{"paper":{"title":"Perfectoid pure singularities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.NT"],"primary_cat":"math.AG","authors_text":"Bhargav Bhatt, Jakub Witaszek, Joe Waldron, Karl Schwede, Kevin Tucker, Linquan Ma, Zsolt Patakfalvi","submitted_at":"2024-09-26T15:41:42Z","abstract_excerpt":"Fix a prime number $p$. Inspired by the notion of $F$-pure or $F$-split singularities, we study the condition that a Noetherian ring with $p$ in its Jacobson radical is pure inside some perfectoid (classical) ring, a condition we call \\emph{perfectoid pure}. We also study a related a priori weaker condition which asks that $R$ is pure in its absolute perfectoidization, a condition we call \\emph{lim-perfectoid pure}. We show that both these notions coincide when $R$ is LCI. Mixed characteristic analogs of $F$-injective and Du Bois singularities are also explored. We study these notions of singu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.17965","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.17965/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}