Cartier algebras through the lens of $p$-families
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914590402543616 |
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| author | Brosowsky, Anna |
| author_facet | Brosowsky, Anna |
| contents | We study $F$-graded systems of ideals in $R$, which are sequences of ideals giving rise to Cartier algebras on $R$. We identify how properties of these systems (or modifications of these systems) affect the singularity properties of the corresponding Cartier algebra. In particular, we show that in a Gorenstein and strongly $F$-regular local ring, strong $F$-regularity and $F$-splitting are the same for a special class of $F$-graded systems called $p$-families. Further, we make use of this and a new operation we introduce called $p$-stabilization to get a criterion that in a Gorenstein and strongly $F$-regular local ring, a system is strongly $F$-regular exactly when its $p$-stabilization is $F$-split. Finally, we associate a combinatorial object to systems built out of monomial ideals and show how this can help compute the $p$-stabilization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_22987 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cartier algebras through the lens of $p$-families Brosowsky, Anna Commutative Algebra Algebraic Geometry 13A35 (Primary) 14B05 (Secondary) We study $F$-graded systems of ideals in $R$, which are sequences of ideals giving rise to Cartier algebras on $R$. We identify how properties of these systems (or modifications of these systems) affect the singularity properties of the corresponding Cartier algebra. In particular, we show that in a Gorenstein and strongly $F$-regular local ring, strong $F$-regularity and $F$-splitting are the same for a special class of $F$-graded systems called $p$-families. Further, we make use of this and a new operation we introduce called $p$-stabilization to get a criterion that in a Gorenstein and strongly $F$-regular local ring, a system is strongly $F$-regular exactly when its $p$-stabilization is $F$-split. Finally, we associate a combinatorial object to systems built out of monomial ideals and show how this can help compute the $p$-stabilization. |
| title | Cartier algebras through the lens of $p$-families |
| topic | Commutative Algebra Algebraic Geometry 13A35 (Primary) 14B05 (Secondary) |
| url | https://arxiv.org/abs/2605.22987 |