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We show that the reduced locally closed subscheme of $S$ whose points are exactly those $x\\in S$ such that $(a,b)$ is a break point of the Newton polygon of the fiber $\\mathcal C_x$ of $\\mathcal C$ at $x$ is pure in $S$, i.e., it is an affine $S$-scheme. This result refines and reobtains previous results of de Jong--Oort, Vasiu, and Yang. As an application, we show that for all $m\\in \\mathbb N$ the reduced locally closed subscheme of"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1801.09593","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-01-29T16:01:01Z","cross_cats_sorted":[],"title_canon_sha256":"e6914e6cff235dcc825c7b541cd86f1b15b70174b3d859bb9654b480a7a48704","abstract_canon_sha256":"58c8dc7ee385533c50e7a048ee0d4eea156fd23cdeaa2519b849cb97a9a74b74"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:05.188914Z","signature_b64":"+41YtzdHY8MuAn1j8uqXgZKLlSglnoIPE9xedZnqcZ0aFbjlm8My+8wwL8N7tmyHNdyOmPYwcqBz10ynFvV+BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"340573ed84ae99d711e110590cf20d516c5813299f8a104009a3caf0b6340d20","last_reissued_at":"2026-05-17T23:58:05.188470Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:05.188470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Purity of Crystalline Strata","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Adrian vasiu, Jinghao Li","submitted_at":"2018-01-29T16:01:01Z","abstract_excerpt":"Let $p$ be a prime. Let $n\\in\\mathbb N-\\{0\\}$. Let $\\mathcal C$ be an $F^n$-crystal over a locally noetherian $\\mathbb F_p$-scheme $S$. Let $(a,b)\\in\\mathbb N^2$. We show that the reduced locally closed subscheme of $S$ whose points are exactly those $x\\in S$ such that $(a,b)$ is a break point of the Newton polygon of the fiber $\\mathcal C_x$ of $\\mathcal C$ at $x$ is pure in $S$, i.e., it is an affine $S$-scheme. This result refines and reobtains previous results of de Jong--Oort, Vasiu, and Yang. As an application, we show that for all $m\\in \\mathbb N$ the reduced locally closed subscheme of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09593","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1801.09593","created_at":"2026-05-17T23:58:05.188539+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.09593v3","created_at":"2026-05-17T23:58:05.188539+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.09593","created_at":"2026-05-17T23:58:05.188539+00:00"},{"alias_kind":"pith_short_12","alias_value":"GQCXH3MEV2M5","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_16","alias_value":"GQCXH3MEV2M5OEPB","created_at":"2026-05-18T12:32:25.280505+00:00"},{"alias_kind":"pith_short_8","alias_value":"GQCXH3ME","created_at":"2026-05-18T12:32:25.280505+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GQCXH3MEV2M5OEPBCBMQZ4QNKF","json":"https://pith.science/pith/GQCXH3MEV2M5OEPBCBMQZ4QNKF.json","graph_json":"https://pith.science/api/pith-number/GQCXH3MEV2M5OEPBCBMQZ4QNKF/graph.json","events_json":"https://pith.science/api/pith-number/GQCXH3MEV2M5OEPBCBMQZ4QNKF/events.json","paper":"https://pith.science/paper/GQCXH3ME"},"agent_actions":{"view_html":"https://pith.science/pith/GQCXH3MEV2M5OEPBCBMQZ4QNKF","download_json":"https://pith.science/pith/GQCXH3MEV2M5OEPBCBMQZ4QNKF.json","view_paper":"https://pith.science/paper/GQCXH3ME","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.09593&json=true","fetch_graph":"https://pith.science/api/pith-number/GQCXH3MEV2M5OEPBCBMQZ4QNKF/graph.json","fetch_events":"https://pith.science/api/pith-number/GQCXH3MEV2M5OEPBCBMQZ4QNKF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GQCXH3MEV2M5OEPBCBMQZ4QNKF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GQCXH3MEV2M5OEPBCBMQZ4QNKF/action/storage_attestation","attest_author":"https://pith.science/pith/GQCXH3MEV2M5OEPBCBMQZ4QNKF/action/author_attestation","sign_citation":"https://pith.science/pith/GQCXH3MEV2M5OEPBCBMQZ4QNKF/action/citation_signature","submit_replication":"https://pith.science/pith/GQCXH3MEV2M5OEPBCBMQZ4QNKF/action/replication_record"}},"created_at":"2026-05-17T23:58:05.188539+00:00","updated_at":"2026-05-17T23:58:05.188539+00:00"}