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The category of \"quasi-coherent sheaves\" on it is, by fiat, the quotient category QGr(B):=Gr(B)/Fdim(B) and the category of coherent sheaves on it is qgr(B):=gr(B)/fdim(B), where gr(B) is the category of finitely presented graded modules and fdim(B) is the full subcategory of finite dimensional graded modules. We show that QGr B is equivalent to Mod S, the category"},"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":"1104.3811","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-04-19T17:34:34Z","cross_cats_sorted":[],"title_canon_sha256":"044f9bbd25b723f2d7fa8858f9996c950f3c916f4b30cf788c0101d06acf8bf1","abstract_canon_sha256":"692917654442c88a5908aceb1a490eb4f5473c617d09680e543babe473f1ff2f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:47.350028Z","signature_b64":"18bgxvBbu8bUxOm/ir92Gnaeku/+YcudLgSZPU40nVzBIsmQFp1U+sY71kmxfdWXE/vPyr6VWBBmHHNqyLliCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"60cc804ba61e6191225390201ac4070003a4aee74621ab4a811889829ec17326","last_reissued_at":"2026-05-18T03:51:47.349606Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:47.349606Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The space of Penrose tilings and the non-commutative curve with homogeneous coordinate ring k<x,y>/(y^2)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"S. Paul Smith","submitted_at":"2011-04-19T17:34:34Z","abstract_excerpt":"We construct a non-commutative scheme that behaves as if it is the space of Penrose tilings of the plane.\n  Let k be a field and B=k<x,y>(y^2). We consider B as the homogeneous coordinate ring of a non-commutative projective scheme. The category of \"quasi-coherent sheaves\" on it is, by fiat, the quotient category QGr(B):=Gr(B)/Fdim(B) and the category of coherent sheaves on it is qgr(B):=gr(B)/fdim(B), where gr(B) is the category of finitely presented graded modules and fdim(B) is the full subcategory of finite dimensional graded modules. We show that QGr B is equivalent to Mod S, the category"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3811","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":"1104.3811","created_at":"2026-05-18T03:51:47.349665+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.3811v2","created_at":"2026-05-18T03:51:47.349665+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.3811","created_at":"2026-05-18T03:51:47.349665+00:00"},{"alias_kind":"pith_short_12","alias_value":"MDGIAS5GDZQZ","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_16","alias_value":"MDGIAS5GDZQZCIST","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_8","alias_value":"MDGIAS5G","created_at":"2026-05-18T12:26:34.985390+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/MDGIAS5GDZQZCISTSAQBVRAHAA","json":"https://pith.science/pith/MDGIAS5GDZQZCISTSAQBVRAHAA.json","graph_json":"https://pith.science/api/pith-number/MDGIAS5GDZQZCISTSAQBVRAHAA/graph.json","events_json":"https://pith.science/api/pith-number/MDGIAS5GDZQZCISTSAQBVRAHAA/events.json","paper":"https://pith.science/paper/MDGIAS5G"},"agent_actions":{"view_html":"https://pith.science/pith/MDGIAS5GDZQZCISTSAQBVRAHAA","download_json":"https://pith.science/pith/MDGIAS5GDZQZCISTSAQBVRAHAA.json","view_paper":"https://pith.science/paper/MDGIAS5G","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.3811&json=true","fetch_graph":"https://pith.science/api/pith-number/MDGIAS5GDZQZCISTSAQBVRAHAA/graph.json","fetch_events":"https://pith.science/api/pith-number/MDGIAS5GDZQZCISTSAQBVRAHAA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MDGIAS5GDZQZCISTSAQBVRAHAA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MDGIAS5GDZQZCISTSAQBVRAHAA/action/storage_attestation","attest_author":"https://pith.science/pith/MDGIAS5GDZQZCISTSAQBVRAHAA/action/author_attestation","sign_citation":"https://pith.science/pith/MDGIAS5GDZQZCISTSAQBVRAHAA/action/citation_signature","submit_replication":"https://pith.science/pith/MDGIAS5GDZQZCISTSAQBVRAHAA/action/replication_record"}},"created_at":"2026-05-18T03:51:47.349665+00:00","updated_at":"2026-05-18T03:51:47.349665+00:00"}