{"paper":{"title":"Embeddings and intersections of adelic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"On three-dimensional regular projective varieties over countable fields, the intersection A_I(X,F) ∩ A_J(X,F) equals A_{I∩J}(X,F) for locally free sheaves F.","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Dmitry Badulin","submitted_at":"2025-10-25T19:08:22Z","abstract_excerpt":"We prove embeddings of adelic groups on an excellent scheme of special type and a flat quasicoherent sheaf on it. For a normal excellent scheme of special type we establish the equality $\\mathbb{A}_I(X,\\mathcal{F})\\cap\\mathbb{A}_J(X,\\mathcal{F})=\\mathbb{A}_{I\\setminus0}(X,\\mathcal{F})$ in the case $I\\cap J=I\\setminus0$. We show that the limit of restrictions of global sections of a locally free sheaf on a Cohen-Macaulay projective scheme to power thickenings of integral subschemes equals the group of global sections of this sheaf. Using this result, we deduce a theorem on intersections of adel"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"On a three-dimensional regular projective variety over a countable field the intersection A_I(X,F) ∩ A_J(X,F) equals A_{I∩J}(X,F) for any I,J ⊂ {0,1,2,3} and any locally free sheaf F on X.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The underlying scheme must be normal excellent of special type, or Cohen-Macaulay projective, or three-dimensional regular projective over a countable field (as stated in the respective theorems).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves embeddings and intersection equalities for adelic groups on excellent and projective schemes, plus a limit result for global sections of locally free sheaves.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"On three-dimensional regular projective varieties over countable fields, the intersection A_I(X,F) ∩ A_J(X,F) equals A_{I∩J}(X,F) for locally free sheaves F.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"159748e04b695eaa23a445a5f4fff9a8c8cd3fc14bb0a3e50feaa6bd09b905e4"},"source":{"id":"2510.22408","kind":"arxiv","version":3},"verdict":{"id":"be6c5b1c-644b-4185-ae2c-21b2a2491584","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T03:55:30.599282Z","strongest_claim":"On a three-dimensional regular projective variety over a countable field the intersection A_I(X,F) ∩ A_J(X,F) equals A_{I∩J}(X,F) for any I,J ⊂ {0,1,2,3} and any locally free sheaf F on X.","one_line_summary":"Proves embeddings and intersection equalities for adelic groups on excellent and projective schemes, plus a limit result for global sections of locally free sheaves.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The underlying scheme must be normal excellent of special type, or Cohen-Macaulay projective, or three-dimensional regular projective over a countable field (as stated in the respective theorems).","pith_extraction_headline":"On three-dimensional regular projective varieties over countable fields, the intersection A_I(X,F) ∩ A_J(X,F) equals A_{I∩J}(X,F) for locally free sheaves F."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2510.22408/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"afc7c78ead692af439fd82280c81145177cfab6a5908cd79f544e333021c47a9"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}