Arithmetical rank of squarefree monomial ideals generated by five elements or with arithmetic degree four

TL;DR

This paper proves that for certain squarefree monomial ideals, the arithmetical rank equals the projective dimension, specifically when the ideal has at most five generators or an arithmetic degree of four.

Contribution

It establishes the equality of arithmetical rank and projective dimension for squarefree monomial ideals under specific size and degree constraints.

Findings

Arithmetical rank equals projective dimension for ideals with ≤5 generators.

Equality holds for ideals with arithmetic degree ≤4.

Results extend understanding of algebraic invariants in monomial ideals.

Abstract

Let $I$ be a squarefree monomial ideal of a polynomial ring $S$. In this paper, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $S/I$ when one of the following conditions is satisfied: (1) $\mu (I) \leq 5$; (2) $\arithdeg I \leq 4$.