Monomial resolutions supported by simplicial trees (Q471977)

Let \(I\) be a monomial ideal in a polynomial ring \(S\) over a field \(k\), which is minimally generated by monomials \(m_1,\ldots,m_t\) and \(\Delta\) be a simplicial complex on \(t\) vertices such that each vertex of \(\Delta\) is labeled with one of the generators \(m_1,\ldots,m_t\) and each face is labeled with the least common multiple of the labels of its vertices. In [\textit{D. Bayer} et al., Math. Res. Lett. 5, No. 1--2, 31--46 (1998; Zbl 0909.13010)] it was proved that the chain complex of \(\Delta\) is a free resolution of \(S/I\) if and only if the induced subcomplex \(\Delta_m\) is empty or acyclic for every monomial \(m\in S\) and it is a minimal free resolution if and only if \(m_A\neq m_{A'}\) for every proper subface \(A'\) of a face \(A\). In the paper under review this criterion had been translated for the class of simplicial trees and it was shown that for a simplicial tree \(\Delta\) the chain complex \(C(\Delta;S)\) is a free resolution of \(S/I\) if and only if the induced subcomplex \(\Delta_m\) is connected for every monomial \(m\). Then using the fact that simplicial trees are acyclic, it was shown that every simplicial tree is the Scarf complex of a monomial ideal \(I\) and supports a minimal resolution of \(I\). Also for an eligible simplicial complex \(\Delta\), a Scarf ideal of \(\Delta\) had been constructed with smaller monomial generators compared to the Scarf ideal given in [\textit{I. Peeva} and \textit{M. Velasco}, Trans. Am. Math. Soc. 363, No. 4, 2029--2046 (2011; Zbl 1221.13024)] and for a monomial ideal \(I\) in-between these two Scarf ideals, it was shown that the Scarf complex of \(I\) contains \(\Delta\) as a subcomplex.