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arXiv · 2203.01911

The Cartier core map for Cartier algebras

Anna Brosowsky
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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.

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Anna Brosowsky. 2023-07-12. The Cartier core map for Cartier algebras. https://doi.org/10.1016/j.jalgebra.2023.04.018

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