Computing The Invariants of Intersection Algebras of Principal Monomial Ideals

TL;DR

This paper investigates the algebraic invariants of intersection algebras formed from principal monomial ideals in polynomial rings, providing explicit formulas for key invariants like F-signature and Hilbert-Kunz multiplicity.

Contribution

It introduces explicit formulas for invariants of intersection algebras of principal monomial ideals, utilizing semigroup and toric structures.

Findings

Calculated F-signature, divisor class group, Hilbert-Samuel, and Hilbert-Kunz multiplicities.

Provided explicit formulas for these invariants in specific cases.

Extended the class of rings with known formulas for key algebraic invariants.

Abstract

We continue the study of intersection algebras $\mathcal B = \mathcal B_R(I, J)$ of two ideals $I, J$ in a commutative Noetherian ring $R$. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when $R$ is a polynomial ring over a field and $I,J$ are principal monomial ideals. Specifically, we calculate the $F$-signature, divisor class group, and Hilbert-Samuel and Hilbert-Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formul{\ae}. This provides a new class of rings where formul{\ae} for the $F$-signature and Hilbert-Kunz multiplicity, dependent on families of parameters, are provided.