Algebraic properties of edge ideals of some vertex-weighted oriented \(m\)-partite graphs
This paper continues the study of the edge ideals of \textit{vertex-weighted oriented graphs}. This new concept has been introduced very recently by \textit{P. Gimenez} et al. [in: Singularities, algebraic geometry, commutative algebra, and related topics. Festschrift for Antonio Campillo on the occasion of his 65th birthday, University of Valladolid, Spain, June 19--23, 2017. Cham: Springer. 491--510 (2018; Zbl 1405.13041)] and was inspired by the theory of Reed-Muller codes. Recall that a \textit{weighted oriented graph} is a triple \(D=(V(D),E(D),w)\) where \(V(D)\) is the \textit{vertex set} of \(D\), \(E(D)\) is the \textit{edge set} consisting of ordered pairs \((x_i,x_j)\) with \(x_i,x_j\in V(D)\), and \(w:V(D)\rightarrow\mathbb{N}^+=\{1,2,\dots,n\}\) is the \textit{weight function} that assign to each vertex \(x_i\) the weight \(w_i\). Let \(S=K[x_1,\dots,x_n]\) be a polynomial ring in \(n\) variables with coefficients in a field \(K\). The \textit{edge ideal} of \(D\) is defined as \[ I(D)=(x_ix_j^{w_j}:(x_i,x_j)\in E(D)). \] Contrary to the case of the \textit{classical} edge ideals, introduced by Villarreal, not much is known about the algebraic and homological properties of the edge ideals of vertex-weighted oriented graphs. In this paper, the authors focus their attention on three classes of vertex-weighted oriented \(m\)-partite graphs, and compute the projective dimension and the (Castelnuovo-Mumford) regularity under some suitable hypothesis. A key tool throughout the paper is the notion of \textit{Betti splitting}.
scientific article; zbMATH DE number 7434787
scientific article