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Minimal free resolutions of the \(G\)-parking function ideal and the toppling ideal. (English) Zbl 1310.13022
This paper gives an explicit minimal free resolution \(\mathcal{F}_0\) for the \(G\)-parking function ideal, \(M_G\), of any connected undirected multigraph \(G\) (which is also referred to as a weighted graph or a graph with edge weights in the literature) and an explicit minimal free resolution \(\mathcal{F}_1\) for the toppling ideal \(I_G\) of the graph. These results answer a question of A. Postnikov and B. Shapiro [Trans. Am. Math. Soc. 356, No. 8, 3109–3142 (2004; Zbl 1043.05038)] in the case of undirected multigraphs. As a consequence, the authors settle a conjecture of M. Manjunath and B. Sturmfels [J. Algebr. Comb. 37, No. 4, 737–756 (2013; Zbl 1272.13017)] by showing that \(M_G\) and \(I_G\) have the same Betti numbers. The paper also resolved a conjecture of Perkinson and Wilmes by describing how the combinatorial information of the graph is encoded in the minimal free resolutions. A running example is maintained throughout the paper, which provides a helpful concrete demonstration of many of the results.
In Section 1, the authors give an overview of the results and a nice review of relevant literature. They state Theorem 1.1, which equates the \(k^{th}\) Betti number of both \(I_G\) and \(M_G\) with the sum over all connected \(k+1\)-partitions of the number of acyclic orientations with a unique sink on the graph associated to the partition. In Section \(2\), they give relevant definitions and prove the necessary technical lemmas regarding acyclic \(k\)-partitions, chip-firing, equivalence classes, and divisors. They explain edge contractions, which provide the basis for the differentials of the free resolutions.
In Section 3, the authors define the free modules and differentials that form the resolution \(\mathcal{F}_1\) and show that it is a complex and that the cokernel of the final map is \(I_G\). In Section \(4\) they repeat the process to form \(\mathcal{F}_0\) with the appropriate cokernel being \(M_G\). They clearly explain the differences between the two resolutions, then conclude the section by showing that if \(T\) is a tree, \(\mathcal{F}_0(T)\) is isomorphic to the Koszul complex.
In Section 5, the authors prove the exactness of \(\mathcal{F}_0\) by reducing to the complexes of vector spaces. In Section \(6\) they use a Gröbner basis degeneration argument to show that \(\mathcal{F}_1\) is exact. In fact, they introduce a variable \(t\) to homogenize \(\mathcal{F}_1\) with respect to an integral weight function. They then work with the family of complexes \(\mathcal{F}_t(G)\). Using a quotient module (setting \(t=0\)) gives \(\mathcal{F}_0\) as the free resolution of \(M_G\). Also, localization at \(t\), or setting \(t=1\), yields \(\mathcal{F}_1\) as the minimal resolution of \(I_G\). It was shown by Cori, Rossin, and Salvy [R. Cori et al., Theor. Comput. Sci. 276, No. 1–2, 1–15 (2002; Zbl 1002.68105)] that \(M_G\) is an initial ideal of \(IG\). In this paper, Corollary 6.6 shows that the minimal free resolution of \(M_G\) is a Gröbner degeneration of the minimal free resolution of \(I_G\). The article concludes by showing in Section \(7\) that the complex \(\mathcal{F}_0\) is a cellular resolution, that is, it is supported on a \(CW\)-complex.

13D02 Syzygies, resolutions, complexes and commutative rings
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05E40 Combinatorial aspects of commutative algebra
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
Macaulay2; SageMath
Full Text: DOI arXiv
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