{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:736DRO2RBDXNBGBZS63DNN27UM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"eaeeb3e70b7388a5e33bfce0a7bfd04266fb33db29adf3c4ebf49d63cfc573d2","cross_cats_sorted":["math.AC"],"license":"","primary_cat":"math.AG","submitted_at":"2007-06-27T16:01:36Z","title_canon_sha256":"04cd2854d4cbe33ce9b6320e410dd0263a51b2cb4159000982b8206db779c8be"},"schema_version":"1.0","source":{"id":"0706.4060","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0706.4060","created_at":"2026-05-18T04:28:30Z"},{"alias_kind":"arxiv_version","alias_value":"0706.4060v1","created_at":"2026-05-18T04:28:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0706.4060","created_at":"2026-05-18T04:28:30Z"},{"alias_kind":"pith_short_12","alias_value":"736DRO2RBDXN","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"736DRO2RBDXNBGBZ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"736DRO2R","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:8362cc393bbd8d99cc10ec943ae3d44db8fb46fd8fb008321e9406db46c7fb20","target":"graph","created_at":"2026-05-18T04:28:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this note we show that finitely generated unit $O_X[\\sigma]$--modules for $X$ regular and $F$--finite have a minimal root (in the sense of [Lyubeznik, F-modules] Definition~3.6). This problem was posed by Lyubeznik and answered by himself in the case that $X=\\Spec R$ is a complete local ring.\n  One immediate consequence of this result is that the parameter test module of tight closure theory commutes with localization. As a further application of the methods in this paper we give new proofs of the results on discreteness and rationality of $F$--thresholds [arXiv:0705.1210] and on $D$-module","authors_text":"Manuel Blickle","cross_cats":["math.AC"],"headline":"","license":"","primary_cat":"math.AG","submitted_at":"2007-06-27T16:01:36Z","title":"Minimal $\\gamma$--sheaves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0706.4060","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fa7d51ae2a283c52b18fe4ddd5c6c35fbf61cba7e2ae357257530dbdd1e8ab72","target":"record","created_at":"2026-05-18T04:28:30Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"eaeeb3e70b7388a5e33bfce0a7bfd04266fb33db29adf3c4ebf49d63cfc573d2","cross_cats_sorted":["math.AC"],"license":"","primary_cat":"math.AG","submitted_at":"2007-06-27T16:01:36Z","title_canon_sha256":"04cd2854d4cbe33ce9b6320e410dd0263a51b2cb4159000982b8206db779c8be"},"schema_version":"1.0","source":{"id":"0706.4060","kind":"arxiv","version":1}},"canonical_sha256":"fefc38bb5108eed0983997b636b75fa32d10ffa39cb4419645eb5681eb5179e2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fefc38bb5108eed0983997b636b75fa32d10ffa39cb4419645eb5681eb5179e2","first_computed_at":"2026-05-18T04:28:30.281014Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:28:30.281014Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lMNXsdE7+vuWyl3NSsk2N4BKTKhc4oyK4g5807zyIs1rbGzhvlAfT4xb2WQP1oTMk4zzvKYnuB+m6NZo7tz+BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:28:30.281455Z","signed_message":"canonical_sha256_bytes"},"source_id":"0706.4060","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa7d51ae2a283c52b18fe4ddd5c6c35fbf61cba7e2ae357257530dbdd1e8ab72","sha256:8362cc393bbd8d99cc10ec943ae3d44db8fb46fd8fb008321e9406db46c7fb20"],"state_sha256":"7cb92598fc51bd475390db7c09112ebd5305854703a10c0a572632834e9a0038"}