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We investigate the relative rigid objects, \\ie $R[1]$-rigid objects of $\\mathcal{T}$. Our main results show that the $R[1]$-rigid objects in $\\operatorname{\\mathsf{pr}}(R)$ are in bijection with $\\tau$-rigid $\\Gamma$-modules, and the maximal $R[1]$-rigid objects with respect to $\\operatorname{\\mathsf{pr}}(R)$ a"},"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":"1808.04297","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2018-08-13T15:46:20Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"6b325193be2b1985a2d1c1dc03d2fd1b8d42887dabe952077e8bc2e5b3258043","abstract_canon_sha256":"2f44f726bc735c2e3a70f0b528c2e7dbb77fa343de5066ce594ede3071a4d51f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:13.896022Z","signature_b64":"Qa/mPurCU8nE+7eiruSi1vqiuOxSxJqf62GCT2/xuj3aXUHYgvmuZH7tlJeOlwfc4Ykh3i94pj6+AmpNWxmfCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9770f21f37253046e86df2b1b7710f7f5f6b5e89764ecf56b1769f5b685ab98","last_reissued_at":"2026-05-17T23:58:13.895395Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:13.895395Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Relative rigid objects in triangulated categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Changjian Fu, Pin Liu, Shengfei Geng","submitted_at":"2018-08-13T15:46:20Z","abstract_excerpt":"Let $\\mathcal{T}$ be a Krull-Schmidt, Hom-finite triangulated category with suspension functor $[1]$. Let $R$ be a basic rigid object, $\\Gamma$ the endomorphism algebra of $R$, and $\\operatorname{\\mathsf{pr}}(R)\\subseteq \\mathcal{T}$ the subcategory of objects finitely presented by $R$. We investigate the relative rigid objects, \\ie $R[1]$-rigid objects of $\\mathcal{T}$. Our main results show that the $R[1]$-rigid objects in $\\operatorname{\\mathsf{pr}}(R)$ are in bijection with $\\tau$-rigid $\\Gamma$-modules, and the maximal $R[1]$-rigid objects with respect to $\\operatorname{\\mathsf{pr}}(R)$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.04297","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":"1808.04297","created_at":"2026-05-17T23:58:13.895490+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.04297v2","created_at":"2026-05-17T23:58:13.895490+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.04297","created_at":"2026-05-17T23:58:13.895490+00:00"},{"alias_kind":"pith_short_12","alias_value":"7F3Q6IPTOJJQ","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_16","alias_value":"7F3Q6IPTOJJQI3UG","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_8","alias_value":"7F3Q6IPT","created_at":"2026-05-18T12:32:11.075285+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/7F3Q6IPTOJJQI3UG34VRW5YQ67","json":"https://pith.science/pith/7F3Q6IPTOJJQI3UG34VRW5YQ67.json","graph_json":"https://pith.science/api/pith-number/7F3Q6IPTOJJQI3UG34VRW5YQ67/graph.json","events_json":"https://pith.science/api/pith-number/7F3Q6IPTOJJQI3UG34VRW5YQ67/events.json","paper":"https://pith.science/paper/7F3Q6IPT"},"agent_actions":{"view_html":"https://pith.science/pith/7F3Q6IPTOJJQI3UG34VRW5YQ67","download_json":"https://pith.science/pith/7F3Q6IPTOJJQI3UG34VRW5YQ67.json","view_paper":"https://pith.science/paper/7F3Q6IPT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.04297&json=true","fetch_graph":"https://pith.science/api/pith-number/7F3Q6IPTOJJQI3UG34VRW5YQ67/graph.json","fetch_events":"https://pith.science/api/pith-number/7F3Q6IPTOJJQI3UG34VRW5YQ67/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7F3Q6IPTOJJQI3UG34VRW5YQ67/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7F3Q6IPTOJJQI3UG34VRW5YQ67/action/storage_attestation","attest_author":"https://pith.science/pith/7F3Q6IPTOJJQI3UG34VRW5YQ67/action/author_attestation","sign_citation":"https://pith.science/pith/7F3Q6IPTOJJQI3UG34VRW5YQ67/action/citation_signature","submit_replication":"https://pith.science/pith/7F3Q6IPTOJJQI3UG34VRW5YQ67/action/replication_record"}},"created_at":"2026-05-17T23:58:13.895490+00:00","updated_at":"2026-05-17T23:58:13.895490+00:00"}