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Thus Gersten injectivity for algebraic $K$-theory with $\\mathbf Z/3$-coefficients fails for a two-dimensional ramified regular local ring. The coefficient Bockstein of $a$ is zero, while the map $K_2(A)\\to K_2(F)$ is injective. We also indicate the expected analogous construction for every odd prime. This counterexample does not contradict the "},"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":"2608.05005","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2026-08-05T16:12:07Z","cross_cats_sorted":[],"title_canon_sha256":"d65dd05fd3e2efdd9cdc267e26915490d9128034a1558a5a337941bd17578c09","abstract_canon_sha256":"2375785f3a6df2a31e3026624d578c2ff33531fac25cd1ffa575229c0db4c5b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-06T01:48:02.247114Z","signature_b64":"lEczpynvZ7IITzHiL1ytJzDWJtIuHu8uehlgiHw93ldGl9EexcaBQB2oMiXq63XMfRJozc36kj+Em5WzgYtqBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cc8d3220b7fb0859754710657372a671856de5b9f7e9beb284b142dbcb6e1817","last_reissued_at":"2026-08-06T01:48:02.245738Z","signature_status":"signed_v1","first_computed_at":"2026-08-06T01:48:02.245738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Finite-coefficient Gersten injectivity fails in ramified mixed characteristic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.KT","authors_text":"Niels Feld","submitted_at":"2026-08-05T16:12:07Z","abstract_excerpt":"Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer, and put $A=V[[x,y]]/(3+x^2-y^3)$. We construct a nonzero class $a\\in K_2(A;\\mathbf Z/3)$ whose restriction to the fraction field of $A$ is zero. Thus Gersten injectivity for algebraic $K$-theory with $\\mathbf Z/3$-coefficients fails for a two-dimensional ramified regular local ring. The coefficient Bockstein of $a$ is zero, while the map $K_2(A)\\to K_2(F)$ is injective. We also indicate the expected analogous construction for every odd prime. 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