The Cartier core map for Cartier algebras

TL;DR

This paper introduces a new self-map on the Frobenius split locus of Cartier algebras in prime characteristic rings, demonstrating its properties and applications to ideal compatibility and Stanley-Reisner rings.

Contribution

It defines and analyzes a self-map on the Frobenius split locus for Cartier algebras, extending it to arbitrary ideals and applying it to Stanley-Reisner rings.

Findings

The map is continuous and containment preserving.

It fixes all $ ext{D}$-compatible ideals.

In Stanley-Reisner rings, prime uniformly $F$-compatible ideals are sums of minimal primes.

Abstract

Let $R$ be a commutative Noetherian $F$-finite ring of prime characteristic and let $\mathcal{D}$ be a Cartier algebra. We define a self-map on the Frobenius split locus of the pair $(R,\mathcal{D})$ by sending a point $P$ to the splitting prime of $(R_P, \mathcal{D}_P)$. We prove this map is continuous, containment preserving, and fixes the $\mathcal{D}$-compatible ideals. We show this map can be extended to arbitrary ideals $J$, where in the Frobenius split case it gives the largest $\mathcal{D}$-compatible ideal contained in $J$. Finally, we apply Glassbrenner's criterion to prove that the prime uniformly $F$-compatible ideals of a Stanley-Reisner rings are the sums of its minimal primes.