Determinantal Facet Ideals for Smaller Minors
aa r X i v : . [ m a t h . A C ] J un RESOLUTIONS OF INITIAL IDEALS OF CLOSEDDETERMINANTAL FACET IDEALS
AYAH ALMOUSA AND KELLER VANDEBOGERT
Abstract.
We study resolutions of initial ideals of closed determinantal facet idealswith respect to standard lexicographic order. We show that the multigraded Bettinumbers of these ideals are always 0 or 1, regardless of the characteristic of the field.In addition, we show that the standard graded Betti numbers of closed determinantalfacet ideals and their initial ideals coincide when generators of the ideal come frommaximal minors of a generic n × m matrix with n >
2. Next, we give lower boundson the Betti numbers of certain classes of ideals of initial terms of the generators ofdeterminantal facet ideals with respect to arbitrary term orders. We give an explicitminimal free resolution of the initial ideal of the ideal of maximal minors with respectto standard lexicographic order. We show that the Betti numbers of a certain closeddeterminantal facet ideal and its initial ideal coincide, verifying a conjecture of Ene,Herzog, and Hibi in a new case. We give explicit differentials for the linear strandof the initial ideal with respect to standard lexicographic order of an arbitrary closeddeterminantal facet ideal. Introduction
Let R = k [ x ij | ≤ i ≤ n, ≤ j ≤ m ] where k is any field, and M be a generic n × m matrix of indeterminates. The study of the ideal generated by all minors ofa given size of M has a long history, and is well known (see, for instance, [6]). In asimilar vein, one can instead consider the ideal generated by some of the minors of agiven size of M ; these are known as determinantal facet ideals and were introducedEne, Herzog, Hibi, and Mohammadi in [10]. This problem turns out to be much moresubtle and has seen comparably less attention, even though such ideals arise naturallyin algebraic statistics (see [8] and [12]). In [13], the linear strand of determinantal facetideals is made explicit; in particular, the linear Betti numbers of such ideals may becomputed in terms of the f -vector of an associated simplicial complex. Likewise, in [19],explicit Betti numbers of certain classes of determinantal facet ideals are computed inall degrees; for arbitrary determinantal facet ideals, higher degree Betti numbers haveproven to be nontrivial to compute.Determinantal facet ideals for the case n = 2 were originally introduced as binomialedge ideals independently by Ohtani [15] and Herzog, et. al. [12]; this generalized workof Diaconis, Eisenbud, and Sturmfels in [8]. To study binomial edge ideals, one canassociate each column of M with a vertex of a graph G , and one can associate a minor of Date : June 26, 2020.
Key words and phrases. determinantal facet ideal, binomial edge ideal, initial ideals, Stanley-Reisnertheory, linear strand, free resolutions.AA was partially supported by the NSF GRFP under Grant No. DGE-1650441. M involving two columns i and j with an edge ( i, j ) in the graph. For example, the idealgenerated by all maximal minors of a 2 × m matrix corresponds to a complete graphon m vertices. The relationship between homological invariants of ideals generated bysome maximal minors of M and combinatorial invariants of the associated graph G has been widely studied; see the survey paper [14] for a compilation of such results.Determinantal facet ideals naturally extend this idea by instead associating a simplicialcomplex ∆ on m vertices to the ideal I , where each ( n − I .One method of studying an ideal I is to reduce to the study of its initial ideal,in < ( I ), with respect to some term order < . This has the advantage of gaining accessto the combinatorial and topological techniques developed for the purposes of studyingmonomial ideals. A particularly interesting class of determinantal facet ideals is thosefor which the generators form a Gr¨obner basis. It is well-known that the ideal ofmaximal minors of a matrix is a Gr¨obner basis under any monomial order [17, 2]. Inthe case where the generators of a determinantal facet ideal form a Gr¨obner basis underlex monomial order, we say that the ideal (or its corresponding simplicial complex) is closed .It is well-known that the Betti numbers of an initial ideal in < ( I ) under any term order < is an upper bound for the Betti numbers of I , and it is rare that the Betti numbersof in < ( I ) and I coincide. Ene, Herzog, and Hibi [9] conjecture that the Betti numbersof both a closed binomial edge ideal and its initial ideal with respect to standardlexicographic order coincide. Conca and Varbaro [7] show that for any ideal I in astandard graded homogeneous polynomial ring with a squarefree initial ideal in < ( I )with respect to some term order < , the extremal Betti numbers (see [1]) of I andin < ( I ) coincide. In particular, the regularity and the projective dimension are the samefor both I and its squarefree initial ideal in < ( I ).In this paper, we turn our attention to the case of closed determinantal facet idealsand study properties of resolutions of their initial ideals with respect to < , where < denotes standard lexicographic order. In the case where I corresponds to a closeddeterminantal facet ideal, in < ( I ) is squarefree and corresponds exactly to the leadingterms of the generators of I . This enables us to use tools from combinatorics andtopology to study the Stanley-Reisner complex of in < ( I ) and conclude that all the Z nm -graded Betti numbers of in < ( I ) are either 0 or 1. We observe that consecutivecancellations among the Betti numbers of in( I ) are never possible when n >
2, implyingthat the Betti numbers of J ∆ and in( J ∆ ) coincide in these cases.Next, we employ the construction of trimming complexes as introduced in [19] toso-called sparse Eagon-Northcott complexes, which were introduced by Boocher in [4].In particular, we obtain explicit lower bounds on the Betti numbers for certain classesof initial ideals of determinantal facet ideals with respect to any term order. We thenconstruct an explicit minimal free resolution for the initial ideal of the ideal of allmaximal minors with respect to standard lexicographic order for an arbitrary n × m matrix. This allows us to verify the previously mentioned conjecture of Ene, Herzog,and Hibi when removing a single generator from the ideal generated by all maximalminors. NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 3
Using a slight generalization of the above resolution, we are able to obtain an explicitlinear strand for the initial ideal in < ( I ) of a closed determinantal facet ideal I where < is standard lexicographic order. In particular, this allows us to show that the Bettinumbers of the linear strand for both I and in < ( I ) agree, for all n ≥ I by reducing to the study of its initial idealin < ( I ) with respect to standard lexicographic order < . In particular, we show that the Z nm -graded Betti numbers of in < ( I ) are always either 0 or 1 (see Theorem 3.18). Asan application, it is shown that in general, the Betti numbers of the linear strand of aclosed determinantal facet ideal I and its initial ideal in < ( I ) agree. Even more, we showthat when ∆ is a closed and pure ( n − n > J ∆ and in( J ∆ ) coincide.In Section 4, we reformulate certain types of complexes considered in [4] for the pur-poses of using the trimming complex construction. In particular, we compute explicit q i -maps for sparse Eagon-Northcott complexes with respect to any arbitrary monomialordering and hence are able to bound their rank. Combining these bounds with Corol-lary 2.5 yields lower bounds on the Betti numbers of the initial ideal of certain classesof determinantal facet ideals with respect to any term order.In Section 5, we construct an explicit minimal free resolution of the initial ideal of theideal of maximal minors of a generic n × m matrix with respect to standard lexicographicorder. In particular, this also gives a minimal free resolution of the box polarizationof any power of the standard graded maximal ideal. Moreover, via specialization, weobtain novel minimal free resolutions of both the ideal of all squarefree monomials of aspecified degree and arbitrary powers of the homogeneous maximal ideal. We use thisresolution to verify the conjecture of Ene, Herzog, and Hibi in the case where n = 2when removing a single generator.In Section 6, we pursue a slight strengthening of a result in Section 3. More precisely,we can generalize the complex introduced in Section 5 in order to deduce an explicitlinear strand for the initial ideal in < ( I ) of a closed determinantal facet ideal I , where < is standard lexicographic order. The proofs in this section follow closely that of [13],where a different complex is used in place of the so-called generalized Eagon-Northcottcomplex. In particular, we recover Corollary 3.20 using different methods.2. Trimming Complexes
We recall the construction of trimming complexes. All proofs of the following resultsmay be found in Section 2 and 3 of [19].
Setup 2.1.
Let R = k [ x , . . . , x n ] be a standard graded polynomial ring over a field k .Let I ⊆ R be a homogeneous ideal and ( F • , d • ) denote a homogeneous free resolutionof R/I . AYAH ALMOUSA AND KELLER VANDEBOGERT
Write F = F ′ ⊕ (cid:16) L mi =1 Re i (cid:17) , where each e i generates a free direct summand of F .Using the isomorphismHom R ( F , F ) = Hom R ( F , F ′ ) ⊕ (cid:16) m M i =1 Hom R ( F , Re i ) (cid:17) write d = d ′ + d + · · · + d m , where d ′ ∈ Hom R ( F , F ′ ), d i ∈ Hom R ( F , Re i ). Let a i denote any homogeneous ideal with d i ( F ) ⊆ a i e i , and ( G i • , m i • ) be a homogeneous free resolution of R/ a i .Use the notation K ′ := im( d | F ′ : F ′ → R ), K i := im( d | Re i : Re i → R ), and let J := K ′ + a · K + · · · + a m · K m . Proposition 2.2.
Adopt notation and hypotheses of Setup 2.1. Then for each i =1 , . . . , m there exist maps q i : F → G i such that the following diagram commutes: F q i ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ d i ′ (cid:15) (cid:15) G m i / / a i , where d i ′ : F → R is the composition F d i / / Re i / / R , the second map sending e i . Proposition 2.3.
Adopt notation and hypotheses as in Setup 2.1. Then for each i =1 , . . . , m there exist maps q ik : F k +1 → G ik for all k ≥ such that the following diagramcommutes: F k +1 q ik (cid:15) (cid:15) d k +1 / / F kq ik − (cid:15) (cid:15) G ik m ik / / G ik − NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 5
Theorem 2.4.
Adopt notation and hypotheses as in Setup 2.1. Then the mapping coneof the morphism of complexes (1) · · · d k +1 / / F k q k − ... q mk − (cid:15) (cid:15) d k / / · · · d / / F d ′ / / q ... q m (cid:15) (cid:15) F ′ d (cid:15) (cid:15) · · · L m ik / / L mi =1 G ik − L m ik − / / · · · L m i / / L mi =1 G i − P ℓi =1 m i ( − ) · d ( e i ) / / R is acyclic and forms a resolution of the ideal K ′ + a · K + · · · + a m · K m . As an immediate consequence, one obtains:
Corollary 2.5.
Adopt notation and hypotheses of Setup 2.1. Assume furthermore thatthe complexes F • and G • are minimal. Then for i ≥ , dim k Tor Ri ( R/J, k ) = rank F i + m X j =1 rank G ji − rank q i ... q mi ⊗ k ! − rank q i − ... q mi − ⊗ k ! . Similarly, µ ( J ) = µ ( K ) − m + m X j =1 µ ( a j ) − rank q ... q m ⊗ k ! . (cid:3) The resolution of Theorem 2.4 may be used to construct resolutions of subsets ofgenerating sets of an ideal by the following observation.
Observation . Adopt notation and hypotheses as in Setup 2.1 with m = 1. If d ( F ) = a e , then the resolution of Theorem 2.4 resolves K ′ .With this observation, the following can be shown. Theorem 2.7 ([19], Theorem 5.6) . Let R = k [ x ij | ≤ i ≤ n, ≤ j ≤ m ] where k isany field and M = ( x ij ) denote a generic n × m matrix, with n ≤ m . Choose indexingsets I j = ( i j , . . . , i jn ) for j = 1 , . . . , r pairwise disjoint; that is, I i ∩ I j = ∅ for i = j (this intersection is taken as sets). Definerk ℓ := (cid:18) n + ℓ − ℓ (cid:19) · r X i =1 ( − i +1 (cid:18) ri (cid:19)(cid:18) m − inℓ − ( i − n (cid:19) . If J = ( j , . . . , j n ) with j < · · · < j n , let ∆ J denote the determinant of the matrixformed by columns j , . . . , j n of M . Then the ideal K ′ := (∆ J | J = I j , j = 1 , . . . , r ) AYAH ALMOUSA AND KELLER VANDEBOGERT has Betti table · · · ℓ · · · n ( m − n ) − n ( m − n )0 1 0 · · · · · · n − (cid:0) mn (cid:1) − r · · · (cid:0) n + ℓ − ℓ − (cid:1)(cid:0) mn + ℓ − (cid:1) − rk ℓ − · · · n · · · r · (cid:0) n ( m − n ) ℓ (cid:1) − rk ℓ · · · r · n ( m − n ) r In particular, pd R R/K ′ = n ( m − n ) . Multigraded Betti Numbers of Closed Determinantal Facet Ideals
In this section we study the initial ideals generated by arbitrary collections of maximalminors of an n × m matrix of indeterminates in the case where the set of generatorsforms a Gr¨obner basis for the ideal. In this case, the initial ideal is of degree n andsquarefree, so that Stanley-Reisner theory may be employed to compute the Z nm -gradedBetti numbers. Setup 3.1.
Let ˇ X i = { x i , . . . , x im } for all 1 ≤ i ≤ n , and let S = k [ ˇ X , . . . , ˇ X n ] be apolynomial ring in the variables of the ˇ X i over a field k . Let M be an n × m matrixof variables in S , where the variables of ˇ X i are in row i of the matrix. Let < denotestandard lexicographic order in S ; that is, lexicographic order with x < · · · < x m Adopt notation and hypotheses as in Setup 3.1. For a simplicialcomplex ∆ and an integer i , the i -th skeleton ∆ ( i ) of ∆ is the subcomplex of ∆ whosefaces are those faces of ∆ whose dimension is at most i . Let S denote the set of simplicesΓ with vertices in [ m ] with dim(Γ) ≥ n − ( n − ⊂ ∆.Let Γ , . . . , Γ r be maximal elements in S with respect to inclusion, and let ∆ i :=Γ ( n − i . Each Γ i is called a clique . The simplicial complex ∆ clique whose facets arethe cliques of ∆ is called the clique complex associated to ∆. The decomposition∆ = ∆ ∪ · · · ∪ ∆ r is called the clique decomposition of ∆. Definition 3.3. Adopt notation and hypotheses as in Setup 3.1. A determinantal facetideal J ∆ ⊆ S is the ideal generated by determinants of the form [ a ] where a supportsan n − a ] correspond to the vertices of some facet σ ∈ ∆. Remark . Let I be an ideal generated by a subset of minors of an n × m matrix M . The simplicial complex ∆ associated to a determinantal facet ideal can be viewedas a combinatorial tool to keep track of the generators of such an ideal, since eachfacet corresponds to a minor in the generating set of I . The clique decomposition of∆ = S ri =1 ∆ i keeps track of the largest submatrices M i of M where the ideal of maximalminors of M i is contained in I . Definition 3.5. A simplicial complex ∆ is said to be closed (with respect to a givenlabeling) if it satisfies any one of the following equivalent conditions: NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 7 (1) For any two facets F = { a < · · · < a n } and G = { b < · · · < b n } with a i = b i for some i , the ( n − F ∪ G is containedin ∆.(2) For all i = j and all F = { a < a < · · · < a n } in ∆ i and G = { b < b < · · ·
For a squarefree monomial ideal I , the Stanley-Reisner complex of I is the simplicial complex with faces supported on squarefree monomials not containedin I . For each σ ⊆ [ m ], define the restriction of ∆ to σ by∆ | σ = { τ ∈ ∆ | τ ⊆ σ } . Definition 3.8. The simplicial join of simplicial complexes Γ and Γ , denoted Γ ∗ Γ ,is the simplicial complex with faces σ ∪ σ where σ ∈ Γ and σ ∈ Γ . The cone of asimplicial complex Γ, denoted cone (Γ), is the join Γ ∗ v of Γ with some vertex v not inΓ. Proposition 3.9 (Hochster’s Formula) . Let Γ be a simplicial complex on V = { x , . . . , x n } and let I be the associated Stanley-Reisner ideal in S = k [ x , . . . , x n ] . Then the nonzeroBetti numbers of I Γ lie only in squarefree degrees σ , and β i − ,σ ( I ) = β i,σ ( S/I ) = dim k ˜ H | σ |− i − (Γ | σ ; k ) . Notation 3.10. Let A be a set of pairs ( i j , τ j ) such that τ = [ τ , . . . , τ k ] ∈ ∆ clique and k ≥ n . Associate to A a monomial m A where m A = Y ( i j ,τ j ) ∈A x i,τ j . Let a = ( a , . . . , a n ) ∈ Z n . Define | a | := P ni =1 a i . Define a ≤ i := ( a , . . . , a i ) , where a ≤ i = ∅ if i ≤ a ≤ i = a if i ≥ n . Setup 3.11. Let ∆ be a closed pure ( n − A ( a ; τ ) = { ( i, τ j ) | i ∈ [ n ] , | a ≤ i − | < j ≤ | a ≤ i |} , where τ is a ( k − clique with k ≥ n , and a = ( a , . . . , a n ) ∈ N n , | a | = k . Set m k ( a ; τ ) := m A ( a ; τ ) . Example . Let ∆ be the 2-dimensional simplicial complex with clique decomposition { , , , } ∪ { , , , } , so J ∆ is generated by the maximal minors of a generic 3 × σ = { , , , } and a = (1 , , m ( a ; σ ) = x , x , x , x , . AYAH ALMOUSA AND KELLER VANDEBOGERT Proposition 3.13. Adopt notation and hypotheses as in Setups 3.1 and 3.11. Let Γ be the associated Stanley-Reisner complex of the squarefree monomial ideal in( J ∆ ) andlet u = m k ( a ; τ ) for some τ ∈ ∆ clique and a ∈ N n such that | a | = k and a i = 0 for ≤ i ≤ n . Then Γ | u is homotopy equivalent to S n − . The proof of this proposition requires some notions from simplicial topology. Forfurther reference, see, for example, [3]. Definition 3.14. The nerve of a finite set of simplicial complexes { Λ i } i ∈A is the sim-plicial complex N on vertex set A and with faces { σ ⊆ A| ∩ i ∈ σ Λ i = ∅} . Lemma 3.15 (Nerve Theorem) . Let ∆ be a finite simplicial complex and { Λ } i ∈A be afinite cover (that is, a set of subcomplexes such that ∪ i ∈A Λ i = ∆) . Suppose that everynon-empty intersection ∩ i ∈ σ Λ i is contractible. Then ∆ and the nerve N are homotopic.Proof of Proposition 3.13. Write u = u · · · u n where u i := Y ( i,τ j ) ∈A ( a ; τ ) x i,τ j for a fixed i . Let Γ be the Stanley-Reisner complex of in( J G ). Let ˜ u i = u u i , and letΛ i = Γ | ˜ u i . Each Λ i is a simplex, since there is no monomial of in( J ∆ ) that does notcontain at least one variable from every set of variables ˇ X i . The set { Λ i } i ∈ [ m ] is asimplicial cover for Γ | u .Now consider the nerve N of the simplicial cover { Λ i } . Every non-empty intersection ∩ i ∈ σ Λ i is a simplex (since every nonempty intersection of simplices is a simplex), hencecontractible. By the Nerve Lemma 3.15, N and Γ | u are homotopic.Therefore, N has n vertices corresponding to each Λ i . The intersection of any n − i is nonempty, which will correspond to a simplex on the variables in somemonomial u k , but the intersection of all n simplicies in { Λ i } is empty. So N is homotopicto the boundary of an n − | u ∼ = N ∼ = S n − . (cid:3) Lemma 3.16. Adopt notation and hypotheses as in Setups 3.1 and 3.11. Let w be amonomial that can be written as w = w · · · w ℓ such that each w i satisfies the following:(1) Each w i = m A i where A i = { ( k i,j , τ ij ) | τ i ∈ ∆ clique and k i,j ∈ Z ≥ } ,(2) τ i ∪ τ j / ∈ ∆ clique for i = j , and(3) For all w i and w j with i = j , w i and w j are relatively prime.Then Γ | w = Γ | w ∗ · · · ∗ Γ | w ℓ .Proof. Proceed by induction on ℓ . ℓ = 1 is clear.Let w ′ = w · · · w ℓ − , and assume by induction that Γ | w ′ = Γ | w ∗ · · · ∗ Γ | w ℓ − . Wewish to show that Γ | w = Γ w ′ ∗ Γ ℓ .Take a face σ ∈ Γ w ′ and a face σ ′ ∈ Γ ℓ . It suffices to show that ρ = σ ∪ σ ′ isa face of Γ w . Suppose, seeking contradiction, that ρ is not a face of Γ. Then themonomial corresponding to ρ is in the ideal in( J ∆ ), so it is divisible by some generator x ,i x ,i · · · x n,i n where { i < i < · · · < i n } is a facet of ∆. This implies ρ is a face NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 9 of ∆ clique , contradicting the assumption that the monomial τ r ∪ τ i / ∈ ∆ clique for any i < r . (cid:3) Setup 3.17. Adopt notation and hypotheses of Setups 3.1 and 3.11. Let ∆ = ∆ ∪ ∆ r be a clique decomposition of ∆ with the total order defined by ∆ < ∆ < · · · < ∆ r given by min( V (∆ )) < min( V (∆ )) < · · · < min( V (∆ r )). Observe that this is well-defined, since distinct cliques of a closed simplicial complex must have distinct minimumindexed vertices.Let w be a product of monomials w = w · · · w ℓ where each w i = m k i ( a i ; τ i ) is amonomial satisfying the hypotheses of Setup 3.11 such that:(1) For any i = j , w i and w j are relatively prime.(2) If i < j , then τ i ∪ τ j is not a face of ∆ clique , and any clique that contains τ i isstrictly less than any clique containing τ j in the total ordering on the cliques.(3) | a i | = k for all i , and a ij = 0 for any i, j .(4) Let v ∈ τ j and let ( k v , v ) be the pair associated to v in A ( a j ; τ j ). If v ∪ τ i ∈ ∆ clique for some i < j , then there is some ( p , q ) ∈ τ i such that that q > v but p < k v . Theorem 3.18. Adopt notation and hypotheses as in Setups 3.1 and 3.11. Let w be aproduct of monomials w = w · · · w ℓ satisfying the conditions of Setup 3.17. Then β i, w ( S/ in( J ∆ )) = (cid:26) if i = | w | − ℓ ( n − otherwise . For any monomial m that cannot be written as a product of monomials as in Setup3.17, β i, m ( S/ in( J ∆ )) = 0 for all i . In particular, the Z nm -graded Betti numbers for S/ in( J ∆ ) are either or .Example . Let G be the graph on 4 vertices with edge set E ( G ) = { (1 , , (2 , , (2 , , (3 , } and let J G be its corresponding binomial edge ideal. Observe that the monomial u = x , x , x , x , can never be written as a product of monomials satisfying the conditionsof Setup 3.17. For example, if one writes w = m ( { , } ; { , } ) = x , x , and w = m ( { , } ; { , } ) = x , x , , observe that w an w do not satisfy condition (4) abovebecause x , could instead be placed in w . If w ′ = m ( { , } ; { , , } ) = x , x , x , ,then w ′ = m ( { , } ; { } = x , , which does not satisfy condition (3). Therefore, β i, u ( S/ in( J G )) = 0 for all i .In contrast, one can write the monomial v = x , x , x , x , x , x , as a productof monomials w = m ( { , } ; { , , } ) = x , x , x , and w = m ( { , } ; { , , } ) = x , x , x , satisfying the conditions of Setup 3.17, so β i, v ( S/ in( J ∆ )) = (cid:26) i = 20 otherwise . Proof of Theorem 3.18. By Lemma 3.16, Γ | w ∼ = Γ ∗ Γ ∗ · · · ∗ Γ ℓ . By the proof ofProposition 3.13, the nerve of each Γ i is homotopy equivalent to the boundary of an( n − n − ℓ homology ( m − ℓ ( m − | w ∼ = S ℓ ( n − − , and we obtain the desired result by Hochster’s formula.Now it remains to check that for any monomial m not satisfying the properties above, β i, m ( S/ in( J ∆ )) = 0 for all i . First, observe that since in( J ∆ ) is a squarefree monomialideal, β i, m ( S in( J ∆ )) = 0 for all i if m is not squarefree.Let m be a squarefree monomial where the second indices of all the variables corre-spond to a face of ∆ clique , but there does not appear one variable from each color classˇ X i (i.e. from every row of M ). Then this monomial is not divisible by any generator ofin( J ∆ ), so it is not in in( J ∆ ) and Γ | m corresponds to a contractible face of Γ. Similarly,if m is divisible by x i,j x k,ℓ where i < k but j > ℓ and ( j, ℓ ) is a face of ∆, then m cannot be in in( J ∆ ) and therefore Γ | m must be a contractible face of Γ.Let u be a product of monomials u = u · · · u ℓ where each u i is a monomial satisfyingthe hypotheses of Setup 3.11 in a distinct maximal clique from u j for i = j , except for u ℓ .Then u ℓ corresponds to a contractible face of Γ. By Lemma 3.16, Γ | u ∼ = Γ | u ∗ · · · ∗ Γ | u ℓ .But since u ℓ is contractible, Γ | u ∼ = cone (Γ | u ∗ · · ·∗ Γ | u ℓ − ). Since the cone of a simplicialcomplex is always contractible, Γ | u is contractible. (cid:3) By summing over all possible monomials of the form in Setup 3.11 and applyingProposition 3.13, we obtain the following result. Corollary 3.20. Adopt notation and hypotheses as in Setup 3.1 and suppose ∆ is apure ( n − -dimensional simplicial complex which is closed. Then β i,i + ℓ ( S/ in( J ∆ )) = β i,i + ℓ ( S/J ∆ ) = (cid:18) ℓ − i + 1 ℓ − (cid:19) f ℓ + i − (∆ clique ) where f (∆ clique ) is the f -vector of ∆ clique . In particular, the Betti numbers in the linearstrand of J ∆ and in( J ∆ ) coincide. As an application of Theorem 3.18, we prove the following result. Theorem 3.21. Let ∆ be a pure ( n − -dimensional simplicial complex which is closed.When n > , the standard graded Betti numbers of S/J ∆ and S/ in( J ∆ ) coincide. Before proving the theorem above, we recall the following definition. Definition 3.22 ([16]) . A sequence q i,j of numbers is obtained from a sequence p i,j bya consecutive cancellation if there exist indexes s and r such that q s,r = p s,r − , q s +1 ,r = p s +1 ,r − ,q i,j = p i,j for all other values of i, j. Proof of Theorem 3.21. Observe that the Betti numbers of S/J ∆ can be obtained fromthe Betti numbers of S/ in( J ∆ ) by consecutive cancellations (see [16, Theorem 22.12]).By Theorem 3.18, the basis elements of the modules in the free resolution of S/ in( J ∆ )are given by monomials of the form m k ( a , τ ) · · · m k ℓ ( a ℓ , τ ℓ )which have homological degree P k i − ℓ ( n − 1) and internal degree P k i . NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 11 A consecutive cancellation is only possible when both β i,j ( S/ in( J ∆ )) and β i +1 ,j ( S/ in( J ∆ ))are nonzero. Assume that β i,j ( S/ in( J ∆ )) is nonzero. Then it has basis elements abovewhere j = P k i and i = P k i − ℓ ( n − 1) = j − ℓ ( n − j − i = ℓ ( n − β i +1 ,j ( S/ in( J ∆ )) is also nonzero, then j − i − n − 1. Sincetwo consecutive numbers cannot be divisible by n − n = 2, β i +1 ,j ( S/ in( J ∆ )) = 0and no consecutive cancellations are possible. Therefore, the Betti numbers of S/J ∆ and S/ in( J ∆ ) coincide for n > (cid:3) Remark . Observe that consecutive cancellations may still be possible in the casewhen n = 2. However, it is conjectured that even in the case when n = 2, the Bettinumbers of J ∆ and in( J ∆ ) coincide; see [9].4. Sparse Eagon-Northcott Complexes Definition 4.1. Let P be a logical statement outputting the values true or false .Define χ ( P ) = ( . Example . Let S = { , , } . Then χ (1 ∈ S ) = 1 and χ (5 ∈ S ) = 0. Definition 4.3. Let φ : F → G be a homomorphism of free modules of ranks f and g , respectively, with f ≥ g . Via the isomorphism Hom R ( F, G ) = F ∗ ⊗ G , φ induces anelement c φ ∈ F ∗ ⊗ G . The Eagon-Northcott complex is the complex0 → D f − g ( G ∗ ) ⊗ f ^ F → D f − g − ( G ∗ ) ⊗ f − ^ F → · · · → g ^ F → g ^ G with differentials in homological degree ≥ c φ ∈ F ∗ ⊗ G , and the map V g F → V g G is V g φ . Setup 4.4. 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 towhich M has the above matrix representation. Let w be an integral weight order onthe variables of R and let E • denote the Eagon-Northcott complex resolving R/I n ( M )where I n ( M ) denotes the ideal of maximal minors of M .Consider the ring R [ t ], where t is an arbitrary variable of degree 1. Let I hn ( M ) ⊆ R [ t ]denote the ideal generated by the homogenization ∆ hI of each minor ∆ I with respect tothe variable t . There is an induced homogenization of the Eagon-Northcott complex,denoted E h • , which is a complex of free R [ t ]-modules. Proposition 4.5 ([4], Proposition 3.8 and Corollary 3.9) . Adopt notation and hypothe-ses of Setup 4.4. Then E h • is a minimal free resolution of R [ t ] /I hn ( M ) .Moreover, to obtain the minimal free resolution of R/ in w ( I n ( M )) , simply set t = 0 in the resolution E h • . Notation 4.6. Adopt notation and hypotheses as in Setup 4.4. Let α = ( α , . . . , α n )with α i ≥ | α | = ℓ and I = ( i , . . . , i n + ℓ ) with i < · · · < i n + ℓ .The notation f I will denote f i ∧· · ·∧ f i n + ℓ ∈ V n + ℓ F , and g ∗ ( α ) will denote g ∗ ( α )1 · · · g ∗ ( α n ) n ∈ D ℓ ( G ∗ ). The notation ǫ i denotes the vector with a 1 in the i th spot and 0’s elsewhere.The following is a translation of Proposition 4.5 to a form that will be convenient. Proposition 4.7. Adopt notation and hypotheses of Setup 4.4. Let α = ( α , . . . , α n ) with α i ≥ and | α | = ℓ and I = ( i , . . . , i n + ℓ ) with i < · · · < i n + ℓ . Then the minimalfree resolution E ′• of in < ( I n ( M )) over R is such that E ′ ℓ = D ℓ ( G ∗ ) ⊗ V n + ℓ F , and thedifferential E ′ ℓ → E ′ ℓ − takes the form g ∗ ( α ) ⊗ f I X i,j ( − j +1 χ (( i, j ) ∈ I w ( α, I )) x ij g ∗ ( α − ǫ i ) ⊗ f I \ j , where I w ( α, I ) ⊆ { i | α i > } × I is some subset depending on α and I . For convenience, we use the above Proposition to define sparse Eagon-Northcottcomplexes and the indexing sets I w ( α, I ). Definition 4.8. Adopt notation and hypotheses of Setup 4.4. Let α = ( α , . . . , α n )with α i ≥ | α | = ℓ and I = ( i , . . . , i n + ℓ ) with i < · · · < i n + ℓ . The complex E ′• appearing in Proposition 4.7 will be called the sparse Eagon-Northcott complex.The indexing set I w ( α, I ) ⊂ { i | α i > } × I is defined via the nonvanishing terms inthe differential of the sparse Eagon-Northcott complex, as in Proposition 4.7. Definition 4.9. Adopt notation and hypotheses as in Setup 4.12. Define the indexingset L I to be the indexing set such that a = ( x ij | ( i, j ) ∈ L I ) . Enumerate the set L I = { ( i , j ) , . . . , ( i N , j N ) } , where N is some integer. Consider thevector A L I ∈ Z n defined by setting( A L ) k := |{ j | i j = k }| . Definition 4.10. Let J = ( j , . . . , j ℓ ) be an indexing set of length ℓ with j < · · · < j ℓ .Let α = ( α , · · · , α n ), with α i ≥ i . Define L ( α, J ) to be the subset of size ℓ subsets of the cartesian product { i | α i = 0 } × J, where { ( r , j ) , . . . , ( r ℓ , j ℓ ) } ∈ L ( α, J ) if |{ i | r i = j }| = α j .Assume that w is any weight vector and fix and indexing set I = ( i , . . . , i n ) with i < · · · < i n . Define the indexing set L Iw ( α, J ) to be all elements L ∈ L ( α, J ) with L ⊂ I w ( α, I ∪ J ) ∩ L I . Lemma 4.11. Adopt notation and hypotheses of Setup 4.4. Let I = ( i , . . . , i n ) and J = ( j , . . . , j ℓ ) with i < · · · < i n and j < · · · < i l . Let α = ( α , · · · , α n ) , with α i ≥ for each i . Use the notation α i := ( α , . . . , α i − , . . . , α n ) . Assume that ( i, j k ) ∈ I w ( α, I ∪ J ) ∩ L I . Then any L ′ ∈ L Iw ( α i , J \ j k ) is contained in a uniqueelement L ∈ L Iw ( α, J ) . NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 13 Proof. Given L ′ , simply take L := L ′ ∪ ( i, j k ), ordered appropriately. This elementis unique because the second coordinate must be j k , and since α i differs from α by1 in the i th spot, the first coordinate must be i . Moreover, this is well defined since( i, j k ) ∈ I w ( α, I ∪ J ) ∩ L I , whence L ∈ L Iw ( α, J ) by definition. (cid:3) Setup 4.12. Adopt notation and hypotheses as in Setup 4.4. Fix an indexing set I = ( i , . . . , i n ) with i < · · · < i n and consider the map G ∗ ⊗ V n +1 F → Rf I inducedby the differentials of the sparse Eagon-Northcott complex E ′• .Let a be the ideal such that d ′ ( E ′ ) = a f I . Observe that a is a complete intersectionsince it is generated by a subset of the generating set for the row of the standard Eagon-Northcott complex. Let K • = V • U denote the Koszul complex resolving R/ a , where U is a free R -module with basis e , . . . , e N .Define q : G ∗ ⊗ V n +1 F → U by sending all basis elements of the form g i ⊗ f I,j e L ,where L ∈ L Iw ( i, j ), and all other basis elements to 0. By construction, the followingdiagram commutes: G ∗ ⊗ V n +1 F q y y sssssssssss d ′ (cid:15) (cid:15) U / / a Proposition 4.13. Let R = k [ x , . . . , x m ] and let I nm denote the ideal generated by allsquarefree monomials of degree n in R . Observe that the rows of any minimal presentingmatrix N for I nm is indexed by all indexing sets I = ( i < · · · < i n ) , corresponding tothe generator x i · · · x i n .Then, the I th row of N generates the complete intersection ( x j | j ∈ [ m ] \ I ) . Proof. Let K ′ := (cid:0) x a · · · x a n | a < · · · < a n , ( a , . . . , a n ) = ( i , . . . , i n ) (cid:1) . The I th row of N generates to ideal ( K ′ : x i · · · x i n ), and it is immediate that this isequal to the ideal in the statement of the Proposition. (cid:3) Proposition 4.14. Adopt notation and hypotheses as in Setup 4.12. Assume that w corresponds to a proper monomial order < . Then a is a complete intersection on atleast m − n elements.Moreover, if a = ( x ij | ( i, j ) ∈ L ) , where L is some indexing set, then the set of j such that ( i, j ) ∈ L for some ≤ i ≤ n is precisely [ m ] \ I .Proof. Let E ′• denote the sparse Eagon-Northcott complex of Definition 4.8. By theproof of Theorem 3 . E ′• ⊗ R/ ( σ ), where σ = { x − x , x − x , . . . , x − x n } ∪ { x − x , . . . , x − x n } ∪ . . . ∪{ x m − x m , . . . , x m − x nm } , is a minimal free resolution of the ideal of all monomials of degree n in m variables.Each row of this specialized presenting matrix is a complete intersection on precisely m − n elements by Proposition 4.13, implying that the original presenting matrix had at least m − n nonzero entries. Moreover, as the subset of a generating set of a completeintersection, the ideal generated by this row must be a complete intersection.For the final statement, suppose that a = ( x ij | ( i, j ) ∈ L ). Then a ⊗ R/ ( σ ) ∼ =( x j | j ∈ L ), where L denotes the set of all second coordinates appearing in L . ByProposition 4.13, L = [ m ] \ I . (cid:3) Lemma 4.15. Let α = ( α , . . . , α n ) be a vector with | α | = d and let ℓ ≤ d . Let n ℓα := |{ α ′ ≤ α | | α ′ | = ℓ }| . Then, n ℓα = X S ⊆ [ n ] ( − | S | (cid:18) n + ℓ − P j ∈ S ( α j + 1) − n − (cid:19) . Proof. This is the “balls in bins with limited capacity” counting problem. (cid:3) If w corresponds to a proper monomial order < , Proposition 4.14 implies that L I ∈L ( A L I , [ m ] \ I ), where L ( A L I , [ m ] \ I ) is as in Definition 4.10. The following definitioncombines definitions 4.8, 4.10, and 4.9, and will be needed to define the q i maps ofProposition 2.3. Definition 4.16. Adopt notation and hypotheses of Setup 4.4. Let I = ( i , . . . , i n ) and J = ( j , . . . , j ℓ ) with i < · · · < i n and j < · · · < i l . Let α = ( α , · · · , α n ) with α i ≥ i and | α | = ℓ . Define the indexing set L Iw ( α, J ) ⊆ L ( α, J ) as all L ∈ L ( α, I )such that L ⊆ I w ( α, I ∪ J ) ∩ L I . Lemma 4.17. Define q ℓ : D ℓ ( G ∗ ) ⊗ n + ℓ ^ F → ℓ ^ U by sending g ∗ ( α ) ⊗ f J,I X L ∈L Iw ( α,J ) e L ∧ · · · ∧ e L ℓ , where J = ( j , . . . , j ℓ ) , j < · · · < j ℓ , and all other basis vectors to . Then the followingdiagram commutes: D ℓ ( G ∗ ) ⊗ V n + ℓ F q ℓ (cid:15) (cid:15) d k +1 / / D ℓ − ⊗ V n + ℓ − F q ℓ − (cid:15) (cid:15) V ℓ U / / V ℓ − U Proof. Compute the image of g ∗ ( α ) ⊗ f J,I going clockwise: g ∗ ( α ) ⊗ f J,I X { i | α i =0 } ≤ j ≤ ℓ ( − j +1 χ (( i, J j ) ∈ I w ( α, I ∪ J )) x iJ j g ∗ ( α − ǫ i ) ⊗ f J \ J j ,I + X { i | α i =0 } ≤ j ≤ n ( − m − n + j +1 χ (( i, I j ) ∈ I w ( α, I ∪ J )) x iI j g ∗ ( α − ǫ i ) ⊗ f J,I \ I j NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 15 X { i | α i =0 } ≤ j ≤ ℓ X L ∈L Iw ( α i ,J \ J j ) ( − j +1 χ (( i, J j ) ∈ I w ( α, I ∪ J )) x iJ j e L ∧ · · · ∧ e L ℓ − , where α i = α − ǫ i . According to Lemma 4.11, we may rewrite the above: X { i | α i =0 } ≤ j ≤ ℓ X L ∈L Iw ( α i ,J \ J j ) ( − j +1 χ (( i, J j ) ∈ I w ( α, I ∪ J )) x iJ j e L ∧ · · · ∧ e L ℓ − = X ≤ j ≤ ℓ X L ∈L Iw ( α,J ) ( − j +1 x L j e L ∧ · · · ∧ c e L j ∧ · · · ∧ e L ℓ . Going in the counterclockwise direction, we obtain: g ∗ ( α ) ⊗ f J,I X L ∈L Iw ( α,J ) e L ∧ · · · ∧ e L ℓ X L ∈L Iw ( α,J ) X ≤ j ≤ ℓ ( − j +1 x L j e L ∧ · · · ∧ c e L j ∧ · · · ∧ e L ℓ (cid:3) Proposition 4.18. Adopt notation and hypotheses as in Setup 4.12. Assume that w corresponds to a proper monomial order < . Let q ℓ : D ℓ ( G ∗ ) ⊗ n + ℓ ^ F → ℓ ^ U be the maps of Lemma 4.17. If q ℓ ( g ∗ ( α ) ⊗ f J,I ) = 0 , then α ≤ A L I . In particular, rank( q ℓ ⊗ k ) ≤ n ℓA LI · (cid:18) m − nℓ (cid:19) . Proof. It is clear that if q ℓ ( g ∗ ( α ) ⊗ f J,I ) = 0, then L Iw ( α, J ) = ∅ . Thus, α ≤ A L I and J ⊆ [ m ] \ I .Observe that L ( α, J ) ∩ L ( α ′ , J ) = ∅ for α = α ′ . This means that rank( q ℓ ⊗ k ) is thenumber of g ∗ ( α ) ⊗ f J,I ∈ D ℓ ( G ∗ ) ⊗ V n + ℓ F with nonzero image. By the above, there areat most n ℓA LI · (cid:0) m − nℓ (cid:1) such elements. (cid:3) Setup 4.19. 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.Choose indexing sets I j = ( i j , . . . , i jn ) for j = 1 , . . . , r pairwise disjoint; that is, I i ∩ I j = ∅ for i = j (this intersection is taken as sets). 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. Write n ^ F = n ^ F ′ ⊕ Rf I ⊕ · · · ⊕ Rf I r , where the notation f I j denotes f i j ∧ · · · ∧ f i jn . Recall that the sparse Eagon-Northcottcomplex of Definition 4.8 resolves the initial ideal of n × n minors of M .Observe that the sparse Eagon-Northcott differential d : G ∗ ⊗ V n +1 F → V m F induces homomorphisms d ℓ : G ∗ ⊗ V n +1 F → Rf I j by sending g ∗ i ⊗ f { ℓ } ,I j x iℓ f I j , and all other basis elements to 0. In the notation of Setup 2.1, this means we areconsidering the family of ideals a j = ( x iℓ | ( i, ℓ ) ∈ I j ) . For each j = 1 , . . . , r , a j is a complete intersection generated by at least m − n elements(by Proposition 4.14), hence resolved by the Koszul complex. Let U j = M ( i,ℓ ) ∈ L Ij Re iℓ with differential induced by the homomorphism m j : U j → R sending e iℓ x iℓ . If L = ( i, j ) is a 2-tuple, then the notation e L will denote e ij . Lemma 4.20. Adopt notation and hypotheses of Setup 4.19. Define q jℓ : D ℓ ( G ∗ ) ⊗ n + ℓ ^ F → ℓ ^ U j by sending g ∗ ( α ) ⊗ f J,I j X L ∈L Ijw ( α,J ) e L ∧ · · · ∧ e L ℓ , where J = ( j , . . . , j ℓ ) , j < · · · < j ℓ , and all other basis vectors to . Then the followingdiagram commutes: D ℓ ( G ∗ ) ⊗ V n + ℓ F q jℓ (cid:15) (cid:15) d ℓ +1 / / D ℓ − ⊗ V n + ℓ − F q jℓ − (cid:15) (cid:15) V ℓ U / / V ℓ − U Notation 4.21. Let A I j be the vector associated to the indexing set I j as in Definition4.9. Define the vector min j { A I j } to be the vector with k th component equal tomin j { ( A i j ) k } , for all 1 ≤ k ≤ n . Let P i ([ m ]) denote all size i subsets of [ m ]. NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 17 Theorem 4.22. Adopt notation and hypotheses as in Setup 4.19 and let q jℓ : D ℓ ( G ∗ ) ⊗ V n + ℓ F → V ℓ U j be as in Lemma 4.20. Then, rank q ℓ ... q rℓ ⊗ k ! ≤ r X i =1 ( − i +1 X T ∈P i ([ r ]) n ℓ min j ∈ T { A Ij } (cid:18) m − inℓ − ( i − n (cid:19)! Proof. The proof employs the inclusion-exclusion principle. We first count all g ∗ ( α ) ⊗ f J ∈ D ℓ ( G ∗ ) ⊗ V n + ℓ F with nonzero image under precisely i of the q jℓ maps. It iseasy to see that if such an element has nonzero image under precisely i maps, then α ≤ min j ∈ T { A I j } for some size i subset T ⊆ [ r ], and J = J ′ ∪ (cid:16) S j ∈ T I j (cid:17) .There are precisely n ℓ min j ∈ T { A Ij } such α and (cid:0) m − inℓ − ( i − n (cid:1) such indexing sets J ′ . Summingover all T ∈ P i ([ r ]) gives the number of g ∗ ( α ) ⊗ f J ∈ D ℓ ( G ∗ ) ⊗ V n + ℓ F with nonzero imageunder precisely i of the q jℓ maps. By the inclusion exclusion principle, the number of g ∗ ( α ) ⊗ f J ∈ D ℓ ( G ∗ ) ⊗ V n + ℓ F with nonzero image under some collection of q jℓ is boundedby r X i =1 ( − i +1 X T ∈P i ([ r ]) n ℓ min j ∈ T { A Ij } (cid:18) m − inℓ − ( i − n (cid:19)! . Moreover, since L ( α, J ) ∩ L ( α ′ , J ) = ∅ for α = α ′ , the above is an upper bound ofrank k q ℓ ... q rℓ ⊗ k . (cid:3) Example . Observe that we do not necessarily have equality in the above ranks, asthe following example shows. Let M be a generic 3 × x x x ∈ in < I ( M ), where < denotes standard lexicographic order.The corresponding ideal a is ( x , x , x ), so that A L (1 , , = (1 , , q ⊗ k has rank 2 < n , , · (cid:0) (cid:1) = 3.For the sake of providing more examples, retain the setting above. If we remove thegenerator x x x ∈ in < I ( M ), the corresponding ideal a is ( x , x , x , x ). Thisyields A L (1 , , = (0 , , q ⊗ k ) = 3 = n , , · (cid:0) (cid:1) .If we remove the generator x x x ∈ in < I ( M ), then a = ( x , x , x , x ). Inthis case, A L (1 , , = (1 , , q ⊗ k ) = 4 = n , , · (cid:0) (cid:1) .5. The Case for Standard Lex Order In this section, we specialize to the case of standard lexicographic order. Our firstgoal is to produce an explicit resolution of the initial ideal of maximal minors of an n × m matrix under standard lexicographic order. We then employ Theorem 2.4 toverify a conjecture of Ene, Herzog, and Hibi in a previously unknown case. Lemma 5.1. Adopt notation and hypotheses as in Setup 4.4, and assume that w cor-responds to the standard lexicographic order < . Let N denote any minimal present-ing matrix for in < ( I n ( M )) , whose rows are indexed by indices I = ( i , . . . , i n ) with i < · · · < i n . Then the I th row of N generates the ideal (cid:16) x kℓ k | ≤ ℓ
Let α = ( α , . . . , α n ). Define α ≤ i := ( α , . . . , α i ) , where α ≤ i = ∅ if i ≤ α ≤ i = α if i ≥ n . Definition 5.3. 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 . 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 , } Definition 5.5. Adopt notation and hypotheses as in Setup 4.4, and assume that w corresponds to the standard lexicographic order < . Let E ′• denote the sequence ofmodule 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 sparse Eagon-Northcott differential d ℓ ( g ∗ ( α ) ⊗ f I ) = X { i | α i > } X j ( − j +1 χ (( i, I j ) ∈ I < ( α, I )) x iI j g ∗ ( α − ǫ i ) ⊗ f I \ I j . Proposition 5.6. The sequence of homomorphisms E ′• of Definition 5.5 forms a com-plex. NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 19 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 . 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) Theorem 5.7. Adopt notation and hypotheses as in Setup 4.4, and assume that w corresponds to the standard lexicographic order < . Then the complex of Definition 5.5is a minimal free resolution of the ideal in < ( I n ( M )) . The proof of Theorem 5.7 will follow after a series of Lemmas. The idea for theproof uses Theorem 5.8, which is inspired by the proof of acyclicity of the complexesconstructed in [11]. The proof is given in [18]. Theorem 5.8 ([18]) . Let R be a commutative ring. Let ( F • , d • ) be an n -linear complexof free R -modules such that for all i ≥ , rank( F i ) i + n = β i,i + n ( H ( F • )) . If for all i ≥ , the map ( d i ) i + n : ( F i ) i + n → ( F i − ) i + n is left invertible, then F • is acyclic. Lemma 5.9. Adopt notation and hypotheses of Setup 4.4. Then for all ℓ , j , ( d ℓ ) j : ( E ℓ ) j → ( E ℓ − ) j is left invertible, where d ℓ denotes the standard Eagon-Northcott differential.Proof. We will prove that the matrix representation of ( d ℓ ) n + ℓ with respect to thestandard bases is such that every column has a nonzero entry and every row has atmost 1 nonzero entry, whence ( d ℓ ) n + ℓ contains a full rank permutation matrix as asubmatrix and is hence left invertible.The fact that every column contains a nonzero entry is the statement that d ℓ ( g ∗ ( α ) ⊗ f I ) = 0 for all I , α , which is trivial. Similarly, given any x ij g ∗ ( α ) ⊗ f I with j / ∈ I , therow corresponding to this basis element has entry ± g ∗ ( α + ǫ i ) ⊗ f I ∪{ j } , just by definition of the Eagon-Northcott differential. If j ∈ I , thenall entries corresponding to this row are 0. This proves the statement. (cid:3) Corollary 5.10. Adopt notation and hypotheses of Setup 4.4. Then the restriction ofthe differentials of Definition 5.3 to degree n + ℓ ( d ′ ℓ ) n + ℓ : ( E ′ ℓ ) n + ℓ → ( E ′ ℓ − ) n + ℓ are left invertible.Proof. By construction, the differentials d ′ ℓ satisfy the same property as in the proof ofLemma 5.9, and are hence left invertible. (cid:3) Proof of Theorem 5.7. By [4, Theorem 3.1], the minimal free resolution of in < ( I n ( M ))is n -linear with ranks precisely the ranks of the Eagon-Northcott complex. Thus, com-bining Theorem 5.8 with Corollary 5.10, the complex of Definition 5.5 is acyclic. (cid:3) Corollary 5.11. Adopt notation and hypotheses as in Setup 4.4, and assume that w corresponds to the standard lexicographic order < . Let E ′• denote the minimal freeresolution of Definition 5.5. Then E ′• ⊗ R/σ is a minimal free resolution of the ideal ofall 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 } Corollary 5.12. Adopt notation and hypotheses as in Setup 4.4, and assume that w corresponds to the standard lexicographic order < . Let E ′• denote the minimal freeresolution of Definition 5.5. 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 } Remark . Theorem 5.7 and its corollaries presents a novel construction of explicitminimal free resolutions of powers of the graded maximal ideal and the ideal generatedby all squarefree monomials. Minimal resolutions for both classes of ideals were alreadywell known, however, the resolutions constructed above have the advantage of beingbuilt up by simpler free modules with differentials explicitly computable without theuse of any kind of straightening algorithms for Young tableaux. Theorem 5.14. Adopt notation and hypotheses as in Theorem 2.7 and let < denotestandard lexicographic order. If n > , then the ideals (∆ J | J = I j , j = 1 , . . . , r ) and (in < (∆ J ) | J = I j , j = 1 , . . . , r ) have different Betti tables.Proof. By Lemma 5.1, the second differential in the minimal free resolution of in I n ( M )over R is a complete intersection on at most 2( m − n ) elements. In the notation ofObservation 2.6, this implies that the ideal a corresponding to any one of the removedgenerators has pd R R/ a ≤ m − n ). By Theorem 2.4, the ideal (in < (∆ J ) | J = I j , j =1 , . . . , r ) also has projective dimension ≤ m − n ). Since n > 2, this impliespd R R/ (in < (∆ J ) | J = I j , j = 1 , . . . , r ) ≤ m − n ) < n ( m − n ) = pd R R/I n ( M ) , NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 21 where the rightmost equality is by Theorem 2.7. Since the ideals (∆ J | J = I j , j =1 , . . . , r ) and (in < (∆ J ) | J = I j , j = 1 , . . . , r ) have different projective dimensions, theconclusion follows. (cid:3) Corollary 5.15. Adopt notation and hypotheses of Setup 4.12 with n = 2 and assumethat w corresponds to standard lexicographic order < . Let K ′ = (in < (∆ J ) | J = (1 , m )) .Then K ′ has Betti table · · · ℓ · · · m − − m − · · · · · · (cid:0) m (cid:1) − · · · ℓ (cid:18)(cid:0) mn + ℓ − (cid:1) − (cid:0) m − ℓ − (cid:1)(cid:19) · · · · · · (cid:0) m − ℓ (cid:1) − ( ℓ + 1) (cid:0) m − ℓ (cid:1) · · · m − 2) 1 . Proof. Let E ′• denote the sparse Eagon-Northcott complex resolving in < I n ( M ). Therow corresponding to the initial term of the minor (1 , m ) of the second differential of E ′• is a complete intersection on m − m − · m − 2) elements by Lemma5.1.This implies that A L (1 ,m ) = ( m − , m − 2) (where A L (1 ,m ) is as in Definition 4.9), andhence for all ℓ ≤ m − n ℓA L (1 ,m ) = (cid:0) ℓ − ℓ (cid:1) = ℓ + 1. By Proposition 4.18, rank( q ℓ ⊗ k ) ≤ ( ℓ + 1) (cid:0) m − ℓ (cid:1) . To prove the reverse inequality, it suffices to show that for all α , J with | α | = ℓ , J = ( j < · · · < j ℓ ) , j = 1 , j ℓ = m , there exists L ∈ L ( α, J ) with L ⊆ I < ( α, (1 < j < · · · < j ℓ < m )).Since n = 2, α = ( p, ℓ − p ) for some 0 ≤ p ≤ ℓ . Then, consider L = ((1 , j ) , . . . , (1 , j p ) , (2 , j p +1 ) , . . . , (2 , j ℓ )) . This is clearly an element of L (( p, ℓ − p ) , J ). Moreover, by definition, if K = (1 , j , . . . , j ℓ , m ),then (1 , j s ) ∈ I < ( α, K ). This is simply because j s = K s +1 , and by construction(1 , j s ) ∈ L ⇐⇒ ≤ s ≤ p and (2 , j s ) ∈ L ⇐⇒ p + 1 ≤ s ≤ ℓ .The Betti table is an immediate consequence of Corollary 2.5. (cid:3) Corollary 5.16. Adopt notation and hypotheses of Setup 4.4 with n = 2 and assumethat w corresponds to standard lexicographic order < . Let ( i, j ) be an arbitrary indexingset with ≤ i < j ≤ m . Then the ideals (∆ J | J = ( i, j )) and (in < ∆ J | J = ( i, j )) have the same Betti table if and only if ( i, j ) = (1 , m ) .Proof. The converse is Corollary 5.15 combined with Theorem 2.7. For the forwardimplication, observe that R/ (∆ J | J = ( i, j )) has projective dimension 2( m − n ) and R/ (in < ∆ J | J = ( i, j )) has projective dimension j − i + 1 + m − n . If j − i < m − (cid:3) Corollary 5.17. Adopt notation and hypotheses of Setup 4.4. Then the generators of (∆ J | J = I j , j = 1 , . . . , r ) do not form a Gr¨obner basis if n > . Proof. The projective dimension of in < (∆ J | J = I j , j = 1 , . . . , r ) is an upper boundof the projective dimension of (∆ J | J = I j , j = 1 , . . . , r ), but (in < ∆ J | J = I j , j =1 , . . . , r ) has strictly smaller projective dimension for n > (cid:3) Linear Strand of the Minimal Free Resolution of Lex-InitialDeterminantal Facet Ideals Setup 6.1. 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 standard lexicographic order on R and in < I n ( M ) the initial ideal with respect to < of the ideal of maximal minors of M . Theorem 6.2 ([13], 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 6.3 ([13], 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 • . Definition 6.4. Adopt notation and hypotheses as in Setup 6.1. Let ∆ denote asimplicial complex on the vertex set [ m ]. Define C < (∆ , M ) := V n G . For i ≥ 1, let C
Let ∆ be a simplicial complex. Then an i -nonface σ is an element σ / ∈ ∆ such that for some j ≥ σ \ σ j + k ∈ ∆ for all k = 0 , . . . , i . NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 23 Example . 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 6.2 and Proposition 6.3. Lemma 6.8. Adopt notation and hypotheses as in Setup 6.1. 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 6.2. 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 E ′• is as in Definition 5.5.Assume H i ( C < • (∆ , M )) i + n = 0. Let z ∈ C } X j ( − j +1 χ (( ℓ, σ j ) ∈ I < ( α, σ )) 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 6.8 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 6.6. Then the element z = x f , , − x f , , is a cycle which is not a boundary n C < • (∆ clique , M ). Lemma 6.10. Adopt notation and hypotheses as in Setup 6.1. Then the following areequivalent:(1) H ( C < • (∆ , M )) = 0 ,(2) There are no -nonfaces of cardinality n + 1 .Proof. The implication (2) = ⇒ (1) is Lemma 6.8. 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 C < (∆ , M ) that is not a boundary, since z = d ( g ∗ j ⊗ f σ ), and g ∗ j ⊗ f σ / ∈ C (∆ , M ) byconstruction. (cid:3) Lemma 6.11. Let ∆ be a pure ( n − -dimensional simplicial complex on the vertexset [ m ] . If the simplicial complex ∆ is closed, then the associated clique complex ∆ clique has no -nonfaces of cardinality ≥ n + 1 .Proof. Assume that ∆ clique has a 1-nonface σ / ∈ ∆ clique of cardinality ≥ n + 1. Bydefinition, there exists j ≥ σ \ σ j , σ \ σ j +1 ∈ ∆ clique . Let Γ , . . . , Γ r be thecliques of ∆; then there are two cases:Case 1: σ \ σ j , σ \ σ j +1 ∈ Γ i for some i . Since Γ i is a simplex, | σ | = | Γ | + 1. Thusthere is only 1 element σ not contained in Γ i , in which case there are obviously no1-nonfaces, since if σ j / ∈ Γ i , then σ \ σ j − and σ \ σ j +1 / ∈ Γ i (notice that this case isimpossible, regardless of closedness).Case 2: σ \ σ j ∈ ∆ k , σ \ σ j +1 ∈ Γ i for some i = k . Without loss of generality, we mayassume that | σ | = n + 1 by taking an appropriate subset of σ . But then σ \ σ j ∈ ∆ k , σ \ σ j +1 ∈ ∆ i , and it is clear that all but the j th entries of σ \ σ j and σ \ σ j +1 are equal,whence ∆ is not closed. (cid:3) Recall that the standard Eagon-Northcott complex inherits a Z n × Z m -grading, asdescribed in Section 3 of [13]. Since the sparse Eagon-Northcott complexes of Section 4are obtained by simply setting certain entries in the differentials equal to 0, these maps NITIAL IDEALS OF CLOSED DETERMINANTAL FACET IDEALS 25 will remain multigraded in an identical manner. We tacitly use this multigrading forthe remainder of this section. Theorem 6.12. Adopt notation and hypotheses as in Setup 6.1. Assume that ∆ is an ( n − -pure closed simplicial complex. Let in < J ∆ denote the initial ideal with respectto standard lexicographic order of the determinantal facet ideal associated to ∆ .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 ho-mogeneous 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 5.5), there exists y ∈ C < ( M ) such that d ( y ) = z . By multihomogeneity, y = λg s ⊗ f τ for some constant 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 , sinceotherwise ∆ clique would have a 1-nonface of cardinality n +1, contradicting Lemma 6.11.Thus Z lin is generated by { 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 6.14. Adopt notation and hypotheses as in Setup 6.1. Assume that ∆ is an ( n − -pure closed simplicial complex. 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 generalizedEagon-Northcott complex of [13]. Then, C • and C < • have the same underlying freemodules, so the result follows. (cid:3) References 1. Dave Bayer, Hara Charalambous, and Sorin Popescu, Extremal betti numbers and applications tomonomial ideals , arXiv preprint math/9804052 (1998).2. David Bernstein and Andrei Zelevinsky, Combinatorics of maximal minors , Journal of AlgebraicCombinatorics (1993), no. 2, 111–121. 3. Anders Bj¨orner, Topological methods , Handbook of combinatorics (1995), 1819–1872.4. Adam Boocher, Free resolutions and sparse determinantal ideals , Math. Res. Lett (2012), no. 04,805–821.5. Winfried Bruns and J¨urgen Herzog, Cohen-macaulay rings , no. 39, Cambridge university press,1998.6. Winfried Bruns and Udo Vetter, Determinantal rings , vol. 1327, Springer, 2006.7. Aldo Conca and Matteo Varbaro, Square-free gr¨obner degenerations , Inventiones mathematicae(2020), 1–18.8. Persi Diaconis, David Eisenbud, and Bernd Sturmfels, Lattice walks and primary decomposition ,Mathematical Essays in Honor of Gian-Carlo Rota, Springer, 1998, pp. 173–193.9. Viviana Ene, J¨urgen Herzog, and Takayuki Hibi, Cohen-macaulay binomial edge ideals , NagoyaMathematical Journal (2011), 57–68.10. Viviana Ene, J¨urgen Herzog, Takayuki Hibi, and Fatemeh Mohammadi, Determinantal facet ideals ,arXiv preprint arXiv:1108.3667 (2011).11. Federico Galetto, On the ideal generated by all squarefree monomials of a given degree , arXivpreprint arXiv:1609.06396 (2016).12. J¨urgen Herzog, Takayuki Hibi, Freyja Hreinsd´ottir, Thomas Kahle, and Johannes Rauh, Binomialedge ideals and conditional independence statements , Advances in Applied Mathematics (2010),no. 3, 317–333.13. J¨urgen Herzog, Dariush Kiani, and Sara Saeedi Madani, The linear strand of determinantal facetideals , The Michigan Mathematical Journal (2017), no. 1, 107–123.14. Sara Saeedi Madani, Binomial edge ideals: A survey , The 24th National School on Algebra,Springer, 2016, pp. 83–94.15. Masahiro Ohtani, Graphs and ideals generated by some 2-minors , Communications in Algebra (2011), no. 3, 905–917.16. Irena Peeva, Graded syzygies , vol. 14, Springer Science & Business Media, 2010.17. Bernd Sturmfels and Andrei Zelevinsky, Maximal minors and their leading terms , Advances inMathematics (1993), no. 1, 65–112.18. Keller VandeBogert, Linear strand and minimal free resolutions of certain equigenerated monomialideals , In Preparation (2020).19. , Trimming complexes and applications to resolutions of determinantal facet ideals , arXivpreprint arXiv:2004.06016 (2020). Cornell University E-mail address : [email protected] URL : http://math.cornell.edu/~aalmousa University of South Carolina E-mail address : [email protected] URL ::