Cohen–Macaulay weighted oriented edge ideals and its Alexander dual

The study of the edge ideal I(D<inf>G</inf>) of a weighted oriented graph D<inf>G</inf> with underlying graph G started in the context of Reed–Muller type codes. We generalize some Cohen–Macaulay constructions for I(D<inf>G</inf>), 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 I(D<inf>C<inf>n</inf></inf>) is Cohen–Macaulay if and only if I(D<inf>C<inf>n</inf></inf>) is unmixed and I(C<inf>n</inf>) is Cohen–Macaulay, where Cn denotes the cycle of length n. 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 I(D<inf>G</inf>) and its conditions to be Cohen–Macaulay.