# Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs

## TL;DR

This paper investigates the algebraic properties of monomial ideals and their associated edge ideals in vertex-weighted digraphs, providing classifications, computational methods, and conditions for Cohen-Macaulayness and duality.

## Contribution

It introduces new classifications and computational techniques for symbolic Rees algebras and Cohen-Macaulay properties of vertex-weighted digraphs and their edge ideals.

## Key findings

- Normality of symbolic Rees algebra characterized by primary components.
- Procedure for computing symbolic Rees algebra using Hilbert bases.
- Edge ideals of weighted acyclic tournaments are Cohen-Macaulay.

## Abstract

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy Alexander duality