{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:CG6H5F3X45KDCFBI3LQLQOKIWR","short_pith_number":"pith:CG6H5F3X","schema_version":"1.0","canonical_sha256":"11bc7e9777e754311428dae0b83948b472f5d5c720e4204fc389fe09784f213f","source":{"kind":"arxiv","id":"1403.0640","version":2},"attestation_state":"computed","paper":{"title":"The Grothendieck group of non-commutative non-noetherian analogues of $\\mathbb{P}^1$ and regular algebras of global dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Gautam Sisodia, S. Paul Smith","submitted_at":"2014-03-04T00:34:05Z","abstract_excerpt":"Let $V$ be a finite-dimensional positively-graded vector space. Let $b \\in V \\otimes V$ be a homogeneous element whose rank is $\\text{dim}(V)$. Let $A=TV/(b)$, the quotient of the tensor algebra $TV$ modulo the 2-sided ideal generated by $b$. Let ${\\sf gr}(A)$ be the category of finitely presented graded left $A$-modules and ${\\sf fdim}(A)$ its full subcategory of finite dimensional modules. Let ${\\sf qgr}(A)$ be the quotient category ${\\sf gr}(A)/{\\sf fdim}(A)$. We compute the Grothendieck group $K_0({\\sf qgr}(A))$. In particular, if the reciprocal of the Hilbert series of $A$, which is a pol"},"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":"1403.0640","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-03-04T00:34:05Z","cross_cats_sorted":[],"title_canon_sha256":"18613a2e7e07a6263dbeb7088ef804cd2c3caeace807817c2b93cac5df8e6165","abstract_canon_sha256":"87e19225a16986191bb491816d0fe6a4703313110037299b7be1b6b44b529d51"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:18.406218Z","signature_b64":"9pNj9xT0TLtOVzDBGcSyXnwsFfCWGhqztSBnJc0da1iOpluSVeVGghmUNlcAzRnyr9/NuC92hQUKrCvRvrXeBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11bc7e9777e754311428dae0b83948b472f5d5c720e4204fc389fe09784f213f","last_reissued_at":"2026-05-18T02:40:18.405575Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:18.405575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Grothendieck group of non-commutative non-noetherian analogues of $\\mathbb{P}^1$ and regular algebras of global dimension two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Gautam Sisodia, S. Paul Smith","submitted_at":"2014-03-04T00:34:05Z","abstract_excerpt":"Let $V$ be a finite-dimensional positively-graded vector space. Let $b \\in V \\otimes V$ be a homogeneous element whose rank is $\\text{dim}(V)$. Let $A=TV/(b)$, the quotient of the tensor algebra $TV$ modulo the 2-sided ideal generated by $b$. Let ${\\sf gr}(A)$ be the category of finitely presented graded left $A$-modules and ${\\sf fdim}(A)$ its full subcategory of finite dimensional modules. Let ${\\sf qgr}(A)$ be the quotient category ${\\sf gr}(A)/{\\sf fdim}(A)$. We compute the Grothendieck group $K_0({\\sf qgr}(A))$. In particular, if the reciprocal of the Hilbert series of $A$, which is a pol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.0640","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":"1403.0640","created_at":"2026-05-18T02:40:18.405669+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.0640v2","created_at":"2026-05-18T02:40:18.405669+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.0640","created_at":"2026-05-18T02:40:18.405669+00:00"},{"alias_kind":"pith_short_12","alias_value":"CG6H5F3X45KD","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_16","alias_value":"CG6H5F3X45KDCFBI","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_8","alias_value":"CG6H5F3X","created_at":"2026-05-18T12:28:22.404517+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/CG6H5F3X45KDCFBI3LQLQOKIWR","json":"https://pith.science/pith/CG6H5F3X45KDCFBI3LQLQOKIWR.json","graph_json":"https://pith.science/api/pith-number/CG6H5F3X45KDCFBI3LQLQOKIWR/graph.json","events_json":"https://pith.science/api/pith-number/CG6H5F3X45KDCFBI3LQLQOKIWR/events.json","paper":"https://pith.science/paper/CG6H5F3X"},"agent_actions":{"view_html":"https://pith.science/pith/CG6H5F3X45KDCFBI3LQLQOKIWR","download_json":"https://pith.science/pith/CG6H5F3X45KDCFBI3LQLQOKIWR.json","view_paper":"https://pith.science/paper/CG6H5F3X","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.0640&json=true","fetch_graph":"https://pith.science/api/pith-number/CG6H5F3X45KDCFBI3LQLQOKIWR/graph.json","fetch_events":"https://pith.science/api/pith-number/CG6H5F3X45KDCFBI3LQLQOKIWR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CG6H5F3X45KDCFBI3LQLQOKIWR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CG6H5F3X45KDCFBI3LQLQOKIWR/action/storage_attestation","attest_author":"https://pith.science/pith/CG6H5F3X45KDCFBI3LQLQOKIWR/action/author_attestation","sign_citation":"https://pith.science/pith/CG6H5F3X45KDCFBI3LQLQOKIWR/action/citation_signature","submit_replication":"https://pith.science/pith/CG6H5F3X45KDCFBI3LQLQOKIWR/action/replication_record"}},"created_at":"2026-05-18T02:40:18.405669+00:00","updated_at":"2026-05-18T02:40:18.405669+00:00"}