{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:WKW26IMCS5YEZO7O3KWKG55B3B","short_pith_number":"pith:WKW26IMC","schema_version":"1.0","canonical_sha256":"b2adaf218297704cbbeedaaca377a1d854b90592aa87e30323fdb486470672a6","source":{"kind":"arxiv","id":"1107.1082","version":2},"attestation_state":"computed","paper":{"title":"F-signature of pairs and the asymptotic behavior of Frobenius splittings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Karl Schwede, Kevin Tucker, Manuel Blickle","submitted_at":"2011-07-06T10:09:34Z","abstract_excerpt":"We generalize $F$-signature to pairs $(R,D)$ where $D$ is a Cartier subalgebra on $R$ as defined by the first two authors. In particular, we show the existence and positivity of the $F$-signature for any strongly $F$-regular pair. In one application, we answer an open question of I. Aberbach and F. Enescu by showing that the $F$-splitting ratio of an arbitrary $F$-pure local ring is strictly positive. Furthermore, we derive effective methods for computing the $F$-signature and the $F$-splitting ratio in the spirit of the work of R. Fedder."},"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":"1107.1082","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-07-06T10:09:34Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"ebe456fc8b2527f2f7ddb1a9a47f6fcf7805f50ab01f03948a415c3749effa64","abstract_canon_sha256":"7518d8453b4d769f1d9e559879bd852d2db2414acb0e2b9d37c9ba5f36214f22"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:39:05.426716Z","signature_b64":"ijdcwe5C35YkoCyo/b9zxDRLlA50Lq4OM4tYwPlgNkijsuc3/lzyViah5M+twtZW2VWoOeSvnG94/nkZFYMBBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2adaf218297704cbbeedaaca377a1d854b90592aa87e30323fdb486470672a6","last_reissued_at":"2026-05-18T03:39:05.426144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:39:05.426144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"F-signature of pairs and the asymptotic behavior of Frobenius splittings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Karl Schwede, Kevin Tucker, Manuel Blickle","submitted_at":"2011-07-06T10:09:34Z","abstract_excerpt":"We generalize $F$-signature to pairs $(R,D)$ where $D$ is a Cartier subalgebra on $R$ as defined by the first two authors. In particular, we show the existence and positivity of the $F$-signature for any strongly $F$-regular pair. In one application, we answer an open question of I. Aberbach and F. Enescu by showing that the $F$-splitting ratio of an arbitrary $F$-pure local ring is strictly positive. Furthermore, we derive effective methods for computing the $F$-signature and the $F$-splitting ratio in the spirit of the work of R. Fedder."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1082","kind":"arxiv","version":2},"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":"1107.1082","created_at":"2026-05-18T03:39:05.426213+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.1082v2","created_at":"2026-05-18T03:39:05.426213+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.1082","created_at":"2026-05-18T03:39:05.426213+00:00"},{"alias_kind":"pith_short_12","alias_value":"WKW26IMCS5YE","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"WKW26IMCS5YEZO7O","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"WKW26IMC","created_at":"2026-05-18T12:26:44.992195+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/WKW26IMCS5YEZO7O3KWKG55B3B","json":"https://pith.science/pith/WKW26IMCS5YEZO7O3KWKG55B3B.json","graph_json":"https://pith.science/api/pith-number/WKW26IMCS5YEZO7O3KWKG55B3B/graph.json","events_json":"https://pith.science/api/pith-number/WKW26IMCS5YEZO7O3KWKG55B3B/events.json","paper":"https://pith.science/paper/WKW26IMC"},"agent_actions":{"view_html":"https://pith.science/pith/WKW26IMCS5YEZO7O3KWKG55B3B","download_json":"https://pith.science/pith/WKW26IMCS5YEZO7O3KWKG55B3B.json","view_paper":"https://pith.science/paper/WKW26IMC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.1082&json=true","fetch_graph":"https://pith.science/api/pith-number/WKW26IMCS5YEZO7O3KWKG55B3B/graph.json","fetch_events":"https://pith.science/api/pith-number/WKW26IMCS5YEZO7O3KWKG55B3B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WKW26IMCS5YEZO7O3KWKG55B3B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WKW26IMCS5YEZO7O3KWKG55B3B/action/storage_attestation","attest_author":"https://pith.science/pith/WKW26IMCS5YEZO7O3KWKG55B3B/action/author_attestation","sign_citation":"https://pith.science/pith/WKW26IMCS5YEZO7O3KWKG55B3B/action/citation_signature","submit_replication":"https://pith.science/pith/WKW26IMCS5YEZO7O3KWKG55B3B/action/replication_record"}},"created_at":"2026-05-18T03:39:05.426213+00:00","updated_at":"2026-05-18T03:39:05.426213+00:00"}