Cohen–Macaulay weighted oriented edge ideals and its Alexander dual

Abstract

The study of the edge ideal [Formula: see text] of a weighted oriented graph [Formula: see text] with underlying graph [Formula: see text] started in the context of Reed–Muller type codes. We generalize some Cohen–Macaulay constructions for [Formula: see text], which Villarreal gave for edge ideals of simple graphs. Our constructions can be used to produce large classes of Cohen–Macaulay weighted oriented edge ideals. We use these constructions to classify all the Cohen–Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We also show that [Formula: see text] is Cohen–Macaulay if and only if [Formula: see text] is unmixed and [Formula: see text] is Cohen–Macaulay, where [Formula: see text] denotes the cycle of length [Formula: see text]. Miller generalized the concept of Alexander dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in that direction, we study the Alexander dual of [Formula: see text] and its conditions to be Cohen–Macaulay.