{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:KIDQWLW77ACYGWT3NZEGQKSJBG","short_pith_number":"pith:KIDQWLW7","schema_version":"1.0","canonical_sha256":"52070b2edff805835a7b6e48682a490984523676456eba7e479ffe95c8ac76cd","source":{"kind":"arxiv","id":"1208.1599","version":1},"attestation_state":"computed","paper":{"title":"Algebraic K-theory of endomorphism rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA","math.RT"],"primary_cat":"math.KT","authors_text":"Changchang Xi, Hongxing Chen","submitted_at":"2012-08-08T07:23:08Z","abstract_excerpt":"We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\\mathcal C}$ be an additive category, and let $Y\\ra X$ be a covariant morphism of objects in ${\\mathcal C}$. Then $K_n\\big(_{\\mathcal C}(X\\oplus Y)\\big)\\simeq K_n\\big(_{{\\mathcal C},Y}(X)\\big)\\oplus K_n\\big(_{\\mathcal C}(Y)\\big)$ for all $1\\le n\\in \\mathbb{N}$, where $_{{\\mathcal C},Y}(X)$ is the quotient ring of the endomorphism ring $_{\\mathcal C}(X)$ of $X$ modulo the ideal generated by all those endomorphisms of $X$ which factoriz"},"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":"1208.1599","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2012-08-08T07:23:08Z","cross_cats_sorted":["math.RA","math.RT"],"title_canon_sha256":"f2002e91582a1d44c4931e76300e7431cd62b1343edee3d9493b56d8694667a3","abstract_canon_sha256":"712db275c54bc00f1e12b873cec23cfc5384ef9e3b1415a1fcea8ea9adf4fb04"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:20.252182Z","signature_b64":"CBjcqB/W+OmfdwGLU5jGECm5CdajE358cKY6oMJefhjyGAwu93hfGrjHDc+juXg/AucpTy+S8kyKmCeWzqdgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52070b2edff805835a7b6e48682a490984523676456eba7e479ffe95c8ac76cd","last_reissued_at":"2026-05-18T00:17:20.251569Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:20.251569Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algebraic K-theory of endomorphism rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA","math.RT"],"primary_cat":"math.KT","authors_text":"Changchang Xi, Hongxing Chen","submitted_at":"2012-08-08T07:23:08Z","abstract_excerpt":"We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\\mathcal C}$ be an additive category, and let $Y\\ra X$ be a covariant morphism of objects in ${\\mathcal C}$. Then $K_n\\big(_{\\mathcal C}(X\\oplus Y)\\big)\\simeq K_n\\big(_{{\\mathcal C},Y}(X)\\big)\\oplus K_n\\big(_{\\mathcal C}(Y)\\big)$ for all $1\\le n\\in \\mathbb{N}$, where $_{{\\mathcal C},Y}(X)$ is the quotient ring of the endomorphism ring $_{\\mathcal C}(X)$ of $X$ modulo the ideal generated by all those endomorphisms of $X$ which factoriz"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1599","kind":"arxiv","version":1},"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":"1208.1599","created_at":"2026-05-18T00:17:20.251652+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.1599v1","created_at":"2026-05-18T00:17:20.251652+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.1599","created_at":"2026-05-18T00:17:20.251652+00:00"},{"alias_kind":"pith_short_12","alias_value":"KIDQWLW77ACY","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_16","alias_value":"KIDQWLW77ACYGWT3","created_at":"2026-05-18T12:27:11.947152+00:00"},{"alias_kind":"pith_short_8","alias_value":"KIDQWLW7","created_at":"2026-05-18T12:27:11.947152+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/KIDQWLW77ACYGWT3NZEGQKSJBG","json":"https://pith.science/pith/KIDQWLW77ACYGWT3NZEGQKSJBG.json","graph_json":"https://pith.science/api/pith-number/KIDQWLW77ACYGWT3NZEGQKSJBG/graph.json","events_json":"https://pith.science/api/pith-number/KIDQWLW77ACYGWT3NZEGQKSJBG/events.json","paper":"https://pith.science/paper/KIDQWLW7"},"agent_actions":{"view_html":"https://pith.science/pith/KIDQWLW77ACYGWT3NZEGQKSJBG","download_json":"https://pith.science/pith/KIDQWLW77ACYGWT3NZEGQKSJBG.json","view_paper":"https://pith.science/paper/KIDQWLW7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.1599&json=true","fetch_graph":"https://pith.science/api/pith-number/KIDQWLW77ACYGWT3NZEGQKSJBG/graph.json","fetch_events":"https://pith.science/api/pith-number/KIDQWLW77ACYGWT3NZEGQKSJBG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KIDQWLW77ACYGWT3NZEGQKSJBG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KIDQWLW77ACYGWT3NZEGQKSJBG/action/storage_attestation","attest_author":"https://pith.science/pith/KIDQWLW77ACYGWT3NZEGQKSJBG/action/author_attestation","sign_citation":"https://pith.science/pith/KIDQWLW77ACYGWT3NZEGQKSJBG/action/citation_signature","submit_replication":"https://pith.science/pith/KIDQWLW77ACYGWT3NZEGQKSJBG/action/replication_record"}},"created_at":"2026-05-18T00:17:20.251652+00:00","updated_at":"2026-05-18T00:17:20.251652+00:00"}