Monomials, binomials and Riemann–Roch

Journal of Algebraic Combinatorics, Aug 2012

The Riemann–Roch theorem on a graph G is related to Alexander duality in combinatorial commutative algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration. We also develop a self-contained Riemann–Roch theory for Artinian monomial ideals.

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Monomials, binomials and Riemann–Roch

J Algebr Comb Monomials, binomials and Riemann-Roch Madhusudan Manjunath 0 1 Bernd Sturmfels 0 1 0 B. Sturmfels Department of Mathematics, University of California , Berkeley , USA 1 M. Manjunath ( ) Fachrichtung Mathematik, Universität des Saarlandes , Saarbrücken , Germany The Riemann-Roch theorem on a graph G is related to Alexander duality in combinatorial commutative algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration. We also develop a self-contained Riemann-Roch theory for Artinian monomial ideals. We examine the Riemann-Roch theorem on a finite graph G, due to Baker and Norine [3], through the lens of combinatorial commutative algebra. Throughout this paper, G is undirected and connected, has n nodes, and multiple edges are allowed, but we do not allow loops. Its Laplacian is a symmetric n × n-matrix ΛG with non-positive integer entries off the diagonal and kernel spanned by e = (1, 1, . . . , 1). Divisors on G are identified with Laurent monomials xu = x1u1 x2u2 · · · xnun . The chip firing moves are binomials xu − xv where u, v ≥ 0 and u − v is in the lattice spanned by the columns of ΛG. The lattice ideal IG spanned by such binomials is here called the toppling Riemann-Roch theory for graphs; Combinatorial commutative algebra; Chip firing games; Laplacian matrix of a graph; Lattice ideals and their Betti numbers; Alexander duality of monomial ideals; Scarf complex 1 Introduction ideal of the graph G. It was introduced by Perkinson, Perlman and Wilmes [11, 15], following an earlier study of the inhomogeneous version of IG by Cori, Rossin and Salvy [6]. For any fixed node, the toppling ideal IG has a distinguished initial monomial ideal MG. This monomial ideal was studied by Postnikov and Shapiro [12], and it is characterized by the property that the standard monomials of MG are the G-parking functions. We construct free resolutions for both IG and MG, and we study their role for Riemann–Roch theory on G. For an illustration, consider the complete graph on four nodes, G = K4. The chip firing moves on K4 are the integer linear combinations of the columns of ⎡ 3 ⎢ −1 ΛG = ⎢⎣ −1 −1 −−11⎥⎦⎥ . These correspond to the maximal parking functions studied in combinatorics; see [5, 12]. We claim that the duality seen in Figs. 4.2 and 4.3 of [9] is the same as that expressed in the Riemann–Roch Theorem for G. This will be made precise in Sections 3 and 4. The present article is organized as follows. Section 2 is concerned with the case when G is a saturated graph, meaning that any two nodes i and j are connected by at least one edge. We show that here IG is a generic lattice ideal, and we determine its minimal free resolution and its Hilbert series in the finest grading. The Scarf complex of the initial monomial ideal MG is supported on the barycentric subdivision of the The toppling ideal is the lattice ideal in K[x] = K[x1, x2, x3, x4] that represents imageZ(ΛG): IG = x1 − x2x3x4, x23 − x1x3x4, x33 − x1x2x4, x1x2x3 − x4 , 3 3 x1 x2 − x32x42, x12x32 − x22x42, x22x32 − x1 x4 . 2 2 2 2 This ideal is generic in the sense of Peeva and Sturmfels [10], as each of the seven binomials contains all four variables. The minimal free resolution is given by the Scarf complex 0 ← K[x] ← K[x]7 ← K[x]12 ← K[x]6 ← 0. The seven binomials in ( 2 ) form a Gröbner basis of IG, with the underlined monomials generating the initial ideal MG. That monomial ideal has the irreducible decomposition MG = x1, x22, x33 ∩ x1, x23, x32 ∩ x12, x2, x33 ∩ x12, x23, x3 ∩ x13, x2, x32 ∩ x13, x22, x3 . The ideal MG is the tree ideal of [9, §4.3.4]. Its standard monomials are in bijection with the 16 spanning trees. Its Alexander dual is generated by the six socle elements ( 1 ) ( 2 ) ( 3 ) ( 4 ) (n − 2)-simplex [12, §6], and this lifts to the Scarf complex of the lattice ideal IG by [10, Corollary 5.5]. In Section 3 we revisit the Riemann–Roch formula rank(D) − rank(K−D) = degree(D) − genus +1. We prove this formula in an entirely new setting: the role of the curve is played by a monomial ideal, and that of the divisors D and K is played by monomials xb and xK. The identity ( 5 ) is shown for monomial ideals that are Artinian, level, and reflection invariant. This includes the parking function ideals MG derived from saturated graphs G. In Section 4 we extend our results to the case of graphs G that are not saturated, and we rederive Riemann–Roch for graphs as a corollary. Here MG is still an initial ideal of IG, but the choice of term order is more delicate [11, §5]. One choice is the cost function used by Baker and Shokrieh for the integer program in [4, Theorem 4.1]. The Scarf complexes in Section 2 support cellular free resolutions of IG and MG, but these resolutions are usual (...truncated)


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Madhusudan Manjunath, Bernd Sturmfels. Monomials, binomials and Riemann–Roch, Journal of Algebraic Combinatorics, 2012, pp. 737-756, Volume 37, Issue 4, DOI: 10.1007/s10801-012-0386-9