Commutative algebra of generalised Frobenius numbers

ALGEBRAIC COMBINATORICS

Madhusudan Manjunath & Ben Smith

Commutative algebra of generalised Frobenius numbers Volume 2, issue 1 (2019), p. 149-171.

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https://doi.org/10.5802/alco.31

Commutative algebra of generalised Frobenius numbers

Madhusudan Manjunath & Ben Smith

Abstract We study commutative algebra arising from generalised Frobenius numbers. Thek- th (generalised) Frobenius number of relatively prime natural numbers (a1, . . . , an) is the largest natural number that cannot be written as a non-negative integral combination of (a1, . . . , an) in kdistinct ways. Suppose thatLis the lattice of integer points of (a1, . . . , an)^⊥. Taking our cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modulesM_L^(k) whose Castelnuovo–Mumford regularity captures the k-th Frobenius number of (a1, . . . , an). We study the sequence {M_L^(k)}^∞_k=1 of generalised lattice modules providing an explicit characterisation of their minimal generators. We show that there are only finitely many isomorphism classes of generalised lattice modules. As a consequence of our commuta- tive algebraic approach, we show that the sequence of generalised Frobenius numbers forms a finite difference progression i.e. a sequence whose set of successive differences is finite. We also construct an algorithm to compute thek-th Frobenius number.

1. Introduction

The Frobenius number F(a₁, . . . , a_n) of a collection (a₁, . . . , a_n) of natural numbers such that gcd(a₁, . . . , a_n) = 1 is the largest natural number that cannot be expressed as a non-negative integral linear combination of a₁, . . . , a_n. Note that the condition gcd(a₁, . . . , a_n) = 1 ensures that a sufficiently large integer can be written as an non-negative integral combination ofa₁, . . . , an. Note that throughout the paper, we use the convention that N= {1,2,3, . . .}. The Frobenius number has been studied extensively from several viewpoints including discrete geometry [13], analytic number theory [7] and commutative algebra [17].

The Frobenius number can be rephrased in the language of lattices as follows [18].

We start by lettingL(a1, . . . , an) be a sublattice of the dual lattice (Zⁿ)^?of points that evaluate to zero at (a1, . . . , an)∈Zⁿ. The Frobenius number is precisely the largest integer rsuch that there exists a pointp∈(Zⁿ)^? that evaluates tor at (a1, . . . , an) andpdoes not dominate any point inL(a1, . . . , an). Here the domination is according to the partial order induced by the standard basis on (Zⁿ)^?.

This leads to a commutative algebraic interpretation of the Frobenius number that we now recall. LetKbe an arbitrary field and letS=K[x₁, . . . , x_n] be the polynomial ring innvariables with coefficients in K. Let I_L(a1,...,an) or simplyI_L be the lattice

Manuscript received 3rd July 2018, accepted 4th July 2018.

Keywords. Frobenius number, syzygy, lattice, poset, Castelnuovo–Mumford regularity.

Acknowledgements. Part of this work was carried out while MM was at Queen Mary University of London, the Mathematisches Forschungsinstitut Oberwolfach and IHES, Bures-sur-Yvette. He was funded by the EPSRC at QMUL and by a Leibniz Fellowship at the MFO.

ideal associated toL. Recall that for a sublattice LofZⁿ, the lattice idealI_L is the ideal generated by all binomialsx^u−x^v such that u−v∈L andu,v∈Zⁿ_>0. Note that L(a1, . . . , an) is, by construction, a sublattice of (Zⁿ)^?. We use the standard isomorphism between (Zⁿ)^? and Zⁿ to regard it as a sublattice of Zⁿ and associate a lattice ideal to it. Observe that the Zⁿ-grading on S/IL also yields a Z-grading, corresponding to the evaluation σ→σ((a1, . . . , an)) forσ∈(Zⁿ)^∗. We refer to this grading as the (a1, . . . , an)-weighted grading or simply the weighted grading.

Theorem1.1 ([17]).The Frobenius numberF(a1, . . . , an)is the given by the formula:

reg(S/IL) +n−1−

n

X

i=1

ai

where reg(S/IL) is the Castelnuovo–Mumford regularity of S/IL with respect to its (a1, . . . , an)-weighted grading. In other words, the Frobenius number is the maximum weighted degree of the highest Betti number ofILas anS-module subtracted byP

iai. Remark 1.2.For a moduleM with weighted grading, the invariant reg(S/IL) +n− 1−Pn

i=1a_i is also called thea-invariant of the module, [17].

Example1.3.Consider the latticeL= (3,5,8)^⊥∩Z³. We calculate its corresponding lattice ideal IL =

x3−x1x2, x³₂−x⁵₁

. The Betti table corresponding to the mini- mal free resolution of S/IL has 22 rows and 3 columns, hence reg(S/IL) = 21 and F(3,5,8) = 21 + 2−16 = 7.

Throughout this paper by the Castelnuovo–Mumford regularity of a graded module M, we mean the maximum row index in its graded Betti table minus one. Ifci,j is the twist corresponding to the Betti numberβi,j, the regularity is given by maxi,j{ci,j−i}.

We refer to Eisenbud [11, Chapter 4] for more information on this topic.

Theorem 1.1 motivates studying “explicit” free resolutions ofI_L as anS-module.

By an explicit free resolution, we mean a cell complex onL whose relabeling gives a free resolution. For instance, the hull complex [6] gives an explicit (non-minimal, in general) free resolution. We refer to the first section of Miller and Sturmfels [15] for more information.

1.1. Generalised Frobenius Numbers. Recently, the following generalisation of the Frobenius number called the k-th Frobenius number has been proposed [8].

For a natural number k, the k-th Frobenius number Fk(a1, . . . , an) of a collection (a1, . . . , an) of natural numbers such that gcd(a1, . . . , an) = 1 is the largest natural number that cannot be written as k distinct non-negative integral linear combina- tions ofa1, . . . , an. Hence, the first Frobenius numberF1(a1, . . . , an) is the Frobenius number of (a1, . . . , an). The finiteness of Fk(a1, . . . , an) for all natural numbers k follows by an argument similar to the one for F1(a1, . . . , an). In the language of lattices, the k-th Frobenius number is the largest integer r such that there exists a point p ∈ (Zⁿ)^? that evaluates to r at (a₁, . . . , a_n) and p does not dominate k distinct points inL(a₁, . . . , a_n). As in the casek= 1, the domination is according to the partial order induced by the standard basis on (Zⁿ)^?.

This interpretation allows a generalisation to any finite index sublattice H of L(a₁, . . . , a_n). The k-th Frobenius number of H is the largest integer r such that there exists a point p ∈ (Zⁿ)^? that evaluates to r at (a₁, . . . , an) and p does not dominatekdistinct points inH. The finite index assumption is necessary for thek-th Frobenius number to be finite. All our results hold in this level of generality.

Our goal in this paper is to develop commutative algebra arising from the k-th Frobenius number. A guiding problem for us is the classification of sequences of gen- eralised Frobenius numbers:

Problem 1.4 (Classification of Frobenius Number Sequences).Given a sequence of natural numbers {c_n}^∞_n=1, does there exist a vector (a₁, . . . , a_n) ∈ Nⁿ and a finite index sublatticeH of L(a1, . . . , an)whose sequence of generalised Frobenius numbers is equal to {cn}^∞_n=1?

To the best of our knowledge, this problem is wide open. For instance, previous to this it was not known whether a geometric progression with common ratio strictly greater than one can occur as a sequence of Frobenius numbers. As a corollary to our results, we show that the answer to this question is “no”.

We start by recalling another commutative algebraic interpretation of the Frobenius numberF₁(a₁, . . . , an) following Bayer and Sturmfels [6], [15]. The key concepts here are the group algebraS[L] and the lattice moduleML associated toL.

The group algebraS[L] is the K-algebra generated by Laurent monomialsx^u·z^v such thatu= (u1, . . . , un)∈Zⁿ_>0,v= (v1, . . . , vn)∈Lwherex^u andz^v are Laurent monomialsx^u₁¹. . . x^u_nⁿandz₁^v1. . . z^v_nⁿ. The lattice moduleMLis theS-module gener- ated by Laurent monomialsx^w over allw∈L. Note that as anS-moduleML is not finitely generated. However we can realiseML as a cyclicS[L]-module as follows:

ML∼=S[L]/hx^u−x^v·z^u−v|u,v∈Zⁿ_>0,u−v∈Li

with theS[L]-action given byx^uz^v·x^w=x^w+u+vwherex^uz^v∈S[L] andx^w∈M_L. To see this isomorphism, consider the morphism fromS[L] toML that takesx^u·z^v tox^u+v and use the first isomorphism theorem.

The group algebra S[L] is naturally Zⁿ-graded where the graded piece indexed byw ∈Zⁿ is the K-vector space spanned by {x^u·z^v} such thatu+v=w,v∈L.

The lattice module ML is also naturally Zⁿ-graded, as it is generated by Laurent monomials. For the latticeL(a1, . . . , an)⊂(Zⁿ)^?, we again regard it as a sublattice of Zⁿ via the standard isomorphism between (Zⁿ)^? and Zⁿ to associate the group algebra and the lattice module to it. Observe thatML also carries the (a1, . . . , an)- weighted grading, as does any lattice module whose corresponding lattice is a finite index sublattice ofL(a₁, . . . , a_n).

Note that S is a S[L]-module via the isomorphism S ∼= S[L]/hz^v−1_K|v ∈ Li.

Bayer and Sturmfels show that there is a categorical equivalence betweenZⁿ-graded S[L]-modules to Zⁿ/L-graded S-modules. The functor π realising this equivalence tensors an S[L]-module with S (here S is seen as an S[L]-module). The functor π takes M_L to S/I_L. Hence, a Zⁿ/L-minimal free resolution of S/I_L as an S-module can be obtained by applying the functorπto aZⁿ-graded minimal free resolution of ML as anS[L]-module, we refer to Miller and Sturmfels [15] for an example. Hence, we have the following interpretation of the Frobenius number in terms of the lattice moduleML.

Theorem 1.5 ([6], [17]).The Frobenius numberF(a₁, . . . , a_n)is reg(ML) +n−1−

n

X

i=1

ai

where reg(ML) is the Castelnuovo–Mumford regularity of ML with respect to its (a1, . . . , an)-weighted grading.

The moduleML behaves similar to a monomial ideal. This categorical equivalence can be used to transfer homological constructions fromMLtoS/IL. By applying the functor πto ML and noting that the Zⁿ/L-grading coincides with the (a1, . . . , an)- weighted grading on ML, we obtain Theorem 1.1. We start by generalising Theo- rem 1.5 tok-th Frobenius numbers. We generalise the lattice moduleML to thek-th lattice moduleM_L^(k)as follows.

x1

x2

x3

Figure 1. The staircase diagram corresponding to the 3rd lattice module M_L(3,4,11)⁽³⁾ . The polynomial ring S = K[x1, x2, x3] has the (3,4,11)-weighted grading. The blue dots correspond to minimal gen- erators of degree 15, red dots correspond to minimal generators of degree 20.

Definition 1.6.The k-th lattice moduleM_L^(k) is theS-module generated by Laurent monomials x^w such thatw dominates at leastk lattice points.

Figure 1 shows the staircase diagram corresponding to the 3rd lattice module of the lattice L(3,4,11). By construction, the first lattice module M_L⁽¹⁾ is the lattice moduleML. The moduleM_L^(k)is also not finitely generated as anS-module, however it is a finitely generatedS[L]-module (see Proposition 2.2 for more details) with the S[L]-action given by x^uz^v·x^w =x^w+u+v where x^uz^v ∈S[L] and x^w∈M_L^(k). The generalised lattice module M_L^(k) also carries a Zⁿ-grading since it is generated by Laurent monomials. However, in general the moduleM_L^(k)is not a cyclic module for natural numbersk >1. We have the following commutative algebraic characterisation of generalised Frobenius numbers in terms of the generalised lattice modules.

Proposition1.7.Thek-th Frobenius number of(a₁, . . . , a_n)is given by the formula:

reg(M_L^(k)) +n−1−

n

X

i=1

ai

where L := L(a1, . . . , an) and reg(M_L^(k)) is the Castelnuovo–Mumford regularity of the S[L]-moduleM_L^(k) with respect to its(a1, . . . , an)-weighted grading.

Proposition 1.7 follows from two observations. Let ci,j be the twist of the free module corresponding to the graded Betti number βi,j of π(M_L^(k)). A computation similar to the proof of [17, Theorem 3.1], comparing expressions for the Hilbert series ofπ(M_L^(k)) gives the following expression for thek-th Frobenius number:

Fk = max

i,j ci,j−X

i

ai

The second observation is that π(M_L^(k)) is a Cohen–Macaulay module with both Krull dimension and depth equal to one. This implies that the regularity and maxi,jci,j

are attained at the highest homological degree, in this case n−1. Proposition 1.7 follows as an immediate consequence.

While Proposition 1.7 provides a simple description of the generalised Frobenius number in terms ofM_L^(k), it raises a number of questions. For instance, given a natural numberk, how can we use Proposition 1.7 to determine thek-th Frobenius number.

This requires a more explicit knowledge ofM_L^(k). The generalised lattice modules are naturally related by the filtration:

M_L⁽¹⁾ ⊇M_L⁽²⁾ ⊇M_L⁽³⁾. . .

How does this filtrartion control their Castelnuovo–Mumford regularity? The con- nection between the generalised Frobenius numbers can be better understood by studying the connection between the generalised lattice modules. With this in mind, we delve into a detailed study of the generalised lattice modules: their minimal gen- erating sets, their Hilbert series and their syzygies. We now summarise our results.

We associate a graph GL onL as follows. Fix a binomial minimal generating set of IL. The graph GL is defined as follows: there is an edge between points w1 and w2inLif there exists a binomial minimal generatorx^u−x^v such that the difference of its exponents is equal tow1−w2 i.e.u−v=w1−w2. By construction,GL has anL-action on its edges since if (w₁,w₂) is an edge then (w₁+y,w₂+y) for any y∈L.

Let d_GL be the metric on L induced by the graph G_L. For a point w ∈ L, let N^(k)(w) be the set of all points in L in the ball of radius k centered at w in the metricdG_L.

Theorem 1.8 (Neighbourhood Theorem).For any non-negative integerk, any min- imal generator ofM_L^(k+1) as anS[L]-module is the least common multiple of Laurent monomials corresponding to(k+ 1)lattice points each of which is a point inN^(k)(0) where0= (0, . . . ,0).

We prove Theorem 1.8 by an inductive characterisation of the generalised lattice modules. This characterisation of M_L^(k+1) is in terms of syzygies of M_L^(k) that we believe is of independent interest. We briefly describe this characterisation in the following. Fix a natural number k, a minimal generatorx^w ofM_L^(k)as an S-module is called exceptional if w dominates strictly larger than k points in L. We describe M_L^(k+1) in terms of the exceptional generators of M_L^(k) and the first syzygies of a

“modification” ofM_L^(k)that we now describe. LetM_L,mod^(k) be theS-module generated by every element of M_L^(k) and the element 1_K (the multiplicative identity of K).

Formally,

M_L,mod^(k) =h1_K, m|m∈M_L^(k)iS

Note that fork >1,M_L,mod^(k) is naturally anS-module but not anS[L]-module i.e.

M_L,mod^(k) does not inherit the naturalL-action.

By construction, we have the following characterisation of minimal generators of M_L,mod^(k) :

Proposition 1.9.The (Laurent) monomial minimal generating set of M_L,mod^(k) con- sists of precisely 1_K and every (Laurent) monomial minimal generator of M_L^(k) that is not divisible by1_K (in other words, whose exponent does not dominate the origin).

For each minimal generatorg1ofM_L,mod^(k) , let Syz¹_g1(M_L,mod^(k) ) be theK-vector space Syz¹_g1(M_L,mod^(k) ) generated by syzygies of the form:

m·(0, . . . ,0,lcm(g1, g2)/g1

| {z }

g₁

,0, . . . ,0,−lcm(g1, g2)/g2

| {z }

g₂

,0, . . . ,0)

where g2 6= g1 is a minimal generator of M_L,mod^(k) and m is a monomial in S. Note that multiplication bymis the standard multiplication onS. Consider the direct sum

⊕_gSyz¹_g(M_L,mod^(k) ) whereg varies over all minimal generators ofM_L,mod^(k) .

We define a map φ^(k)_S from ⊕gSyz¹_g(M_L,mod^(k) ) to M_L^(k+1). We first define the map φ^(k)_S from the canonical basis of each piece Syz¹_g(M_L,mod^(k) ) toM_L^(k+1)as follows:

φ^(k)_S ((s₁, s₂)) =x^{degZn((s1,s2))} where deg_Zn(·) is the multidegree.

We extend this mapK-linearly to defineφ^(k)_S . Note that the image ofφ^(k)_S is an ele- ment ofM_L^(k+1). This is because (s1, s2) is of the form (lcm(g1, g2)/g2,−lcm(g1, g2)/g1) for two distinct minimal generators of M_L,mod^(k) . By construction, φ^(k)_S ((s₁, s₂)) = lcm(g₁, g₂). By Proposition 2.6, we have the following two cases: either both g₁ and g₂ are minimal generators of M_L^(k) or one of them g₁= 1_K, say and g₂ is a minimal generator ofM_L^(k)that is not divisible by 1_K. In both cases, the support of lcm(g1, g2) contains at least (k+ 1) points in L (by support of Laurent monomial, we mean the set of points in L that its exponent dominates). It contains (potentially among others) the unions of the supports ofg₁andg₂. Hence, the image ofφ^(k)_S is inM_L^(k+1). We show the following converse to this.

Theorem 1.10.Up to the action of L, every minimal generator of M_L^(k+1) is either in the image ofφ^(k)_S or is an exceptional generator of M_L^(k).

Example 1.11.Consider the lattice (3,5,8)^⊥ ∩Z³ with corresponding lattice ideal IL=hx3−x1x2, x³₂−x⁵₁i. As 1_Kis a minimal generator ofM_L⁽¹⁾, the lattice module is not altered under the modification construction. The minimal first syzygies ofM_L⁽¹⁾, up to the action ofL, are of the form (lcm(x^u,1_K)/1_K,−lcm(x^u,1_K)/x^u) where u∈ N⁽¹⁾(0). The mapφ⁽¹⁾_S sends the minimal first syzygies to{x3, x1x2, x³₂, x⁵₁}, precisely the monomials in each minimal binomial of IL. This gives us an explicit description ofM_L⁽²⁾=hx3, x³₂ias anS[L]-module and a minimal generating set.

Theorem 1.10 characterises the minimal generators ofM_L^(k), a natural next ques- tion is about the syzygies of M_L^(k) as an S[L]-module. Is there a similar inductive characterisation of the syzygies ofM_L^(k)? What are the possible Betti tables ofM_L^(k)? Both these questions are wide open in general. As a first result in this direction, we show the following finiteness result. Recall that for a Zⁿ-graded module M (either anS-module or anS[L]-module) and for anyb∈Zⁿ, we have the twistM(b) of M defined by (Mb)c=Mb+c for every c∈Zⁿ.

Theorem1.12.LetLbe a finite index sublattice of(a₁, . . . , a_n)^⊥∩Zⁿ. For eachk∈N, letx^uk be any element ofM_L^(k)of the smallest(a₁, . . . , an)-weighted degree. There are finitely many classes among the generalised lattice modules {ML(k)(u_k)}k∈N up to isomorphism of bothZⁿ-gradedS[L]-modules andZⁿ-gradedS-modules. Hence, there are only finitely many distinct Betti tables for the generalised lattice modules ofL.

The key object in the proof of Theorem 1.12 is a poset that we refer to as the structure poset associated toL. The elements of the structure poset ofLare elements

in Zⁿ/L of (a₁, . . . , a_n)-weighted degree in the range [0, F₁] where F₁ is the first Frobenius number of L. Note that there are precisely ind(L)·(F₁+ 1) elements in the structure poset, where ind(L) is the index ofLin (a1, . . . , an)^⊥∩Zⁿ. The partial order in this poset is defined as follows: for elements [a],[b] in the structure poset we say that [a]>[b] if for every representativea∈Zⁿof [a] there exists a representative bof [b] such thata>bunder the standard partial order onZⁿ.

Letmk be the minimum weighted degree of any element ofM_L^(k). The key obser- vation in the proof of Theorem 1.12 is that M_L^(k) is completely determined (up to isomorphism ofZⁿ-gradedS[L]-modules) by “filling” the structure poset ofL. More precisely, to determineM_L^(k) we need to know the elements in Zⁿ/Lof weighted de- gree [mk, mk+F1] that dominate at leastkpoints inL. These elements determine a subposet of the structure poset ofL that we refer to as thestructure poset ofM_L^(k).

The structure poset ofM_L^(k) determines it up to isomorphism of Zⁿ-gradedS[L]- modules (andZⁿ-gradedS-modules). Since the structure poset ofLis finite it has only finitely many subposets. Hence, there are only finitely manyZⁿ-graded isomorphism classes of generalised lattice modules. A classification of the structure poset ofLand subposets of the structure poset ofLthat can occur as structure posets of generalised lattice modules is wide open. In Section 3, we provide a detailed example illustrating this phenomenon. As a corollary to Theorem 1.12 we obtain the following:

Corollary1.13.There exists a finite set of integers{b1, . . . , bt} ⊂Z>0∪ {−1}such that for everyk there exists a natural numberj such that the k-th Frobenius number can be written as:

F_k=m_k+b_j

where m_k is the minimum (a₁, . . . , a_n)-weighted degree of any element in M_L^(k). This finite set {b₁, . . . , b_t} is precisely the set of integers that can be realised as reg(M_L^(k)(uk)) +n−1−Pn

i=1a_i.

For k = 1, note thatmk = 0 andbj = reg(M_L⁽¹⁾) +n−1−Pai. Suppose that Lis a finite sublattice of (a1, a2)^⊥∩Z² where (a1, a2) are relatively prime numbers.

The set {b1, . . . , b_t} consists of one element and the generalised lattice moduleM_L^(k) is generated by one element. WhenL= (a1, a2)^⊥∩Z², thek-th lattice moduleM_L^(k) is generated byx^a₁²x^(k−2)a₂ ¹. Hence,m_k= (k−1)a₁a₂and the set{b1, . . . , b_t}consists of only one element F₁ = a₁a₂−a₁−a₂. The k-th Frobenius number F_k will be m_k +F₁ =ka₁a₂−a₁−a₂, exactly as in [8]. Obtaining a formula along the lines of Corollary 1.13 was suggested as an open problem in [8]. Finally, note that if the structure poset ofM_L^(k) is equal to the structure poset ofLthenFk=mk−1 and in this casebj =−1.

As an application of Theorem 1.12, we show that the sequence of Frobenius num- bers (Fk)^∞_k=1 is a finite difference progression. A sequence (ck)^∞_k=1 is called a finite difference progression if there exists a finite set of differences such that for every k∈Nthe differenceck+1−ck is contained in this set. This provides a partial answer to Problem 1.4.

Theorem 1.14.For any finite index sublatticeL of (a1, . . . , an)^⊥∩Zⁿ, the sequence of Frobenius numbers(Fk)^∞_k=1 is a finite difference progression.

We prove Theorem 1.14 using Corollary 1.13 along with the fact that the sequence (mk)^∞_k=1 is a finite difference progression. As a corollary, a geometric progression with common ratio strictly greater than one cannot occur as a sequence of Frobenius numbers.

As an another application of our results, we use the neighbourhood theorem (The- orem 1.8) to construct an algorithm that takes the lattice in terms of a basis and a natural numberkas input, computes thek-th lattice module and thek-th Frobenius number.

1.2. Related Work. There is a vast literature on the Frobenius number, we refer to Alfonsín’s book [1] for more information. Work on the generalised Frobenius numbers has so far primarily used analytic methods and methods from polyhedral geometry.

The work of Beck and Robins [8] uses analytic methods to derive an explicit formula for the coefficients of the Hilbert series of K[x, y] with the (a1, a2)-weighted grading.

Aliev, Fukshanksy and Henk [3] give bounds generalising a theorem of Kannan for the first Frobenius number. They relate thek-th Frobenius number to thek-covering radius of a simplex with respect to the lattice (a1, . . . , an)^⊥∩Zⁿ, giving bounds on the generalised Frobenius number as a corollary.

A recent work of Aliev, De Loera and Louveaux [2] considered the semigroup Sg_>k((a1, . . . , an)) ={b| ∃x1, . . . ,xk∈Zⁿ_>0such that (a1, . . . , an)·xi=b}

where x₁, . . . ,x_k are distinct. In this framework, the k-th Frobenius number is the largest non-negative integer b /∈ Sg_>k((a₁, . . . , a_n)). They study this semigroup by considering a monomial ideal I^(k)((a1, . . . , an)) such that the set of (a1, . . . , an)- weighted degrees of its elements is equal to Sg_>k((a1, . . . , an)) [2, Theorem 1]. They use the Gordan–Dickson Lemma to deduce the finite generation ofI^(k)((a1, . . . , an)) and hence, Sg_>k((a1, . . . , an)). In fact, the work [2] studies a more general version where (a₁, . . . , a_n) is replaced by anyd×nmatrix with integer entries.

The monomial ideal I^(k)((a1, . . . , an)) is, in fact, the intersection of M_L^(k) with the polynomial ring S (both are submodules of S[L]). We note that this ideal I^(k)((a₁, . . . , a_n)) does not carry anL-action and this seems to make it less amenable to study compared toM_L^(k).

2. Generalised Lattice Modules

In this section, we discuss generalised lattice modules in detail including the neigh- bourhood theorem (Theorem 1.8) and the inductive characterisation ofM_L^(k) (Theo- rem 1.10) in the introduction. We start by recalling the definition of generalised lattice modules. Fix a non-zero vector (a₁, . . . , a_n)∈Nⁿ. Let Lbe a finite index sublattice of the lattice of integer points in (a₁, . . . , a_n)^⊥∩Zⁿ and S = K[x₁, . . . , x_n] be the polynomial ring innvariables.

Definition2.1.The k-th lattice moduleM_L^(k) is theS-module generated by Laurent monomials x^w where w is an element in Zⁿ that dominates at least k points in L.

Formally,

M_L^(k)=hx^w|w∈Zⁿ dominates at leastk points inLiS.

By construction,M_L^(k) is a Zⁿ-gradedS-module and is Z-graded by (a₁, . . . , a_n)- weighted degree. On the other hand,M_L^(k)is not a finitely generatedS-module. How- ever M_L^(k) carries an L-action via the map v·x^w = x^w+v for every v ∈ L and x^w∈M_L^(k). This action makesM_L^(k) into aL-module and furthermore, into anS[L]- module whereS[L] is the group algebra ofL.

Recall that the group algebraS[L] is defined as theK-algebra generated by symbols x^uz^v where u ∈ Zⁿ_>0 and v ∈ L with multiplication given by x^u1z^v1 ·x^u2z^v2 = x^u1+u2z^v1+v2. The action byS[L] on thek-th lattice moduleM_L^(k)is given byx^uz^v·

x^w =x^u+v+w where x^uz^v ∈S[L] and x^w ∈M_L^(k). We refer to [6], [15] for a more detailed discussion on this topic. In the following, we show that M_L^(k) is a finitely generatedS[L]-module.

Proposition 2.2.For any natural number k, thek-th lattice moduleM_L^(k) is a finite generatedS[L]-module.

Proof. By the action of S[L] on M_L^(k), it suffices to consider orbit representatives of theL-action onM_L^(k) that dominate the origin. These representatives are monomials inS(rather than Laurent monomials) and define a monomial ideal in the polynomial ringS. By the Gordan–Dickson Lemma, this monomial ideal is finitely generated and henceM_L^(k)is finitely generated as an S[L]-module.

The proof of Proposition 2.2 is based on an argument in [2], it is however not constructive in the sense that it does not give bounds on the degrees of the minimal generators ofM_L^(k). The methods in Section 3 give a constructive proof that shows that the (a1, . . . , an)-weighted degree of any minimal generator of M_L^(k) is in the interval [m_k, m_k +F₁] where m_k is the minimum (a₁, . . . , a_n)-weighted degree of a Laurent monomialx^wsuch that wdominates at leastklattice points.

Example 2.3.The case k = 1 is precisely the notion of lattice module studied by Bayer and Sturmfels [6]. The lattice module M_L⁽¹⁾ as an S-module is generated by Laurent monomialsx^wwherew∈L. As anS[L]-module,M_L⁽¹⁾ is cyclic and is gener- ated by the element 1_K. Furthermore, [6] show thatM_L⁽¹⁾∼=S[L]/hx^u−x^vz^u−v|u,v∈ Zⁿ_>0,u−v∈Li.

For k >2, the lattice modules M_L^(k) are in general not cyclic S[L]-modules. For k= 2, we give a simple description of a minimal generating set of M_L⁽²⁾ in terms of the first syzygies of M_L⁽¹⁾ (Theorem 2.5). For k > 3, a generalisation of this result is more involved and is the content of Theorem 2.7. One source of complication is that fork >2, the lattice modulesM_L^(k) have exceptional generators i.e. those that dominate strictly greater thankpoints inL, whereasM_L⁽¹⁾ does not have exceptional generators. Another complication is for k > 3, we may get minimal generators of M_L^(k)that do not arise as a syzygy between two minimal generators ofM_L^(k−1), rather as a “syzygy between a minimal generator and a lattice point”. This motivates us to consider the syzygies ofM_L,mod^(k) .

2.1. Inductive Characterisation of M_L⁽²⁾. We discuss the simplest generalised lattice moduleM_L⁽²⁾. We start with a description of the minimal generators ofM_L⁽²⁾. Recall from the introduction that the key to this description is the morphism φ^(k)_S between ⊕gSyz¹_g(M_L,mod^(k) ) and M_L^(k+1). We now describe the specialisation of this map fork= 1. We first note that M_L⁽¹⁾ =M_L,mod⁽¹⁾ and that each piece Syz¹_xu(M_L^(k)) has a basis of the form:

m·(0, . . . ,0,lcm(x^u,x^v)/x^u

| {z }

x^u

, . . . ,−lcm(x^u,x^v)/x^v

| {z }

x^v

,0, . . . ,0),

where x^v 6= x^u is a Laurent monomial minimal generator inM_L⁽¹⁾ as an S-module, lcm(·,·) is the least common multiple andm is a monomial in S. Note that multi- plication bymis the standard multiplication onS.

We define a mapφ⁽¹⁾_S on this basis of Syz¹_xu(M_L⁽¹⁾) and extend itK-linearly. The map φ⁽¹⁾_S : Syz¹_xu(M_L⁽¹⁾)→M_L⁽²⁾takes the elements= (lcm(x^u,x^v)/x^u,−lcm(x^u,x^v)/x^v) tox^degZn(s)where deg_Zn(s) is theZⁿ-graded degree ofs. In fact, deg_Zn(s) = max(u,v) where max is the coordinate-wise maximum. Furthermore, x^degZn(s) ∈ M_L⁽²⁾ since the point max(u,v) dominates at least two lattice points, namely u and v. In the following, we note that the mapφ⁽¹⁾_S is surjective.

Proposition 2.4.The map φ⁽¹⁾_S is surjective.

Proof. It suffices to prove that every Laurent monomial inM_L⁽²⁾can be realised as the image of an element in Syz¹_xu(M_L⁽¹⁾) for some minimal generator x^u ofM_L⁽¹⁾. To see this, consider a Laurent monomialx^w inM_L⁽²⁾. By the definition ofM_L⁽²⁾, the point w dominates at least two points inL. Consider any two pointsu₁ andu₂in Lthat wdominates and consider the Laurent monomial lcm (x^u1,x^u2). This is contained in M_L⁽²⁾ and is the image of (lcm(x^u1,x^u2)/x^u1,−lcm(x^u1,x^u2)/x^u2)∈Syz¹_xu1(M_L⁽¹⁾) underφ⁽¹⁾_S . Hence, by multiplying this syzygy by the monomialx^w/lcm(x^u1,x^u2) we

conclude thatx^wis also in the image of φ⁽¹⁾_S .

Proposition 2.4 is not directly amenable for computational purposes sinceM_L⁽²⁾ is not finitely generated as anS-module. However,M_L⁽²⁾is finitely generated as anS[L]- module. Note that there is a naturalL-action onL

gSyz¹_g(M_L⁽¹⁾) and a surjective map between the first syzygy module ofM_L⁽¹⁾as anS[L]-module and the piece Syz¹₀(M_L⁽¹⁾).

Composing this withφ⁽¹⁾_S gives a surjective mapφ⁽¹⁾_S[L]between the first syzygy module ofM_L⁽¹⁾ andM_L⁽²⁾ asS[L]-modules. To explicitly describe the mapφ⁽¹⁾_S[L], we first note that the first syzygy module ofM_L⁽¹⁾as anS[L]-module has aK-vector space basis of the form:

x^u−x^vz^u−v

whereu,v∈Zⁿ_>0andu−v∈L. The mapφ⁽¹⁾_S[L] takesx^u−x^vz^u−vtox^u∈M_L⁽²⁾. As the functor π takes M_L⁽¹⁾ to S/IL and Syz¹(S/IL) = IL (along with the categorical equivalence betweenZⁿ-graded S[L]-modules and Zⁿ/L-graded S-modules), this in- duces a map from any binomial minimal generating set ofI_L to M_L⁽²⁾, which we also refer to asφ⁽¹⁾_S[L]. We obtain the following.

Theorem 2.5.The lattice moduleM_L⁽²⁾ as anS[L]-module is generated by the image of φ⁽¹⁾_S[L] on a binomial minimal generating set of the lattice ideal IL.

As Example 1.11 shows, the map φ⁽¹⁾_S is not injective. In general, take a Koszul syzygy between two minimal generators in M_L⁽²⁾ and note that it does not lift to a syzygy between some corresponding minimal generators in Syz¹(M_L⁽¹⁾). This shows thatφ⁽¹⁾_S has a non-trivial kernel and hence is not injective.

2.2. Inductive Characterisation ofM_L^(k). We generalise Proposition 2.4 to ar- bitrary lattice modules to obtain an inductive characterisation ofM_L^(k). Let us briefly recall the relevant objects from the introduction.

The modificationM_L,mod^(k) of M_L^(k) is theS-module generated by every element of M_L^(k) and the element 1_K. Hence,

M_L,mod^(k) =h1_K, m|m∈M_L^(k)i_S

By the construction ofM_L,mod^(k) , we have the following characterisation of its mini- mal generators.

Proposition 2.6.The (Laurent) monomial minimal generating set of M_L,mod^(k) con- sists of precisely 1_K and every (Laurent) monomial minimal generator of M_L^(k) that is not divisible by1_K (in other words, whose exponent does not dominate the origin).

For a minimal generatorg ofM_L,mod^(k) , the mapφ^(k)_S from Syz¹_g(M_L,mod^(k) ) toM_L^(k+1) is defined on the canonical basis of Syz¹_g(M_L,mod^(k) ) as:

φ^(k)_S ((s1, s2)) =x^{degZn((s1,s2))} where deg_Zn(·) is the multidegree of the syzygy.

We extend the above mapK-linearly to defineφ^(k)_S . As noted in the introduction, the image ofφ^(k)_S is contained inM_L^(k+1). Theorem 2.7 is a converse to this.

Suppose that x^w ∈ M_L^(k+1) is a minimal generator and let U ={u1, . . . ,ur} be the set of points inLthat w dominates. For a subsetT ⊂L of sizek, let`T be the least common multiple of the Laurent monomials associated to points inT.

Theorem 2.7.Up to the action of L, any minimal generator x^w ofM_L^(k+1)is either in the image ofφ^(k)_S or is an exceptional generator ofM_L^(k). Furthermore, we have the following classification of minimal generators ofM_L^(k+1).

(1) If`T is the same for every subsetT ofU of sizekthen, x^wis an exceptional generator ofM_L^(k).

(2) If there exist subsets T1 andT2 of U of size k such that their least common multiples do not divide each other then, x^w is in image of φ^(k)_S on a syzygy between two minimal generators inM_L^(k).

(3) Otherwise, x^w is in image ofφ^(k)_S on a syzygy between a minimal generator inM_L^(k)and1_K.

Proof. By definition,wdominates at least (k+1) points inL. Consider the ^r_k

subsets ofU of sizek and note that ^r_k

>2. For each subsetT of sizek, let`<sub>T</sub> be the least common multiple of the set of points inT. If the least common multiple`T is the same for all subsetsT of sizek, then we claim thatx^wis an exceptional generator ofM_L^(k). To see this, note thatx^w∈M_L^(k) and any minimal generator` ofM<sub>L</sub><sup>(k)</sup> that divides x<sup>w</sup> dominates every point in some subset of U of size k and ` is the least common multiple of the Laurent monomials corresponding to points inU. However, this least common multiple isx^w. Hence,`=x^wand is an exceptional generator ofM_L^(k).

Otherwise, consider two subsets T₁ and T₂ of U of size k such that their least common multiples`<sub>T1</sub> and`_T2 respectively, are different. There are two cases:

Either`<sub>T1</sub> and`_T2 do not divide each other. Then both`<sub>T1</sub> and`_T2 are not equal to x^w but divide it. Their supports (the set of lattice points that their exponents dominate) are precisely T₁ and T₂ respectively (otherwise, this would contradictx^w being a minimal generator of M_L^(k+1)). Hence, `T<sub>1</sub> and `T₂ are minimal generators of M_L^(k) as any Laurent monomial that divides either `T1 or `T2 must have strictly smaller support. The mapφ^(k)_S takes their syzygy to a monomialmthat dividesx^w. Furthermore, since this monomialm is inM_L^(k+1) and x^w is a minimal generator of M_L^(k+1), we conclude that m= x^w. Finally, note that by Proposition 2.6 there is a lattice pointq∈Lsuch that`<sub>T1</sub>·x<sup>−q</sup>and`_T2·x^−qare minimal generators ofM_L,mod^(k) . Their syzygy maps to an element in the same orbit asx^w under the action ofL.