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It expresses $\\mathcal{D}_{pur}(R)$, the pure derived category of $R$, as an attachment of the usual derived category $\\mathcal{D}(R)$ with Emmanouil's quotient category $\\mathcal{D}_{K-flat}(R):=K(R)/K-Flat$, which here we call the K-flat derived category. It follows that this Verdier quotient is a compactly generated triangulat"},"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":"2107.13042","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AT","submitted_at":"2021-07-27T18:59:04Z","cross_cats_sorted":["math.CT","math.RA"],"title_canon_sha256":"648f1d73275434b814929dd1a6a9b245e796d437589aef13c5ce3f6d81e6c0f1","abstract_canon_sha256":"0b79eeb5d33ef821f12de639902e9709ffd88a47644118f7660321352544cb93"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:27:34.543681Z","signature_b64":"hnJlNW0YX/+O+Vku7oXXLMkqK3bIE9oXeJJJY82h4brBtTHeTWTZ0+TuFCSq0EVtLQzMdm1zb9/nDA8peEcODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"681d741776939869f76c3a0d384f0a76586dc3be416b880b7a907024b9bfaf35","last_reissued_at":"2026-07-05T03:27:34.543136Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:27:34.543136Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"K-flat complexes and derived categories","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT","math.RA"],"primary_cat":"math.AT","authors_text":"James Gillespie","submitted_at":"2021-07-27T18:59:04Z","abstract_excerpt":"Let $R$ be a ring with identity. 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