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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","authors_text":"Dmitry Badulin","cross_cats":["math.AC"],"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.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2025-10-25T19:08:22Z","title":"Embeddings and intersections of adelic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.22408","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-18T03:55:30.599282Z","id":"be6c5b1c-644b-4185-ae2c-21b2a2491584","model_set":{"reader":"grok-4.3"},"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","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.","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.","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)."}},"verdict_id":"be6c5b1c-644b-4185-ae2c-21b2a2491584"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:335cabda1215ea8721ce5cf77e4def224cc96b506b7b39c0f92a1db9ccb1a933","target":"record","created_at":"2026-07-09T00:19:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"71ad210e5578fd65b556a1ea9e0fa1240052bc005e74d4a8e8d60cc91944726e","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2025-10-25T19:08:22Z","title_canon_sha256":"b26d2dde21e16d04693a5144bb17f9fa987cf7a86fb24b5a4bce23c1c5a7e32c"},"schema_version":"1.0","source":{"id":"2510.22408","kind":"arxiv","version":3}},"canonical_sha256":"b197ebe347fd67aa3d148eb0d3468c938e2585f98bee638d75718ecb134ed156","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b197ebe347fd67aa3d148eb0d3468c938e2585f98bee638d75718ecb134ed156","first_computed_at":"2026-07-09T00:19:11.476717Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-09T00:19:11.476717Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VbqUjycDC5WDKm/fXbIo2bSsaIHyYAEz5Tc5etVF5v3VtEBRKSmsMRXAG0T0gHb/5ZRZ/FsMem1NXzQRtfCCCA==","signature_status":"signed_v1","signed_at":"2026-07-09T00:19:11.477365Z","signed_message":"canonical_sha256_bytes"},"source_id":"2510.22408","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:335cabda1215ea8721ce5cf77e4def224cc96b506b7b39c0f92a1db9ccb1a933","sha256:f35ff39fccd9d2902a4863b02894636e54970e26f40956cb502ab44adff1ee4d"],"state_sha256":"daf63ebb486b23339c8e5d5c36327e7f2afb373055ad1f4d75179fe5a9ce0ac7"}