Computing the Betti table of a monomial ideal: a reduction algorithm
Computing minimal free resolutions of ideals and their associated graded Betti numbers is a central theme of research in commutative algebra. In the paper under review, the authors introduce a new algorithm for computing the minimal free resolution of an arbitrary monomial ideal. They demonstrate that their algorithm is more efficient than others currently built into Macaulay 2 and CoCoA (see Table 1). In Theorem 6.2, this algorithm is applied to extend a result of \textit{J. Herzog} and \textit{H. Srinivasan} [Math. Sci. Res. Inst. Publ. 68, 245--249 (2015; Zbl 1359.13012)]: for an ideal \(I\) generated by monomials of degree at most \(d\) \[ \beta_{i,k} (S/I)= 0 \text{ for all } k = j, \dots, j+d-1 \quad \Longrightarrow \quad \beta_{i+1,j+d}(S/I) = 0. \] That is, if the Betti table of \(I\) has \(d\) 0's in a column of the Betti table, an entry of the next column of its Betti table is also 0. To make their work more readable, the authors present a preliminary algorithm incorporating the key ideas of their main algorithm that they later refine. Given a monomial ideal \(I\), the preliminary algorithm begins with the (not necessarily minimal) Lyubeznik resolution of \(I\) and a directed graph \(G(I)\) associated with \(I\). The graph \(G(I)\) is then changed via a reduction procedure on its invertible edges. As the graph changes, corresponding changes take place in the resolution according to a result of \textit{M. Jöllenbeck} and \textit{V. Welker} [Mem. Am. Math. Soc. 923, 74 p. (2009; Zbl 1160.13007)] from algebraic discrete Morse theory. When the reduction procedure can no longer be applied to \(G(I)\) the corresponding resolution is a minimal free resolution of \(I\). The preliminary algorithm is made more efficient by ignoring maps of the resolution (considering only Betti tables), ordering the edges of \(G(I)\), only considering invertible edges of \(G(I)\), and building up the Lyubeznik resolution step-by-step rather than constructing it at the beginning. The final algorithm is presented as the MAIN algorithm in Section 4.
scientific article; zbMATH DE number 6825751
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