The Cartier core map for Cartier algebras
Anna Brosowsky

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 be a commutative Noetherian -finite ring of prime characteristic and let be a Cartier algebra. We define a self-map on the Frobenius split locus of the pair by sending a point to the splitting prime of . We prove this map is continuous, containment preserving, and fixes the -compatible ideals. We show this map can be extended to arbitrary ideals , where in the Frobenius split case it gives the largest -compatible ideal contained in . Finally, we apply Glassbrenner's criterion to prove that the prime uniformly -compatible ideals of a Stanley-Reisner rings are the sums of its minimal primes.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Algebraic Geometry and Number Theory · Algebraic structures and combinatorial models
