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This generalizes the fundamental formula for the core of an integrally closed ideal in a two-dimensional regular local ring due to Huneke and Swanson. As an application, we show that for integrally closed modules $M$ and $N$ over a two-dimensional regular local ring with $M\\subset N$ and $M^{**}=N^{**}$, the core of $M$ is containe"},"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":"1907.05093","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-07-11T10:29:47Z","cross_cats_sorted":[],"title_canon_sha256":"3eb456d38aa6b3184e849d9d1d3a086033c2690cbf4c58c039f24d4f500f7a7b","abstract_canon_sha256":"df3070677a13b11b9173e621732ffaf343045190e5a436dfbab87b2ac9aba390"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:53.966938Z","signature_b64":"yiNhhYrnrggu2tInQwaEPgQ+xJj1iQ1EiMIMupNzpZoepe2+Yxm9OXD9EtjxR28rLa2SWdyPvnHESfivGxk/Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7257c8914afe9d9cb9c6502306b5fe79fbb79e008af515477ecf44ac7f181d8","last_reissued_at":"2026-05-17T23:40:53.966275Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:53.966275Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The core of a module and the adjoint of an ideal over a two dimensional regular local ring","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Kohsuke Shibata","submitted_at":"2019-07-11T10:29:47Z","abstract_excerpt":"The core of an module is the intersection of all its reductions. 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