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We examine connections between topological features of $X$ and the algebraic structure of the ring ${A}(X)$. We show that if ${J}(X) \\ne 0$ and $A(X)$ is a completely integrally closed local ring that is not a valuation ring of $F$, then there is a subspace $Y$ of the space of valuation rings of $F$ that is perfect in the patch topology such that ${A}("},"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":"1710.01774","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2017-10-04T19:28:42Z","cross_cats_sorted":[],"title_canon_sha256":"33e700d04e979d2ba90ae7b59fdcb8bb54f02183594c47393734ba87a0925069","abstract_canon_sha256":"42723d8cc5b2a465157b35fc3aabea78b22b1b681202205766be703ed5e80b98"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:38.548663Z","signature_b64":"PhGvRxpOVsLOsV9iB+HyTQvpicw/158wdqmT/7tBbD4DTz4MTwpl1Tj8ImEQLjCQnuYmgHQ4EhFVq6FhKfovDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"28b3de5a50e824600e0c4209b256894edba70e24348ed3d0c81123cc7b3fb2b2","last_reissued_at":"2026-05-18T00:33:38.547937Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:38.547937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the topology of valuation-theoretic representations of integrally closed domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Bruce Olberding","submitted_at":"2017-10-04T19:28:42Z","abstract_excerpt":"Let $F$ be a field. For each nonempty subset $X$ of the Zariski-Riemann space of valuation rings of $F$, let ${A}(X) = \\bigcap_{V \\in X}V$ and ${J}(X) = \\bigcap_{V \\in X}{\\mathfrak M}_V$, where ${\\mathfrak M}_V$ denotes the maximal ideal of $V$. We examine connections between topological features of $X$ and the algebraic structure of the ring ${A}(X)$. We show that if ${J}(X) \\ne 0$ and $A(X)$ is a completely integrally closed local ring that is not a valuation ring of $F$, then there is a subspace $Y$ of the space of valuation rings of $F$ that is perfect in the patch topology such that ${A}("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.01774","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":"1710.01774","created_at":"2026-05-18T00:33:38.548057+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.01774v1","created_at":"2026-05-18T00:33:38.548057+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.01774","created_at":"2026-05-18T00:33:38.548057+00:00"},{"alias_kind":"pith_short_12","alias_value":"FCZ54WSQ5ASG","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_16","alias_value":"FCZ54WSQ5ASGADQM","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_8","alias_value":"FCZ54WSQ","created_at":"2026-05-18T12:31:15.632608+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/FCZ54WSQ5ASGADQMIIE3EVUJJ3","json":"https://pith.science/pith/FCZ54WSQ5ASGADQMIIE3EVUJJ3.json","graph_json":"https://pith.science/api/pith-number/FCZ54WSQ5ASGADQMIIE3EVUJJ3/graph.json","events_json":"https://pith.science/api/pith-number/FCZ54WSQ5ASGADQMIIE3EVUJJ3/events.json","paper":"https://pith.science/paper/FCZ54WSQ"},"agent_actions":{"view_html":"https://pith.science/pith/FCZ54WSQ5ASGADQMIIE3EVUJJ3","download_json":"https://pith.science/pith/FCZ54WSQ5ASGADQMIIE3EVUJJ3.json","view_paper":"https://pith.science/paper/FCZ54WSQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.01774&json=true","fetch_graph":"https://pith.science/api/pith-number/FCZ54WSQ5ASGADQMIIE3EVUJJ3/graph.json","fetch_events":"https://pith.science/api/pith-number/FCZ54WSQ5ASGADQMIIE3EVUJJ3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FCZ54WSQ5ASGADQMIIE3EVUJJ3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FCZ54WSQ5ASGADQMIIE3EVUJJ3/action/storage_attestation","attest_author":"https://pith.science/pith/FCZ54WSQ5ASGADQMIIE3EVUJJ3/action/author_attestation","sign_citation":"https://pith.science/pith/FCZ54WSQ5ASGADQMIIE3EVUJJ3/action/citation_signature","submit_replication":"https://pith.science/pith/FCZ54WSQ5ASGADQMIIE3EVUJJ3/action/replication_record"}},"created_at":"2026-05-18T00:33:38.548057+00:00","updated_at":"2026-05-18T00:33:38.548057+00:00"}