.7. Let ∆ be a pure ( n − -dimensional simplicial complex on m vertices and let P (∆) be as in Notation 5.10. Then β i,i +1 ( R/ in( J ∆ )) = f i ( P (∆)) . x x x x x x x x x x x x Figure 1.
Complex of boxes P for in( I ( M )) where M is a 2 × x x x x x x x x x x Figure 2. P ( G ) where G has clique decomposition { , , } ∪ { , , } as in Example 4.6. Example 5.13.
Let G be the graph in Example 4.6 with clique decomposition { , , }∪{ , , } . The complex of boxes P for in( I ( M )) where M is a 2 × P ( G ) of P on the vertices corresponding to edgesin G is depicted in Figure 2. The f -vector for this subcomplex is (5 , , J G ). In particular, P ( G )has 1-cells corresponding to the 1-nonfaces { , , } , { , , } and 2-cells correspondingto the 1-nonface { , , , } in Example 4.6.Note, however, that this cell complex cannot support the full free resolution of theideal generated by the labels on its vertices, since P (∆) ≤ α is not in general acyclyicfor any multidegree α . Consider, for example, the multidegree α = x x x x . Then P ( G ) ≤ α consists of the disjoint vertices labeled by x x and x x , so ˜ H ( P ( G ) ≤ α ) = 1and this complex does not satisfy the hypotheses of Lemma 5.3. Remark . Nagel and Reiner show that other strongly stable and squarefree stronglystable ideals have a minimal, linear, cellular resolution given by the complex of boxes P . However, ideals generated by a subset of generators in these other cases do not, ingeneral, have a linear strand supported on the induced subcomplex of P in the samemanner. The key issue is that in other cases, the multigraded Betti numbers may notbe 0 or 1, so the proof of Theorem 5.11 does not apply. INEAR STRANDS OF INITIAL IDEALS OF DETERMINANTAL FACET IDEALS 17
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Cornell University
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