{"paper":{"title":"Remarks on diagonal dimension for algebraic stacks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The diagonal dimension of a variety with mild singularities is at most twice its Krull dimension in arbitrary characteristic.","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Fei Peng, Pat Lank","submitted_at":"2026-05-13T12:10:54Z","abstract_excerpt":"This note is concerned with the Rouquier dimension of the bounded derived category of coherent complexes on a Noetherian algebraic stack. Specifically, we study the diagonal dimension of a morphism, which can be used to produce upper bounds on Rouquier dimension. First, we obtain an explicit upper bound for smooth morphisms with a regular target. Second, we identify strong generators of a fiber product, recovering a result of Elagin--Lunts--Schn\\\"{u}rer. Finally, we show that the diagonal dimension of a variety in arbitrary characteristic with mild singularities is at most twice its Krull dime"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"the diagonal dimension of a variety in arbitrary characteristic with mild singularities is at most twice its Krull dimension","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"the precise definition of 'mild singularities' and the technical conditions on the Noetherian algebraic stack that allow the diagonal dimension to be defined and bounded as stated","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Diagonal dimension of a variety with mild singularities is at most twice its Krull dimension; explicit upper bounds are given for smooth morphisms to regular targets.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The diagonal dimension of a variety with mild singularities is at most twice its Krull dimension in arbitrary characteristic.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"58d5b9e7b3fb185f3ba25428924c7cb4e245154a5c0b3ea9054fad94c14f8785"},"source":{"id":"2605.13416","kind":"arxiv","version":1},"verdict":{"id":"b26fef4b-d5a7-4f95-8f22-603afdce5325","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:07:30.756572Z","strongest_claim":"the diagonal dimension of a variety in arbitrary characteristic with mild singularities is at most twice its Krull dimension","one_line_summary":"Diagonal dimension of a variety with mild singularities is at most twice its Krull dimension; explicit upper bounds are given for smooth morphisms to regular targets.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"the precise definition of 'mild singularities' and the technical conditions on the Noetherian algebraic stack that allow the diagonal dimension to be defined and bounded as stated","pith_extraction_headline":"The diagonal dimension of a variety with mild singularities is at most twice its Krull dimension in arbitrary characteristic."},"references":{"count":14,"sample":[{"doi":"","year":2012,"title":"Hochschild dimensions of tilting objects.Int","work_id":"11e580d9-ee8f-4817-8581-3172cd0409de","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Dimension theory of non- commutative curves","work_id":"a27ece50-6ae2-4e90-ada8-329625f24cdb","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Descending strong generation in algebraic geometry","work_id":"7aba0dbd-4547-47e8-9ca7-db7c3eaf907f","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Approximability and rouquier dimension for noncommutative algebras over schemes","work_id":"ab2fd6bb-8336-4ee1-a9dd-446d0e0e4c18","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Categorical characterizations of regularity for algebraic stacks","work_id":"45075b50-bf1e-461b-ac6c-a0c47f3f1362","ref_index":5,"cited_arxiv_id":"2504.02813","is_internal_anchor":true}],"resolved_work":14,"snapshot_sha256":"16e34613a5d63eeef0ddcbdfb6502b25c4a69dfec9c10dd632da06d8263a7cea","internal_anchors":1},"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"}