Arithmetical rank of strings and cycles

TL;DR

This paper establishes that for certain hypergraphs called strings and cycles, the arithmetical rank of the associated squarefree monomial ideal equals the projective dimension of the quotient ring, linking algebraic invariants to combinatorial structures.

Contribution

It proves the equality of arithmetical rank and projective dimension for ideals associated with string and cycle hypergraphs, a specific class of combinatorial structures.

Findings

Arithmetical rank equals projective dimension for string hypergraphs.

Arithmetical rank equals projective dimension for cycle hypergraphs.

Provides a combinatorial characterization of algebraic invariants.

Abstract

Let $R$ be a polynomial ring over a field $K$. To a given squarefree monomial ideal $I \subset R$, one can associate a hypergraph $H(I)$. In this article, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $R/I$ when $H(I)$ is a string or a cycle hypergraph.