Lyubeznik's resolution and rooted complexes
Let \(R\) be a polynomial ring, \(I\) be an ideal of \(R\) generated by monomials \(m_1,\dots,m_n\), \(L=L(I)\) be the lattice of the least common multiples for any of \(m_1,\dots,m_n\), \(\Delta\) be a simplicial complex whose faces are labeled with these least common multiples. Based on results of \textit{D. Bayer} and \textit{B. Sturmfels} [J. Reine Angew. Math. 502, 123--140 (1998; Zbl 0909.13011)] concerning \(\Delta\), the author indicates that the routed complex \(RC(L,\pi)\) (with respect to a routed map \(\pi)\), introduced by \textit{A. Björner} and \textit{G. Ziegler} [J. Comb. Theory, Ser. B 51, No. 1, 96--126 (1991; Zbl 0753.05019)], provides a free cellular resolution of \(I\). This resolution is minimal if \(L\) is a geometric lattice. In particular, this result can be applied to matroid ideals [see \textit{I. Novik}, \textit{A. Postnikov} and \textit{B. Sturmfels}, Duke Math. J. 111, No. 2, 287--317 (2002; Zbl 1022.13002)] for modular matroids and thus a generalization of Lyubeznik's resolution [see \textit{G. Lyubeznik}, J. Pure Appl. Algebra 51, No. 1/2, 193--195 (1988; Zbl 0652.13012)] is obtained.
scientific article; zbMATH DE number 1879612