{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ZTFZ35XAYY2KXJFDG5QPBY2J46","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":"80aae8b7f3106b2a417935a3144870c516e91b3622d7d2fdf7f726b2915258fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-10-18T12:16:58Z","title_canon_sha256":"91c7c5b62b8369181c8007dd57cdd490295522ed7d8ead64e2e3ca13cb315934"},"schema_version":"1.0","source":{"id":"1310.4987","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.4987","created_at":"2026-05-18T03:01:10Z"},{"alias_kind":"arxiv_version","alias_value":"1310.4987v2","created_at":"2026-05-18T03:01:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.4987","created_at":"2026-05-18T03:01:10Z"},{"alias_kind":"pith_short_12","alias_value":"ZTFZ35XAYY2K","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"ZTFZ35XAYY2KXJFD","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"ZTFZ35XA","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:296585c71e7401bf6751affdba6981c43ce926c91686f4b36c03b6118cf0075c","target":"graph","created_at":"2026-05-18T03:01:10Z","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":"We define left relative H-separable tower of rings and continue a study of these begun by Sugano. It is proven that a progenerator extension has right depth two if and only if the ring extension together with its right endomorphism ring is a left relative H-separable tower. In particular, this applies to twisted or ordinary Frobenius extensions with surjective Frobenius homomorphism. For example, normality for Hopf subalgebras of finite-dimensional Hopf algebras is also characterized in terms of this tower condition.","authors_text":"Lars Kadison","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-10-18T12:16:58Z","title":"A tower condition characterizing normality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4987","kind":"arxiv","version":2},"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:c3a837e48a5e90db393f9738addbbd9c745fd0e136a736b3e70ccc974707e1e2","target":"record","created_at":"2026-05-18T03:01:10Z","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":"80aae8b7f3106b2a417935a3144870c516e91b3622d7d2fdf7f726b2915258fc","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-10-18T12:16:58Z","title_canon_sha256":"91c7c5b62b8369181c8007dd57cdd490295522ed7d8ead64e2e3ca13cb315934"},"schema_version":"1.0","source":{"id":"1310.4987","kind":"arxiv","version":2}},"canonical_sha256":"cccb9df6e0c634aba4a33760f0e349e799c248a3f5f004cf4a7cf820d6dd7153","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cccb9df6e0c634aba4a33760f0e349e799c248a3f5f004cf4a7cf820d6dd7153","first_computed_at":"2026-05-18T03:01:10.100285Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:01:10.100285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8elXSfdxLACTqBokzDY5ZWiG22fAigDyQQv1+MxgVWz+Y9kq8iuoV9hdhy4L2pSrx6b7Jn6N4VUghr2Ccg1+AA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:01:10.101110Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.4987","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3a837e48a5e90db393f9738addbbd9c745fd0e136a736b3e70ccc974707e1e2","sha256:296585c71e7401bf6751affdba6981c43ce926c91686f4b36c03b6118cf0075c"],"state_sha256":"572c92da0d968c6498884b580566d8468f34484d3ba20aef7f08c2977594e3cd"}