Big Cohen-Macaulay Test Ideals on Mixed Characteristic Toric Schemes

TL;DR

This paper introduces a formula for computing big Cohen-Macaulay test ideals in mixed characteristic toric rings with monomial ideals, aligning with known formulas in positive and zero characteristic cases.

Contribution

It provides a new explicit formula for big Cohen-Macaulay test ideals in mixed characteristic toric schemes, bridging characteristic zero and positive characteristic theories.

Findings

Formula for big Cohen-Macaulay test ideals in mixed characteristic toric rings.

Consistency with formulas for test ideals in positive characteristic.

Alignment with multiplier ideals in characteristic zero.

Abstract

We provide a formula to compute the big Cohen-Macaulay test ideal for triples $((R,\Delta),\mathfrak{a}^{t})$ where $R$ is a mixed characteristic toric ring and $\mathfrak{a}$ is a monomial ideal. Of particular interest is that this result is consistent with the formulas for test ideals in positive characteristic and multiplier ideals in characteristic zero.