ON THE GENERALIZED SCARF COMPLEX OF LATTICE IDEALS 9 HARACHARALAMBOUSANDAPOSTOLOSTHOMA 0 0 2 Abstract. Letkbeafield,L⊂Zn bealatticesuchthatL∩Nn={0},and n IL⊂k[x1,...,xn]thecorrespondinglatticeideal. Wepresentthegeneralized a Scarf complex of IL and show that it is indispensable in the sense that it is J contained ineveryminimalfreeresolutionofR/IL. 9 ] C 1. Introduction A Let k be a field, let L ⊂ Zn be a lattice and I the corresponding lattice ideal . L h ink[x ,...,x ]. In a seriesof recentpapers,see [1], [5], [9], [10], [13], [16], [17] and 1 n t [25], the problem of getting different minimal generating systems for lattice ideals a m I was investigated. One of the motivations was a related question from Algebraic L Statistics that asked when a prime lattice ideal possesses a unique minimal system [ of binomial generators. As a result the notion of indispensable binomials was in- 1 troduced by Ohsugi and Hibi [16] to describe the binomials that up to constant v multiples are part of all minimal systems of generators of I . The corresponding 8 L 0 notion for higher syzygies is the motivating question for the present article. Three 2 landmarkpapers dealing with free resolutions oflattice ideals by Bayer,Peeva and 1 Sturmfels providecrucialleads: see [4], [20], and[21]. The algebraicScarfcomplex . 1 isdefinedin[20]anditisshowntobe aminimalfreeresolutionwhenIL isgeneric. 0 TheTaylorcomplexdefinedin[4]isalwaysaresolution,alashardlyeveraminimal 9 one. The complexity of computing minimal resolutions becomes even more appar- 0 ent in [21], and the importance ofthe topologicalstructureof the fibers is strongly : v hinted. Apart from the above, the problem of computing syzygies of lattice ideals i has also been addressed by many recent articles, see for example [6], [8], [18], [19]. X The structure of this paper is as follows: in section 2 we introduce the basic r a notions and terminology. In section 3 we introduce simple minimal resolutions of I : these are complexes such that the syzygies determined by Zn/L-homogeneous L baseshaveminimalsupport. Insection4wegeneralizetheScarfsimplicialcomplex and basic fibers of [20]. In Section 5 we introduce the generalized algebraic Scarf complex and show that it is an indispensable complex. The last section contains examples and open problems. 2. Notation Let L ⊂ Zn be a lattice such that L∩Nn = {0}. The polynomial ring R = k[x ,...,x ] is positively multigraded by the group Zn/L, see [15]. Let a = e + 1 n i i L where {e : 1 ≤ i ≤ n} is the canonical basis of Zn. By A we denote the i 1991 Mathematics Subject Classification. 13D02,13D25. Key words and phrases. Resolutions, lattice ideal, syzygies, indispensable syzygies, Scarf complex. 1 2 H.CHARALAMBOUSANDA.THOMA subsemigroupofthegroupZn/Lgeneratedby{a :1≤i≤n}. SinceL∩Nn ={0}, i the semigroup A is pointed meaning that {x : x ∈ A and − x ∈ A} = {0}. Equivalently if A is a pointed semigroup generated by a ,...,a then L ⊂ Zn is 1 n the lattice of relations of a ,...,a and L∩Nn ={0}. We set deg (x )=a . The 1 n A i i A-degree of the monomial xv =xv1···xvn is 1 n v deg (x ):=v a +···+v a ∈A. A 1 1 n n When we want to put the emphasis on L we occasionally write deg (xv) := L deg (xv). The lattice ideal I (or I ), associated to L is the ideal generated A L A by all the binomials xu+ −xu− where u+,u− ∈Nn and u=u+−u− ∈L. Prime latticeidealsarecalledtoricideals[24]andarethe definingidealsoftoricvarieties. For binomials in IL, we define degA(xu+ −xu−) := degAxu+. Lattice ideals are A-homogeneous. For b ∈ A we let R[−b] be the A-graded free R-module of rank 1 whose generator has A-degree b. Let (F ,φ): 0−→F −φ→p ······−→F −φ→1 F −→R/I −→0, L p 1 0 L be a minimal A-graded free resolution of R/I . The i-Betti number of R/I of L L A-degree b is equal to the rank of the R-summand of F of A-degree b: i βi,b(R/IL)=dimkTori(R/IL,k)b and is denoted by βi,b(R/IL). This is an invariant of IL, see [15]. For b ∈ A, we let Cb equal the fiber Cb :=deg−A1(b)=deg−L1(b):={xu : degA(xu)=b} and a a ∆b :={F ⊂{1,...,n}: ∃ x ∈Cb, F ⊂suppx }. It is well known that βi,b(R/IL)=dimkH˜i(∆b) see [2, 6, 7, 23]. The degrees b for which H˜i(∆b)6=0 are called i-Betti degrees. The semigroup A is pointed, so we can partially order A with the relation c≥d⇐⇒ there is e∈A such that c=d+e. The minimal elements ofthe set{b: βi,b(R/IL)6=0} withrespectto ≥ arecalled minimal i-Betti degrees. 3. Simple syzygies Inthissectionwepresentthe theoryofsimple syzygiesofA-homogeneousideals for a positive grading A. We will apply these results for lattice ideals. Let I be an A-homogeneous ideal and (F,φ) be a minimal A-graded free resolution of R/I, where F = R·E . i M ti 1≤t≤si In particular E is the basis element of F ∼= R of A-degree 0. The elements of 10 0 Imφ =kerφ arecalledi-syzygies. InthesequelallsyzygiesareA-homogeneous. i+1 i We note that the zero syzygies of R/I are the elements of I. Let h be an A- homogeneouselement ofF . We write h asa combinationofthe basiselements E i ti with nonzero coefficients: a (1) h= X (X catx t)Eti . 1≤t≤si cat6=0 SYZYGIES 3 Since h is A-homogeneous, degAxat +degA(Eti) = degA(h). When h ∈ kerφi we define S(h), the syzygy support of h to be the set a S(h)={x tEti : cat 6=0}, and partially order the A-homogeneous i-syzygies by h′ ≤h if and only if S(h′)⊂ S(h). WenotethatE mayappearinS(h)morethanoncewithdifferentmonomial ti coefficients. Definition 3.1. We say that a nonzero A-homogeneous h ∈ kerφ is simple if i there is no nonzero A-homogeneous h′ ∈kerφ such that h′ 1. J Suppose that m E ∈/ S(h) while m E ∈S(h); where f =m −n ,f = i J\{ki} j J\{kj} i i i j m −n . Note that since R has no zero divisors and the coefficient of E j j J\{ki,kj} in θ(h) has to be zero, it follows that n E ∈ S(h). Therefore the coefficient i J\{ki} of E in θ(h) is (c n f −(c m −d n )f ) for some c ,c ,d ∈ k which J\{ki,kj} i i kj j j j j ki i j j cannot be zero since f ,f is an R-sequence, a contradiction. ki kj We note that a simple syzygy is determined by its syzygy support: Theorem 3.8. If h and h′ are two simple i-syzygies and S(h)=S(h′), then there exists a c6=0 in k such that h=ch′. SYZYGIES 5 Proof. Since h′ 6= 0 there exists ca′ 6= 0 such that h′t = ca′xaEa+Pb6=acbxbEb. Therefore ca 6=0 and h′′ =h− ca h′ ∈kerφ , ca′ i while S(h′′) S(h). Therefore h′′ =0 and h=ca/ca′h′. (cid:3) Next we give the definition of an indispensable complex. We note the related notion of rigidity, see [14]. Definition 3.9. Let I be an A-homogeneous ideal. We say that an A-graded complex (G,θ) is an indispensable complex for R/I if for any simple minimal A-graded free resolution (F,φ) of R/I (with respect to bases B of F ), there is an inclusion j j map i:(G,θ)−→(F,φ) so that the image of G is a subcomplex of F. In particular for each j there is a subset B ′ of B so that the set i−1(B ′) is an A-homogeneous j j j basis of G . If H ∈i−1(B ′) we call θ (H ) an indispensable j−1 syzygy of R/I. j tj j j tj We say that G is a strongly indispensable complex for R/I if the above holds without the requirement for (F,φ) to be simple. Let I be a toric ideal. A polynomial of I is simple, as a zero syzygy, if and L L onlyifitisabinomial. Inthiscasetheindispensablezero-syzygiesofR/I arealso L strongly indispensable. More precisely in [10] it was shown that the binomial f of degreebiscontainedineveryminimalsetofbinomialgeneratorsofI ifandonlyif L bis a minimal 1-Bettidegreeandthe fiber Cb consists ofjust two monomialswith no common divisor: the difference of these two monomials is f up to a constant multiple. Such a binomial is not only contained in every minimal set of binomial generators of I ; it is necessarily contained in every minimal system of generators L of I and thus is strongly indispensable, see [20]. In [20], Peeva and Sturmfels L introduced the algebraic Scarf complex for R/I . They showed that this complex L iscontainedineveryminimalresolutionofR/I . ItfollowsthatthealgebraicScarf L complex is a strongly indispensable complex. We will generalize this construction and show that the generalized algebraic Scarf complex is also indispensable. 4. The generalized Scarf complex Let max (∅)=(−∞,...,−∞). For J ⊂L, 0<|J|<∞, we let max (J)=(max{a : a∈J},··· ,max{a : a∈J})∈Zn . 1 n Wehavemax (J′)≤max (J)ifandonlyifmax (J′) ≤max (J) whilemax (∅) i i isthe smallestelement. ForJ ⊂K ⊂Lwedefinev-supp (J), thevariable support K of J in K: if J =∅ we set v-supp (∅)=∅ and for all other J we set K v-supp (J):={i: ∃ a∈J, such that max (K) −a >0}. K i i We note that v-supp (J)= supp(xmax (K)−a), K [ a∈J where supp(xa)={i:x |xa}. i In [20] the complex ∆ was defined to be the collection of all finite subsets J of L L with unique max (J). We extend this complex to ∆ as follows: L e Definition 4.1. Let ∆ be the collection of all finite subsets J of L that satisfy L the following conditiones: (1) if J′ (J then max (J′)2 and a≤max (J) then supp(xmax (J)−a) v-supp (J)=∅ . \ J We note that max (J) is determined by at most n elements of J. If J ∈∆ then L the first condition implies that |J| ≤ n. On the other hand if |J| = 2 tehen the first two conditions imply that max (J) is unique. Finally all sets J with unique max (J) are in ∆ and thus ∆ ⊂∆ . L L L e e Proposition 4.2. ∆ is a simplicial complex. L e Proof. We have {a} ∈ ∆ for every a ∈ L: b ≤ a =⇒ a − b ≥ 0 ∈ L, a L contradiction. The case |eJ| = 2 is trivial. We now examine the case |J| > 2, J ∈ ∆ . L e Let a ∈ J; we will show that J = J \ {a} ∈ ∆ . Let J ( J such that 1 L 2 1 max (J2)=max (J1). Itfollowsthatmax (J2∪{a})e=max (J),acontradiction. Suppose now that c ∈/ J and c ≤ max (J ). Therefore c 6= a, c < max (J) 1 1 and supp(xmax (J)−c) v-supp (J) = ∅. On the other hand since max (J ) < T J 1 max (J), it follows that for some i, max (J ) < max (J) . Therefore a = 1 i i i max (J) while for all b∈J , b 2. If Cb satisfies the above properties then we call Cb a basic fiber. We note that not all fibers contain subsets that are basic fiber components. As a matteroffactwewillshow,seeTheorem4.12,thatthesetofbasicfibercomponents is finite. The definition of a basic fiber first appeared in [20]. It follows directly from definition 4.7 that if Cb is a basic fiber and |Cb| > 2 then Cb has only one connected component. Moreover if |Cb| = 2 then Cb is a basic fiber if and only if Cb isdisconnected. WealsonotethatifGisabasiccomponentofCb and|Cb|=2 thenG=Cb isabasicfiber. ThisistheonlywayC canbeabasicfibercomponent when |C|=2 as the following Lemma shows: Lemma 4.8. If |J| = 2 and CJ is a basic component of Cb then CJ is a basic fiber. Proof. We have|C |=2. ByLemma4.4C isdisconnectedin∆ (b). Thethird J J gcd condition of definition 4.7 implies that CJ =Cb. (cid:3) In [20, Theorem 3.2, Lemma 3.3] the following was shown: Lemma 4.9. If Cb is a basic fiber with i+1 elements then dimH˜i(∆b)=1 and b is a minimal i-Betti degree. 8 H.CHARALAMBOUSANDA.THOMA Let T be a subset of a fiber Cb. We denote by [T] the set of monomials in T divided by gcd(T). Let I′ =I∪{m}⊂T, m∈/ I. It is clear that m [T \I′]=[[T \I]\{ } ]. gcd(T \I) Lemma 4.10. If CJ is a basic component of Cb and ∅ =6 I ⊂ CJ then [CJ \I] is a basic fiber. Proof. By the preceding comment it suffices to prove the statement when |I| = 1. Let d = deg (gcd(C \I)) and b′ = b−d. Since gcd(C \I) 6= 1 it follows that A J J b′ 6= b. We will show that [CJ \I] = Cb′. Indeed if m ∈ Cb′ then mxd ∈ Cb and clearly mxd is in same connected component of Cb as CJ \I, thus mxd ∈CJ and [CJ \I]=Cb′. It is immediate that gcd[CJ \I]=1. The remaining condition follows as in [20, Lemma 2.4]. (cid:3) The next lemma helps in computing gcd(C \{m}). J Lemma 4.11. Let CJ be a subset of Cb, a∈J and m=xmax (J)−a. Then gcd(C \{m})=xmax (J)−max (J\{a}) . J Proof. AnarbitraryelementofC \{m}isoftheformxmax (J)−bwhereb∈J\{a}. J We note that b≤max (J \{a}). It follows that xmax (J)−b =xmax (J)−max (J\{a})x(max (J\{a})−b) . For each 1 ≤ i ≤ n there exist a b ∈ J \{a} such that b = (max (J \{a}) . i i Therefore gcd(C \{m})=xmax (J)−max (J\{a}). (cid:3) J The set of basic fiber components forms a poset by setting C ≤ C if and J′ J only if there exists a monomial xr such that xrC ⊂ C . We state and prove the J′ J analogue of [20, Theorem 2.5]. Theorem 4.12. The poset of basic fiber components is isomorphic to the generalized Scarf complex ∆ /L and is finite. L e Proof. Let F be an element in ∆ /L. Choose a representative J of F. Let b = L max (J)+L. For any other repreesentativeJ′ of F, CJ =CJ′ ⊂Cb. We will show that CJ is a basic component of Cb. Let m ∈ C . We will show that gcd(C \ {m}) 6= 1. Suppose that m = J J xmax (J)−a, for a∈J. Since J ∈∆ , we have that max (J\{a}) 2. Since gcd(C \ {m}) 6= 1, it follows that J J C \{m} is a face of ∆ (b) and thus C is connected. To show that C is a J gcd J J connectedcomponentofCb weconsideranelementxu ∈Cb\CJ. Thenu+L=b and therefore u=max (J)−a where a∈L and a∈/ J. Since a≤max (J) and supp(xmax (J)−a) v-supp (J)=∅ \ J we have that gcd(xu,m) = 1, ∀ m ∈ C . It follows that C is a connected com- J J ponent of ∆ (b). Set C = C . It is immediate that ψ : F 7→ C is order gcd F J F preserving. Wewillshowthatψisbijective. TheinjectivityfollowsimmediatelyfromLemma 4.6. To show that ψ is surjective we let CJ be a basic component of Cb. We need SYZYGIES 9 to show that J ∈ ∆ . Let J′ ( J. For I = {xmax (J)−a : a ∈/ J′} we have L that gcd(C \ I) =egcd({xmax (J)−a : a ∈ J′}) 6= 1. Therefore there is an i J such that max (J) > a for all a ∈ J′. It follows that a < max (J), ∀ a ∈ J′ i i and max (J′) < max (J). Suppose now that a ∈ L is such that a ∈/ J and a≤ max (J). It follows that max (J ∪{a})= max (J) and CJ ( CJ∪{a} ⊂ Cb. If |J|=2 then Lemma 4.8 gives a contradiction, so in this case max (J) is unique and J ∈∆L. Suppose now that |J|>2. Since m=xmax (J)−a ∈Cb\CJ and CJ is a conneected component of ∆gcd(b), it follows that no variable in the support of m is in the support of any monomial in C and thus J supp(xmax (J)−a) v-supp (J)=∅ \ J as required. (cid:3) 5. The generalized algebraic Scarf complex We generalize the notion of the algebraic Scarf complex introduced in [20]. Definition 5.1. The generalized algebraic Scarf complex is the complex of free R-modules (G ,θ) := R·E L M C e C∈∆L/L where E denotes a basis vector in homological degree |C|−1 and the sum runs C over all basic fiber components C, identified as elements of ∆ /L. We let L e θ(E )= sign(m,C)gcd(C\{m})E , C X [C\{m}] m∈C where sign(m,C) is (−1)l+1 if m is in the lth position in the lexicographic ordering of C. Our firstremarkis that (G ,θ) is a subcomplex of the Taylorresolution,see [4, L Proposition3.10]. Indeed the canonicalbasis of the Taylorcomplex in homological degree i consists of vectors E where C is a subset of a fiber such that gcd(C)=1 C and|C|=i+1. Thedifferentialof(G ,θ)istherestrictionofthedifferentialofthe L Taylor complex on the elements of (G ,θ). Moreover we note that the algebraic L Scarfcomplex of [20] is a subcomplex of (G ,θ). Indeed the canonicalbasis of the L Scarf complex in homological degree i consists of vectors E where C is a basic C fiber and the differential coincides for these elements. We also note that for i ≤ 1 the algebraic Scarf complex is identical to the generalizedalgebraic Scarf complex. Theorem 5.2. The complex G is an indispensable complex for R/I . L L Proof. Let (F ,φ) be a simple A-graded minimal resolution of R/I with respect L L toanA-homogeneousbasis{H , t=1,...,b }ofF . Wewilluseinductiononthe tj j j homological degree i, the case i = 0 being trivial. For i = 1, let C = {m ,m } be 1 2 a basic fiber. Then θ (E )=m −m is an indispensable binomial of I meaning 1 C 2 1 L thatup to a constantmultiple, θ (E ) is partofanyminimal systemofgenerators 1 C of the I , see [10], [20]. Note that binomials in a lattice ideal are always simple. L Thus for i≤1, θ (E ) are indispensable. i C Suppose now that G is indispensable for homological degrees less than i and L thus φj|Gj = θj for j < i. Let C be a basic component of Cb with cardinality i+1. It is clear that θ (E ) ∈ kerφ . We will show that θ (E ) is a simple i C i−1 i C 10 H.CHARALAMBOUSANDA.THOMA syzygy. The proof is essentially the same as in Example 3.7. Indeed suppose that 0 6= h ∈ kerφ and h < θ (E ). Since S(h) ⊂ S(θ (E )) it follows that h ∈ i−1 i C i C G and φ (h) = θ (h). Moreover since S(h) ( S(θ (E )) and S(θ (E )) = i−1 i−1 i−1 i C i C { gcd(C\{m})E : m∈C}, it follows that for some monomial m ∈C, [C\{m}] 1 gcd(C\{m })E ∈/ S(h). 1 [C\{m1}] On the other hand since h6=0 we can find m ∈C such that 2 gcd(C\{m })E ∈S(h). 2 [C\{m2}] It follows that gcd(C \{m ,m })E ∈ S(φ (h)), a contradiction since 1 2 [C\{m1,m2}] i−1 φ (h)=0. i−1 Next suppose that θ (E ) is an R-linear combination of i-syzygies of strictly i C smaller A-degree. It follows that F in homological degree i has a basis generator L h of A-degree b1, where b1 i+1, a contradiction. WestillneedtoshowthatwecanidentifyE withaconstantmultipleofabasis C element H for some t. Suppose that θ (E )= c φ (H ) where deg (H )=b ti i C t i ti A ti P for at least one t. Then for some t with deg (H )=b, we have that A ti gcd(C\{m})E ∈S(θ (E ))∩S(φ (H )) , [C\{m}] i C i ti for a monomial m∈C. Therefore m-supp(θ (E ))∩m-supp(φ (H ))6=∅. i C i ti Since m-supp(θ (E )) = C, m-supp(H ) = m-supp(φ (H )) and by Lemma 3.3 i C ti i ti m-supp(φ (H )) is connected, it follows that m-supp(H )⊂C. Suppose that i ti ti φ (H )= p H i ti X s s,i−1 s where Hs,i−1 are basis generators of Fi−1 of A-degree bs and ps ∈ R is A-homogeneous of A-degree b−bs. It is clear that bs < b and that Cb < C. Moreover s H˜i−1(∆bs)6=0andthereforeCbs =[C\I]forsomesubsetI ofC. Wenotethatthe set I need not be unique. By Lemma 4.10 it follows that Cb is a basic fiber. By s inductionE isindispensableandistheuniquebasiselementofF ofA-degree Cbs i−1 bs, up to a constant multiple. By Lemma 4.9 |Cb | ≥ i. Since Cb = [C \I], it s s followsthat|Cbs|≤i. Therefore|Cbs|=iandCbs =[C\{ml}]foramonomialml. Let cxγ be a monomial term of p where c∈k−{0}. Since m-supp(φ (H ))⊂C, s i ti gcd(C) = 1 and gcd(Cbs) = 1 it follows that xγ = gcd(C \{ml}), so actually p = cxγ for a c 6= 0. Thus φ (H ) ≤ θ (E ) and since both are simple we s i ti i C get that S(φ (H )) = S(θ (E )). We apply Theorem 3.8 to obtain the desired i ti i C conclusion. (cid:3) Next we consider a complex that sits between the algebraic Scarf complex and the generalized algebraic Scarf complex.

On the generalized Scarf complex of lattice ideals

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