{"paper":{"title":"Revising Auslander-Gruson-Jensen duality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A simple description of the free abelian category clarifies Auslander-Gruson-Jensen duality for modules over a ring.","cross_cats":["math.CT","math.RA"],"primary_cat":"math.RT","authors_text":"Jiaqun Wei, Ramin Ebrahimi, Rasool Hafezi","submitted_at":"2026-05-05T07:45:24Z","abstract_excerpt":"For a ring $A$ there is a well-known duality between definable subcategories of right $A$-modules and definable subcategories of left $A$ modules. This is a consequence of Auslander-Gruson-Jensen duality $\\rm mod\\text{-}(mod\\text{-}A)\\rightarrow mod\\text{-}(mod\\text{-}A^{op})$. The existence of this duality arises from the fact that $\\rm mod\\text{-}(mod\\text{-}A)$ is the free abelian category over the pre-additive category $A$ with a single object.\n  In this note, first, we give a simple description of the free abelian category. This description clarifies Auslender-Gruson-Jensen duality and al"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we give a simple description of the free abelian category. This description clarifies Auslander-Gruson-Jensen duality and also the duality between definable subcategories of right A-modules and those of left A-modules.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence of this duality arises from the fact that mod-(mod-A) is the free abelian category over the pre-additive category A with a single object.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Gives a simple description of the free abelian category to clarify Auslander-Gruson-Jensen duality between definable subcategories of right and left modules over a ring.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A simple description of the free abelian category clarifies Auslander-Gruson-Jensen duality for modules over a ring.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3786924c8e07e851dd59c45e7a2555d8b39154bcbbca3275e3d8636e321bf4f1"},"source":{"id":"2605.03458","kind":"arxiv","version":2},"verdict":{"id":"289d1978-47ab-4dc1-9e03-91f47629fb66","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:08:01.794358Z","strongest_claim":"we give a simple description of the free abelian category. This description clarifies Auslander-Gruson-Jensen duality and also the duality between definable subcategories of right A-modules and those of left A-modules.","one_line_summary":"Gives a simple description of the free abelian category to clarify Auslander-Gruson-Jensen duality between definable subcategories of right and left modules over a ring.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence of this duality arises from the fact that mod-(mod-A) is the free abelian category over the pre-additive category A with a single object.","pith_extraction_headline":"A simple description of the free abelian category clarifies Auslander-Gruson-Jensen duality for modules over a ring."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.03458/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T15:23:21.859858Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"58d0578624a76cf4de0627a4370136cd26d5169454081bb64488356acc467bd4"},"references":{"count":12,"sample":[{"doi":"","year":1973,"title":"Abelian categories over additive ones","work_id":"f02b906c-63c6-4a3e-b311-4bd3210b5148","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1965,"title":"Coherent functors","work_id":"895e33f6-49d8-4fca-a70f-36a3198579cf","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1984,"title":"Isolated singularities and existence of almost split sequences","work_id":"925939a5-b483-439f-ade9-3cbcaf5cad7e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2008,"title":"When are definable classes tilting and cotilting classes?","work_id":"aba15ea8-49ff-48eb-b033-7bf65ee81261","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"On definable subcategories","work_id":"374f6565-e6cb-4310-8092-20acfee3e51c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":12,"snapshot_sha256":"9696be085e1709e9b2ac260054fd06be866830f7d6c128d968e53a08dc447974","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"4fb821ebb15672760066458e86112a1267191637d4c42791e3154a7ebdfb6342"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}