# Strong test ideals associated to Cartier algebras

## TL;DR

This paper explores the relationship between test ideals and Cartier algebras in positive characteristic local rings, demonstrating the abundance of strong test ideals and applying these concepts to Stanley-Reisner rings.

## Contribution

It introduces new methods to analyze strong test ideals via Cartier algebras and provides concrete computations, extending previous fundamental results.

## Key findings

- Proves the abundance of strong test ideals in certain rings
- Recovers classical results using Cartier algebra techniques
- Provides detailed analysis of Stanley-Reisner rings

## Abstract

In this note, we use the theory of test ideals and Cartier algebras to examine the interplay between the tight and integral closures in a local ring of positive characteristic. Using work of Schwede, we prove the abundance of strong test ideals, recovering some older fundamental results, and use this approach in concrete computations. In the second part of the paper, the case of Stanley-Reisner rings is fully examined.