Explicit Deformation of Lattice Ideals via Chip Firing Games on Directed Graphs

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1 Explicit Deformation of Lattice Ideals via Chip Firing Games on Directed Graphs arxiv: v [math.co] 4 Jan 204 Spencer Backman and Madhusudan Manjunath For a finite index sublattice L of the root lattice A n, we construct a deterministic algorithm to deform the lattice ideal I L to a nearby generic lattice ideal, answering a question posed by Miller and Sturmfels. Our algorithm is based on recent results of Perkinson, Perlman and Wilmes concerning commutative algebraic aspects of chip firing on directed graphs. As an application of our deformation algorithm, we construct a cellular resolution of the lattice ideal I L by degenerating the Scarf complex of its deformation. Introduction Let k be an arbitrary field and let k[x 0,..., x n ] be the polynomial ring in (n + )- variables. Given a lattice L Z n+, the lattice ideal I L is a binomial ideal associated to L is defined as I L = x u x v u v L, u, v N n+. Lattice ideals generalize toric ideals and are among the most well-studied objects in combinatorial commutative algebra [7]. In particular, the problem of existence and construction of cellular minimal free resolutions for lattice ideals i.e., resolutions for lattice ideals that are supported on a cellular complex, has been a source of great interest in the recent past. The problem of constructing a cellular resolution for any lattice ideal was solved by the hull complex developed by Bayer and Sturmfels [3]. The hull complex is in general not a minimal free resolution and the problem of existence and construction of minimal cellular free resolutions of lattice ideals remains open. Sturmfels and Peeva [8] define a notion of generic lattice ideals and construct a cellular minimal free resolution called the Scarf complex for generic lattice ideals. A lattice ideal is called generic if it is generated by binomials x u x v such that the vector u-v has full support. Madhusudan Manjunath is very grateful to the Alexander von Humboldt-Foundation for their support through the Feoder-Lynen Fellowship during this work

2 The term generic is justified by a theorem of Barany and Scarf that the lattices corresponding to generic lattice ideals are dense in the space of all lattices [2]. Miller and Sturmfels in their book on Combinatorial Commutative Algebra [7, Page 89] remark that despite this abundance of generic lattice ideals, most lattice ideals one encounters in commutative algebra seem to be nongeneric and they ask for a deterministic algorithm to deform an arbitrary lattice ideal into a nearby generic lattice ideal. Our main result in this paper, presented in Section 3, is an algorithm (cf. Algorithm ) to deform a lattice ideal I L where L is a finite index sublattice of A n and is summarized by the following theorem: Theorem. Given a lattice ideal I L, where L is a finite index sublattice of A n and a real number δ > 0, the deterministic algorithm (c.f Algorithm ) outputs a generic lattice ideal I λ Lgen with the lattice L gen at distance at most δ from L, where the distance is measured by a metric in the space of sublattices of R n+ (c.f Definition 8) and the scaling λ L gen of L gen is a sublattice of Z n+. Our deformation algorithm is based on the recent results of Perkinson, Perlman and Wilmes on lattice ideals corresponding to the lattice generated by the rows of the Laplacian matrix of a directed graph. In particular, they showed that every full rank sublattice of Z n is generated by the rows of the reduced Laplacian of a directed graph. This allows us to reduce the problem of deforming lattice ideals to the problem of deforming Laplacian lattice ideals of directed graphs. Perkinson, Perlman and Wilmes in the same paper also described a certain distinguished Gröbner basis of the Laplacian lattice ideal of a directed graph whose equivalent chip-firing interpretation was independently discovered by the first author and Arash Asadi. The key observation about these Gröbner bases which we exploit in our deformation algorithm is that, in a precise sense, they respect perturbations of the lattice which preserve the left kernel of the Laplacian matrix. In Section 4, we use our deformation algorithm to construct a cellular free resolution, in fact a free resolution supported on a simplicial complex, for any lattice ideal I L where L is a finite index sublattice of A n. The hull complex of Bayer and Sturmfels [3] is also a (non-minimal) cellular free resolution of any lattice ideal I L. For any lattice ideals I L where L is a finite index sublattices of A n, the cellular resolution we construct is an alternative to the hull complex. More precisely, we show the following: Theorem 2. (cf. Theorem 3) For any finite index sublattice L of A n, the complex of free k[x 0,..., x n ]-modules obtained from the labelled simplicial complex Scarf def (I L ) is supported on a simplicial complex and is a free, in general nonminimal, resolution of the lattice ideal I L. 2

3 The complex Scarf def (I L ) is constructed by degenerating the Scarf complex of a deformation of I L. 2 Preliminaries 2. Chip Firing on Directed Graphs Let G be a directed graph with vertex set {v 0,..., v n } and adjacency matrix A whose entry A i,j for 0 i, j n is the number of edges directed from v i to v j. Let V ( G) and E( G) be the vertex set and edge set of G respectively. Let D = diag( deg(v 0 ),..., deg(vn )) where deg(v) denotes the number of edges leaving vertex v V ( G). We call the matrix Q = D A the Laplacian matrix of the directed graph G. We note that this definition of the Laplacian is the transpose of the Laplacian appearing in the work of Perkinson et al. [9]. We now describe the associated chip-firing game on the vertices of G coming from the rows of the Laplacian matrix. Let C Z n+, which we call a chip configuration whose i-th coordinate C i is the number of chips at vertex v i. We say that a vertex fires if it sends a chip along each of its outgoing edges to its neighbors. We say that a vertex v i is in debt if C i < 0. Note that the process is commutative in the sense that the order of firings does not effect the final configuration. For f Z n+, we may interpret the configuration C = C Q T f as the configuration obtained from C by a sequence of moves in which the vertex v i fires f i times, and we call f a firing. We restrict our attention to strongly connected directed graphs, directed graphs for which there is a directed path between every ordered pair of distinct vertices. As we show in the following lemma, for a strongly connected directed graph, the left kernel (and the right kernel) of the Laplacian matrix is a one dimensional vector space. The following lemma is an algebraic characterization of the strongly connected property. Lemma 3. A directed graph G is strongly connected if and only if there exists a row vector Σ T = (Σ 0,..., Σ n ) with strictly positive integer entries that spans the left kernel of Q. Proof. Let G be strongly connected. By construction, Q = 0 where = (,..., ), which just says that directed chip-firing moves preserve the total number of chips in the graph, therefore the Laplacian is not of full rank and has some nontrivial left kernel. Suppose there exists some firing strategy Σ 0 which has no effect on chip configurations, i.e., such that Q T Σ = 0. Let V + be the set of vertices of G such that Σ i > 0 for every vector v i V +. We may assume that 3

4 V + by taking the negative of Σ if necessary. Since the net amount of chips leaving V + is positive, there must exist some integer j such that v j V + and (Q T Σ) j < 0, a contradiction. Assume that there exist two linearly independent firing strategies f > 0 and f 2 > 0, such that Q T f = 0 and Q T f 2 = 0, then there exists a non-zero linear combination λ f + λ 2 f 2 0, which is in the kernel, a contradiction. Conversely, suppose G is not strongly connected, but that there exists some firing strategy Σ > 0 such that Q T Σ = 0. Let V,..., V t be the decomposition of vertices of G into maximal strongly connected components. If we construct a meta graph with vertices V,..., V t and an edge between (V i, V j ) if there exists v i V i and v j V j with (v i, v j ) E( G). This meta graph has at least two vertices G is not strongly connected. Furthermore, it is acyclic since otherwise we could find a larger strongly connected component. Hence, there exists some source vertex V i, i.e., some component with no edges (u, v) in G where u V i, 2 j t and v V i. The total number of chips leaving V i is positive, therefore there must exist some vertex v k V i such that (Q T Σ) k < 0, a contradiction. Recall that the reduced Laplacian matrix of a directed graph is the Laplacian matrix with the zero-th row and the zero-th column deleted Theorem 4. (Perkinson-Perlman-Wilmes [9]) Fix any integer m > 0, every full rank sublattice of Z m has a basis whose elements are the columns of a reduced Laplacian matrix of a directed graph. Furthermore, we can take this graph to be such that v 0 is globally reachable meaning that there is a directed path from v i to v 0 for each i. When L is a full dimensional sublattice of the root lattice A n, Theorem 4 can be reformulated as follows: there exists a strongly connected directed graph with Σ 0 = whose rows of the Laplacian form a basis for L. The following theorem gives a combinatorial description for a Gröbner basis for I L. Theorem 5. (Asadi -Backman [], Perkinson-Perlman-Wilmes [9]) Let G be a strongly connected directed graph with Laplacian Q, a primitive positive vector Σ = (Σ 0,..., Σ n ) in the left kernel and a configuration of chips C which is nonnegative away from v 0. If we perform firings f such that 0 f Σ with f 0 = 0 which does not cause any vertex to be sent into debt, this process will eventually terminate and the configuration obtained is independent of the firing choices made. These configurations of chips obtained by the process from the previous theorem are referred to in the literature as v 0 -reduced divisors, superstable configurations, or G-parking functions. 4

5 Corollary 6. [Perkinson-Perlman-Wilmes] The set G = {x u+ x u, u = Q T x, 0 x Σ} is a grevlex Gröbner basis for I L for any linear extension of a rooted spanning tree order with root v 0. In the undirected case Cori-Rosen-Salvy [5] showed that Theorem 5 translates into the statement that the binomials defined by the cuts in the graph, which correspond to firing moves, are a Gröbner basis for the Laplacian lattice ideal with respect to any linear extension of a spanning tree term order. This is due to the fact that a characterization of a Gröbner basis is a generating set such that division with respect to the given term order is unique. Perkinson et al. [9] observed that this result of Cori-Rosen-Salvy [5] extends to the case of directed graphs where Σ 0 = via Theorem 5. A slight difference in the approaches of [] and [9] is that the latter more often work with the reduced Laplacian of a graph where v 0 is globally reachable (a sandpile graph) while the former work with the full Laplacian of a strongly connected directed graph. In [], the authors worked with general strongly connected graphs for investigation of Riemann-Roch theory for directed graphs. Translating between the settings of [] and [9] requires a little finesse. In contrast to the case of undirected graphs where passage between reduced and full Laplacians is completely transparent as both the row and columns sums are zero, in the directed case the column sums are not necessarilly zero: this corresponds to the situation when our directed graph is Eulerian. If we take the lattice generated by the rows of the reduced Laplacian Q of a digraph with v 0 globally reachable and then homogenize with respect to the coordinate corresponding to v 0, the sink vertex, we obtain a full dimensional sublattice of A n. This lattice is generated by the rows of the full Laplacian of a strongly connected directed graph where the row corresponding to v 0 can be canonically obtained as Q T σ, where σ is the minimal script vector. We do not explain the minimal script vector for the reduced Laplacian but refer to [9] for its precise definition, but note that if one has the full Laplacian, it is the vector obtained from Σ by deleting first entry. On the other hand, if we take the lattice generated by the rows of the full Laplacian of a strongly connected directed graph, dehomogenization with respect to the coordinate corresponding to the vertex v 0 gives the lattice spanned by the reduced Laplacian if and only if Σ 0 =. Using a simple variant of Theorem 4, we can always take a generating set for the same lattice coming from a different strongly connected directed graph with Σ 0 =. Example 7. Consider the directed graph G shown in Figure. It has Laplacian matrix: 5

6 v v 2 2 Figure : Example 7 v Q = The left kernel Σ of Q is (, 2, 3). The Laplacian lattice ideal I L i.e., the lattice generated by the rows of Q reads I L = x 2 2 x x 0, x 4 2 x 2 0x 2, x 6 2 x 3 0x 3, x 3 x 0 x 2 2, x 2 x 2 0, x x 2 2 x 3 0, x 4 2 x 4 0, x 6 x 2 0x 4 2, x 5 x 3 0x 2 2, x 4 x 4 0, x 3 x 2 2 x 5 0. In fact, the above generating set is a Gröbner basis of I L with respect to any order with x 0 minimum. Since the exponent vectors ( 2, 2, 0), (4, 0, 4) and ( 4, 4, 0) of the binomials x 2 x 2 0, x 4 2 x 4 0 and x 4 x 4 0 do not have full support, the lattice ideal is not evidently generic (in fact, we do not know an algorithm to decide if a lattice is generic). In Example, we deform I L into a generic lattice ideal. 6

7 3 Explicit Deformation of Lattice Ideals 3. Deformation of Lattice Ideals We first define a metric on the space of lattices and use this metric to give a precise definition of a deformation of a lattice ideal. Definition 8. (Metric on the Space of Lattices and Convergence on Lattices) For sublattices L and L 2 of R n, we define the distance d(l, L 2 ) between L and L 2 as the minimum of B B 2 2 over all bases B and B 2 of L and L 2 respectively expressed as matrices and. 2 is the l 2 -norm on matrices. A sequence of lattices {L k } is said to converge to a lattice L l if for every δ > 0 there exists a positive integer K(δ, q) such that d(l k, L l ) δ for all k K(δ, q). See the book of Cassels [4, Page 27] for a detailed discussion on sequences of lattices and a proof that d(.,.) is a metric on the space of sublattices of R n. Barany and Scarf in [2] show that the generic lattices are dense in the spaces of all lattices with respect to the topology induced by this metric. Definition 9. (Deformation of a Lattice Ideal) Given a lattice L Z n+, a deformation of I L is a sequence of lattices {L k } that converge to L and for every lattice L i in the sequence, there is a non-zero real number λ i such that λ i L i of L i is a sublattice of Z n+ and the lattice ideal I λi L i is generic. We call the sequence {I λk L k } a deformation of the lattice ideal I L. A lattice L gen such that d(l, L gen ) = δ and there exists a λ R such that λ L gen Z n+ is generic is called a δ-deformation of L and δ is called the deformation parameter. 3.2 A Naive Approach to Explicit Deformation A natural first approach, but an unsucessful one, to deform a lattice ideal I L would be to take an arbitrary generating set of I L and deform its exponents to obtain an ideal generated by binomials all of whose exponents have full support. But the pitfall to this approach is that the resulting ideal need not be a lattice ideal as the following example shows: consider the lattice ideal I A4 where A 4 is the root lattice of dimension three. The ideal I A4 is minimally generated by the binomials x 0 x, x x 2 and x 2 x 3. One deformation of these binomials is x 0 x ɛ 2x ɛ 3 x +2ɛ, x x ɛ 0x ɛ 3 x +2ɛ 2 and x 2 x ɛ x ɛ 0 x +2ɛ 3 respectively where ɛ = /k for some large integer k. Scaling the exponent vector of each binomial by k we obtain x k 0x 2 x 3 x k+2, x k x 0 x 3 x k+2 2 and x k 2x x 0 x k+2 3. But, for any natural number k the ideal I k generated by these binomials is not saturated with respect to the 7

8 product of all the variables and is hence not a lattice ideal [7, Lemma 7.6]. To see this, note that x 2 x k+ 2 x k+ 0 x 2 3 is a binomial in the saturation of I k with respect to the product of all variables. This is because the vector ( k, 2, k +, 2) is a point in the lattice generated by the exponents of the binomial generators of I k. But x 2 x k+ 2 x k+ 0 x 2 3 is not contained in I k since x 2 x k+ 2 is not divisible by any monomial term in the binomial generators x k 0x 2 x 3 x k+2, x k x 0 x 3 x k+2 2 and x k 2x x 0 x k+2 3 of I k. In fact, a key step in our deformation algorithm is to deform the exponents of a (Gröbner) basis of the lattice ideal in such a way that the resulting ideal remains a lattice ideal. 3.3 Deformation Algorithm Algorithm. Deformation Algorithm Input: A lattice ideal I L where L is a full rank sublattice of A n and a real number δ > 0. Output: A Gröbner basis of a generic Laplacian lattice ideal I λ Lgen d(l gen, L) δ and λ L gen Z n+. with As a preprocessing step, apply the lattice reduction algorithm from Theorem 4 to compute a strongly connected directed graph G whose corresponding Laplacian Q has rows generating L and has left kernel Σ with Σ 0 =. Begin with Q 0 := Q and at the rth iteration of this step, let Q r be our current Laplacian matrix. If the Gröbner basis given by Theorem 4 applied to Q r has full support, output this Gröbner basis and take L gen to be the lattice spanned by the rows of Q r. Otherwise, there exists some vector x with 0 x Σ and some index i such that (Q T x) i = 0, find a coordinate j such that x j /Σ j x i /Σ i, which exists since Σ is a primitive vector and x Σ. Let ˆQ r be the Laplacian matrix of the directed graph H i,j on (n + )-vertices labelled v 0,..., v n with two directed edges, one from v i to v j of weight /Σ i and one from v j to v i of weight /Σ j. Take Q r+ = Q r + ɛ r ˆQr with ɛ r < min {0 y Σ,( ˆQ T r y) k (Q T r y) k <0,k {i,j}} { (QT r y) k δ ( ˆQ, T r y) k ˆQ r (n + ) s Σ }. s Theorem 0. Given any lattice ideal I L, where L is a full rank sublattice of A n and any real number δ > 0, Algorithm outputs a generic lattice ideal I Lgen such 8

9 that d(l gen, L) δ. Proof. We first show that the Algorithm terminates. In every iteration, by choosing ɛ r smaller than the first term in the minimum, for every vector 0 z Σ such that (Q r z) j 0, we have (Q r+ z) j 0, and there exists a vector 0 y Σ and an index i such that (Q r y) i = 0 and (Q r+ y) i 0. Hence, the algorithm terminates after at most ( j Σ j)(n + ) iterations. The matrix Q r is the Laplacian of a directed graph obtained from G by adding edges, hence this graph is strongly connected and by Lemma 3 and Theorem 4, we have obtained a Gröbner basis for a Laplacian lattice ideal I Lgen. Moreover, this Gröbner basis has full support because (Q T x) i 0 for all 0 x Σ and indices 0 i n from which it follows that I Lgen is generic. Since we chose ɛ r at each step to be less than the second term in the minimum, every entry in Q gen Q has absolute value at most δ/n +. Hence (by Definition 8) d(l, L gen ) δ. We note that the second term in the minimum makes use of our knowledge that the algorithm terminates in at most (n + ) s Σ s iterations. Algorithm generalizes to sublattices of A n of higher codimension as follows: given any sublattice L of A n, we first extend a basis B of the lattice L to a basis B of a lattice L Q n of dimension n such that B B 2 δ/2. Apply Algorithm to the lattice L with parameter δ/2. Note that this approach is somewhat unsatisfactory for lattices of co-dimension strictly greater than one since the dimension of the deformed lattice will not be the same as the given lattice. If we impose that the deformed lattice has the same dimension as the given lattice then our approach does not directly generalize as the following example shows. Consider the one dimensional sublattice spanned by the vector (,,, ) of A 3, since every element in this lattice has two positive coordinates, this lattice is not generated by the rows of the Laplacian of a directed graph. Hence, Theorem 4 and Step i of Algorithm do not apply to this lattice. Example. In Example 7, the generating set of the lattice ideal I L has three binomials x 2 x 2 0, x 4 x 4 0 and x 4 2 x 4 0 whose exponents do not have full support. Following the explicit deformation algorithm (Algorithm ), we deform the Laplacian Q to the digraph shown in Figure 2 to Q: Q ɛ = 3 + ɛ/2 2 ɛ/2 ɛ/3 2 + ɛ/3 9

10 v v 2 +ϵ/3 2 +ϵ/2 v Figure 2: Example Suppose ɛ = p/q, for non-zero natural numbers p and q. We can scale this matrix by the integer 6q to obtain a matrix with integer entries. 30q 8q 2q 6q Q ɛ = 6q 8q + 3p 2q 3p 6q 6q 2p 2q + 2p This matrix is the Laplacian matrix of a directed graph. Let L gen be the lattice generated by the rows of the matrix 6q Q ɛ. The lattice I λlgen is: I Lgen = x 2q+2p 2 x 6q 0 x 6q+2p, x 24q+4p 2 x 2q 0 x 2q+4p, x 36q+6p 2 x 8q 0 x 8q+6p, x 8q+3p x 6q 0 x 2q+3p 2, x 2q+p x 2q 0 x p 2, x 2q+p 2 x 6q p x 8q 0, x 24q p 2 x 24q 0 x p, x 36q+6p x 2q 0 x 24q+6p 2, x 30q+4p 0. x 8q 0 x 2q+4p 2, x 24q+2p x 24q 0 x 2p 2, x 8q x 2q 2 x 30q The above generating set is a Gröbner basis of I Lgen as described in Corollary 6. The lattice ideal I Lgen is generic and I Lgen is a δ-deformation of I L. The choice of the deformation (Figure 2) is not unique. In this example, adding the Laplacian matrix of one directed graph to Q was sufficient for the deformation, 0

11 but in general we might have to perform this operation several times. 3.4 Geometric Aspects of the Explicit Deformation The set of Laplacians coming from a strongly connected directed graph with a onedimensional left kernel spanned by a vector Σ N n+ is a cone of matrices viewed as points in R (n+)2. We denote this cone by C Σ. The cone C Σ is polyhedral, lying in the space of matrices with row sums equal to zero and has facet defining inequalities given by the nonpositivity constraints of the off-diagonal entries. The cone C Σ is neither closed nor open and its interior is given by the collection of Laplacians of graphs with no non-zero entry and whose left kernel is spanned by Σ. For each Laplacian matrix in C Σ, we associate a lattice ideal as follows: this is the (lattice) ideal corresponding to the lattice generated by the rows of the Laplacian matrix. Theorem 0 implies that the subset of Laplacian matrices whose lattice ideal is generic is dense in C Σ. The closure of C Σ is the set of all Laplacians with left kernel containing Σ, i.e., those Laplacians coming from digraphs comprised of a disjoint collection of strongly connected components, whose left kernel is given by the corresponding restriction of Σ. Lemma 2. The rays of C Σ are generated by the Laplacians coming from weighted cycles obtained in the following way: take some subset S of V ( G) and cyclically order the elements of S. If v i is followed by v j in the cyclic order, add an edge (v i, v j ) with weight Σ i. Proof. First observe that we can scale the ith rows in a Laplacian by Σ i to linearly map C Σ to the collection of Laplacians with left kernel spanned by. These are the Laplacians of Eulerian directed graphs, i.e., those having an Eulerian circuit, and it is a classical fact that such directed graphs decompose into directed cycles. We may now map these directed cycles back to the aforementioned weighted directed cycles by rescaling the rows. 4 Free Resolutions of Lattice Ideals By Degeneration We construct a (non-minimal) cellular free resolution of lattice ideal I L arising from finite index sublattices of A n. This free resolution is constructed by degenerating the Scarf complex of a deformation the lattice ideal. Algorithm can be used to perform this deformation. Our construction is an adaptation of the free

12 resolutions of monomial ideals constructed from a generic deformation in [7, Section 6.3] to lattice ideals. We start with the framework of Bayer and Sturmfels [3] where they consider a Laurent monomial module associated with a lattice, in fact monomial modules associated with a lattice are categorically equivalent to lattice ideals. We then deform the exponents of this Laurent monomial modules such that the resulting lattice ideal is generic. The Scarf complex of a deformed lattice ideal is hence a minimal free resolution and we relabel this Scarf complex to obtain a free resolution for I L. Unlike in the monomial ideal case in [7, Section 6.3], the Laurent monomial module has infinitely many minimal generators and this makes the choice of the deformation parameter δ 0 more involved. More precisely, we choose the deformation parameter δ 0 such that for every 0 < δ δ 0, the lattice ideal I Lδ has the same Scarf complex ( as a simplicial complex.) We compute the deformation parameter δ 0 using Theorem 5.4 of [8] which provides a description of the Scarf complex of I L in terms of the Scarf complex of the initial ideal with respect to a degree reverse lexicographic order. We compute the initial ideal of the Laplacian lattice ideal with respect to a degree reverse lexicographic order using Corollary 6 and choose a small enough δ 0 -deformation so that the poset defined by the least common multiples of the non-empty subsets of the initial ideal is stable for all 0 < δ δ 0. We now describe the relabelling procedure to construct a free resolution of I L from the Scarf complex of its deformation. Given a lattice ideal I L, we first deform the lattice ideal I L into a generic lattice ideal I Lgen using the Algorithm. For simplicity of exposition, we assume that L gen is a δ-deformation of L as computed in the last paragraph. Let B = {b 0,..., b n } and B gen = {b 0,δ,..., b n,δ } be the first n rows of the Laplacian matrix of a directed graph whose rows generate the lattice L and its deformation L gen respectively. We construct the Scarf complex of the generic lattice L gen as described in [3]. By construction, the vertices of the Scarf complex are precisely the points of L gen. We label the Scarf complex of L gen as follows: suppose that α δ is the lattice point with α δ = n k=0 α kb k,δ, we label the vertex α δ of the Scarf complex by x α, where α = n k=0 α kb k and x α = n i=0 xα i i. This labeling of the vertices of the Scarf complex induces a labelling of the faces by labelling each face with the least common multiple of the labels of its vertices. We denote this labelled simplicial complex by Scarf def (I L ). Given such a labelled simplicial complex we can associate a complex of free k[x 0,..., x n ]-modules as described in [7, Chapter 9.3]. For a labelled complex of k[x 0,..., x n ]-modules C and for a vector b R n+, let C b be the subcomplex of all faces of C such that the exponents of their labels are dominated coordinatewise by b. Theorem 3. The complex of free k[x 0,..., x n ]-modules associated to Scarf def (I L ) 2

13 is exact and hence this complex is a free resolution of I L. Proof. Consider a deformation of I L as performed by Algorithm and note that the exponents of the monomials in the initial ideal vary continuously in the deformation parameter. We use Theorem 5.4 of [8] that provides a description of the Scarf complex of I L in terms of the Scarf complex of the initial ideal with respect to a degree reverse lexicographic order. In particular, Theorem 5.4 shows that the abstract simplicial complex underlying the Scarf complex depends only on the poset defined by the least common multiples of all non-empty subsets of the initial ideal. Hence, there exists sufficiently small δ 0 > 0 such that for every 0 < δ δ 0 for any δ-deformation of Laplacian matrix, the abstract simplicial complex underlying the Scarf complex of the lattice ideal is the same. By the criterion for exactness of a labelled complex described in [3, Proposition 4.2], it suffices to show that the reduced homology group H j (Scarf def (I L ) b, k) = 0 for every vector b in Z n+. Since the abstract simplicial complex underlying the Scarf complex of the lattice ideal is the same for sufficiently small deformations, we know that for every b Z, there exists an ɛ > 0 such that H j (Scarf def (I L ) b, k) = H j (Scarf(I Lgen) bɛ, k)) where b ɛ is the coordinate-wise maximum of all points α ɛ in L gen such that α is a point in L that is dominated coordinate-wise by b (recall the construction of α ɛ from α described in the paragraph before Theorem 3). From [3, Proposition 4.2], Hj (Scarf(I Lgen) bɛ, k) = 0 for all b ɛ R n. Hence, the complex Scarf def (I L ) is exact. A similar method is used by Manjunath and Sturmfels [6] to construct a nonminimal free resolution for toppling ideals i.e., the special case where the lattice is generated by the rows of the Laplacian of an undirected connected graph. The complex Scarf def (I L ) is not minimal as shown by the following example shows: Example 4. Consider the lattice ideal: I L = x 2 2 x x 0, x 4 2 x 2 0x 2, x 6 2 x 3 0x 3, x 3 x 0 x 2 2, x 2 x 2 0, x x 2 2 x 3 0, x 4 2 x 4 0, x 6 x 2 0x 4 2, x 5 x 3 0x 2 2, x 4 x 4 0, x 3 x 2 2 x 5 0 from Example 7. Its deformation constructed in Example with p =, q = 2. I Lgen = x 26 2 x 2 0 x 4, x 52 2 x 24 0 x 28, x 78 2 x 36 0 x 42, x 39 x 2 0 x 27 2, x 25 x 24 0 x 2, x x 25 2 x 36 0, x 5 2 x 48 0 x 3, x 78 x 24 0 x 54 2, x 64 x 36 0 x 28 2, x 50 x 48 0 x 2 2, x 60 0 x 36 x Using [8, Theorem 5.2], we can determine the number of faces of a given dimension in its Scarf complex from its initial ideal with respect to any degree reverse lexicographic order. For the degree reverse lexicographic term order with the order x 0 > x > x 2 on the variables, the initial ideal is: 3

14 M = x 25, x 2 0 x 4, x 36 0 Since M is a monomial ideal in two variables, we conclude that its Scarf complex has dimension one. Furthermore, since M has three minimal generators, its Scarf complex has three vertices and two edges. Hence, by [8, Theorem 5.2], the Scarf complex of I Lgen has one vertex, three edges and two faces and is a triangulation of the two dimensional torus R 3 /L gen. We degenerate the Scarf complex of I Lgen to construct a free resolution of the lattice ideal k[x 0, x, x 2 ]/I L and hence, the rank of free modules of this free resolution is one, three and two in homological degrees zero, one and two respectively. This free resolution is non-minimal since the Betti numbers of k[x 0, x, x 2 ]/I L as computed using the package Macaulay 2 read one, two and one respectively. Acknowledgments: We thank Anders Jensen, Bernd Sturmfels and Josephine Yu for their helpful comments and discussions on this project. We are very grateful to Matt Baker and Frank-Olaf Schreyer for their guidance and support. References [] A. Asadi and S. Backman: Chip-firing and Riemann-Roch theory for Directed Graphs, arxiv: [2] I. Barany and H. Scarf: Matrices with Identical Sets of Neighbors, Mathematics of Operations Research, , (998). [3] D. Bayer and B. Sturmfels: Cellular Resolutions of Monomial Modules, Journal für die Reine und Angewandte Mathematik, 502 (998) [4] J.W.S. Cassels: An Introduction to the Geometry of Numbers, Springer- Verlag (959). [5] R. Cori, D. Rossin and B. Salvy: Polynomial Ideals for Sandpiles and their Gröbner bases, Theoretical Computer Science 276 (2002) 5. [6] Madhusudan Manjunath and Bernd Sturmfels: Monomials, Binomials and Riemann-Roch, Journal of Algebraic Combinatorics, 37(4), (203) [7] E. Miller and B. Sturmfels: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, Vol. 227, Springer, New York (2005). 4

15 [8] I. Peeva and B. Sturmfels: Generic Lattice Ideals, Journal of the American Mathematical Society (998) [9] D. Perkinson, J. Perlman and J. Wilmes: Primer for the Algebraic geometry of Sandpiles, arxiv:2.663 (20). School of Mathematics, Georgia Institute of Technology Department of Mathematics, University of California Berkeley 5