Computing the support of monomial iterated mapping cones
Let \(I=\langle f_1,\dots,f_r\rangle\subseteq \mathbb{K}[x_1,\dots,x_n]\) and let \(I_i=\langle f_1,\dots,f_i\rangle\) be the subideal of \(I\) generated by the first \(i\) generators of \(I\). The following short exact sequence \[ 0\rightarrow \overline{I}_i \overset{\varphi}{\rightarrow} I_{i-1}\oplus \langle f_i\rangle \overset{i}{\rightarrow} I_i\rightarrow 0 \] is defined as Mayer-Vietoris sequence, where \(\overline{I}_i=I_{i-1}\cap \langle f_i\rangle\). In this paper the author constructs the corresponding Mayer-Vietoris tree, in order to study the monomial iterated mapping cones. By constructing this tree the author succeeds in computing the monomial support of a free resolution of a monomial ideal, avoiding the hard computation of the resolution itself. Moreover by this approach algorithms are provided to analyze this support to obtain homological and numerical invariants of a monomial ideal, such as Betti numbers, Hilbert series, irreducible decompositions, regularity and Euler characteristic of simplicial complexes. It is also worth mentioning that the author in the last section of the article, concludes in the effectiveness of the Mayer-Vietoris tree, by comparing it with other commonly used computer algebras systems, such as CoCoA, Singular, and Macaulay2.
scientific article; zbMATH DE number 5782199
scientific article