{"paper":{"title":"Derived complete intersections and polynomial growth of Betti numbers over dg-algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A dg-algebra has polynomially growing Betti numbers for its modules if and only if it is a derived complete intersection.","cross_cats":[],"primary_cat":"math.AC","authors_text":"Justin Lyle, Michael K. Brown","submitted_at":"2026-05-11T18:11:27Z","abstract_excerpt":"A theorem of Gulliksen states that a local ring is a complete intersection if and only if the Betti numbers of its finitely generated modules grow polynomially. We prove a derived version of Gulliksen's Theorem. More precisely, we prove a structure theorem for dg-algebras whose modules exhibit polynomial Betti growth. As a key ingredient in the proof, we establish the existence and uniqueness of minimal models and acyclic closures of morphisms of dg-algebras in a broader setting than was previously known. We also extend to dg-algebras a theorem of Halperin on the vanishing of deviations of loc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove a structure theorem for dg-algebras whose modules exhibit polynomial Betti growth.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The dg-algebras are considered in a broader setting than previously known that permits existence and uniqueness of minimal models and acyclic closures of morphisms.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Structure theorem for dg-algebras whose modules have polynomial Betti growth, as a derived analogue of Gulliksen's theorem.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A dg-algebra has polynomially growing Betti numbers for its modules if and only if it is a derived complete intersection.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"bcc26455327714c6d9d4aa365d5ea83647e2e50e5da4417df3ccea10d288fb11"},"source":{"id":"2605.11105","kind":"arxiv","version":2},"verdict":{"id":"11382b87-629c-4521-83e2-40f8358aeecf","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T00:50:00.546679Z","strongest_claim":"We prove a structure theorem for dg-algebras whose modules exhibit polynomial Betti growth.","one_line_summary":"Structure theorem for dg-algebras whose modules have polynomial Betti growth, as a derived analogue of Gulliksen's theorem.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The dg-algebras are considered in a broader setting than previously known that permits existence and uniqueness of minimal models and acyclic closures of morphisms.","pith_extraction_headline":"A dg-algebra has polynomially growing Betti numbers for its modules if and only if it is a derived complete intersection."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.11105/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:36:06.189983Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T10:31:16.853799Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T08:46:58.317743Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"fe4e2f62198c8bf3c7656d520dec2248e87d047c5c719644b80d2401ecca497b"},"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"}