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It follows that if $\\mathbb F$ is a (possibly unbounded) complex of flat $R$-modules and $\\kappa(\\mathfrak p)\\otimes_R \\mathbb F$ is exact for every $\\mathfrak p\\in\\Spec R$, then $\\mathbb G\\otimes_R^\\bullet\\mathbb F$ is exact for every $R$-complex $\\mathbb G$. If, moreover, $\\m"},"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":"1012.1394","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-12-07T03:58:24Z","cross_cats_sorted":[],"title_canon_sha256":"048e529e557a69555d9a4877270b4b00900b55e33d78c7ccdfe12b7a8cdabda8","abstract_canon_sha256":"863540948500bbc6db31abafd5e0c6d352cdb861fe43eddca4291531cb8617d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:01.110600Z","signature_b64":"4piylHolxhHV6J0rTPQQiwLEzgz0cHxrNO9sLbkpFyBnXLcKBqmWX0QVq5KjxcXtAu+vjdm6jHsbk3q4eRNrDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ae1ff5d0a8a74e1aacb7a240e78a5fd4c995133ba90a01acbd80eb9338a69a1","last_reissued_at":"2026-05-18T04:34:01.110181Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:01.110181Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Acyclicity of complexes of flat modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Mitsuyasu Hashimoto","submitted_at":"2010-12-07T03:58:24Z","abstract_excerpt":"Let $R$ be a noetherian commutative ring, and \\[ \\mathbb F: ...\\rightarrow F_2\\rightarrow F_1\\rightarrow F_0\\rightarrow 0 \\] a complex of flat $R$-modules. We prove that if $\\kappa(\\mathfrak p)\\otimes_R\\mathbb F$ is acyclic for every $\\mathfrak p\\in\\Spec R$, then $\\mathbb F$ is acyclic, and $H_0(\\mathbb F)$ is $R$-flat. It follows that if $\\mathbb F$ is a (possibly unbounded) complex of flat $R$-modules and $\\kappa(\\mathfrak p)\\otimes_R \\mathbb F$ is exact for every $\\mathfrak p\\in\\Spec R$, then $\\mathbb G\\otimes_R^\\bullet\\mathbb F$ is exact for every $R$-complex $\\mathbb G$. 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