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Minimal free resolutions of lattice ideals of digraphs. (English) Zbl 1445.13015
Let \(I(\mathcal L)=\langle x^{m^+}-x^{m^-}| m \in \mathcal L\rangle \subset K[x]\), \(x=(x_1,\ldots,x_n)\), be the lattice ideal associated to the lattice \(\mathcal L\) spanned by the columns of the Laplacian matrix \(L\) of a finite, strongly connected, weighted, directed graph \(G\). A free resolution of \(I(\mathcal L)\) is computed. It is proved that this resolution is minimal if and only it \(G\) is strongly complete.
13D02 Syzygies, resolutions, complexes and commutative rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05E40 Combinatorial aspects of commutative algebra
05C20 Directed graphs (digraphs), tournaments
05C57 Games on graphs (graph-theoretic aspects)
Full Text: DOI arXiv
[1] Asadi, Arash; Backman, Spencer, Chip-firing and Riemann-Roch theory for directed graphs, (2010) · Zbl 1274.05189
[2] Backman, Spencer; Manjunath, Madhusudan, Explicit deformation of lattice ideals via chip-firing games on directed graphs, J. Algebraic Combin., 42, 4, 1097-1110, (2015) · Zbl 1328.05119
[3] Bak, Per; Tang, Chao; Wiesenfeld, Kurt, Self-organized criticality, Phys. Rev. A (3), 38, 1, 364-374, (1988) · Zbl 1230.37103
[4] Berkesch, Christine; Schreyer, Frank-Olaf, Syzygies, finite length modules, and random curves, (2014) · Zbl 1359.13029
[5] Brualdi, Richard A.; Ryser, Herbert J., Combinatorial matrix theory, 39, x+367 pp., (1991), Cambridge University Press, Cambridge · Zbl 0746.05002
[6] Bruns, Winfried; Herzog, Jürgen, Cohen-Macaulay rings, 39, xii+403 pp., (1993), Cambridge University Press, Cambridge · Zbl 0788.13005
[7] Cori, Robert; Rossin, Dominique; Salvy, Bruno, Polynomial ideals for sandpiles and their Gröbner bases, Theoret. Comput. Sci., 276, 1-2, 1-15, (2002) · Zbl 1002.68105
[8] Corrales, Hugo; Valencia, Carlos E., Arithmetical structures on graphs, (2017) · Zbl 1428.05188
[9] Cox, David; Little, John; O’Shea, Donal, Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, xiv+536 pp., (1997), Springer-Verlag, New York · Zbl 0756.13017
[10] Decker, Wolfram; Schreyer, Frank-Olaf, Varieties, Gröbner Bases and Algebraic Curves, (2011)
[11] Eisenbud, David, Commutative algebra. With a view toward algebraic geometry, 150, xvi+785 pp., (1995), Springer-Verlag, New York · Zbl 0819.13001
[12] Eröcal, Burçin; Motsak, Oleksandr; Schreyer, Frank-Olaf; Steenpaß, Andreas, Refined algorithms to compute syzygies, J. Symbolic Comput., 74, 308-327, (2016) · Zbl 1405.14138
[13] Gantmacher, F. R., The theory of matrices. Vols. 1, 2, Vol. 1, x+374 pp. Vol. 2, ix+276 pp., (1959), Chelsea Publishing Co., New York · Zbl 0927.15002
[14] Godsil, Chris; Royle, Gordon, Algebraic graph theory, 207, xx+439 pp., (2001), Springer-Verlag, New York · Zbl 0968.05002
[15] Greuel, Gert-Martin; Pfister, Gerhard, A Singular introduction to commutative algebra, xx+689 pp., (2008), Springer, Berlin · Zbl 1133.13001
[16] Loughry, J.; van Hemert, J. I.; Schoofs, L., Efficiently Enumerating the Subsets of a Set, (2000)
[17] Manjunath, Madhusudan; Schreyer, Frank-Olaf; Wilmes, John, Minimal free resolutions of the \(G\)-parking function ideal and the toppling ideal, Trans. Amer. Math. Soc., 367, 4, 2853-2874, (2015) · Zbl 1310.13022
[18] Manjunath, Madhusudan; Sturmfels, Bernd, Monomials, binomials and Riemann-Roch, J. Algebraic Combin., 37, 4, 737-756, (2013) · Zbl 1272.13017
[19] Mohammadi, Fatemeh; Shokrieh, Farbod, Divisors on graphs, connected flags, and syzygies, Int. Math. Res. Not., 24, 6839-6905, (2014) · Zbl 1305.05132
[20] O’Carroll, Liam; Planas-Vilanova, Francesc, The primary components of positive critical binomial ideals, J. Algebra, 373, 392-413, (2013) · Zbl 1274.13002
[21] O’Carroll, Liam; Planas-Vilanova, Francesc; Villarreal, Rafael H., Degree and algebraic properties of lattice and matrix ideals, SIAM J. Discrete Math., 28, 1, 394-427, (2014) · Zbl 1334.13017
[22] Perkinson, David; Perlman, Jacob; Wilmes, John, Tropical and non-Archimedean geometry, 605, Primer for the algebraic geometry of sandpiles, 211-256, (2013), Amer. Math. Soc., Providence, RI · Zbl 1320.05060
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