{"paper":{"title":"A projective resolution of the symplectic Steinberg module","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The symplectic Steinberg module admits an explicit projective resolution over Sp_{2n}(R).","cross_cats":["math.GR","math.GT"],"primary_cat":"math.AT","authors_text":"Urshita Pal","submitted_at":"2026-05-07T16:17:56Z","abstract_excerpt":"Borel--Serre proved that for a number ring $R$ with fraction field $K$, the symplectic group $\\text{Sp}_{2n}(R)$ is a virtual duality group of degree quadratic in $n$, and that the symplectic Steinberg module $\\text{St}^\\omega_{2n}(K)$ is its dualizing module. We construct a projective resolution of this symplectic Steinberg module as an $\\text{Sp}_{2n}(R)$-representation, that is similar in form to a resolution of Lee--Szczarba for the special linear group, but whose construction is more involved. When $R$ is a Euclidean number ring, we use this resolution to compute the top degree cohomology"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct a projective resolution of this symplectic Steinberg module as an Sp_{2n}(R)-representation, that is similar in form to a resolution of Lee--Szczarba for the special linear group, but whose construction is more involved.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The construction of the projective resolution is valid and produces a genuine resolution of the Steinberg module as an Sp_{2n}(R)-module for general number rings R (with the cohomology application restricted to Euclidean R).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Constructs a projective resolution of the symplectic Steinberg module St^ω_{2n}(K) for Sp_{2n}(R), analogous but more involved than Lee-Szczarba's for SL_n, and applies it to compute top cohomology of principal level-p congruence subgroups over Euclidean rings.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The symplectic Steinberg module admits an explicit projective resolution over Sp_{2n}(R).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"0312e7d715ed63d78ddd4919a54d75683ec245b0ca4cd1cd2b98e16069d7a794"},"source":{"id":"2605.06499","kind":"arxiv","version":2},"verdict":{"id":"4df567ad-8201-476a-af38-b6448d79af80","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T03:50:58.298179Z","strongest_claim":"We construct a projective resolution of this symplectic Steinberg module as an Sp_{2n}(R)-representation, that is similar in form to a resolution of Lee--Szczarba for the special linear group, but whose construction is more involved.","one_line_summary":"Constructs a projective resolution of the symplectic Steinberg module St^ω_{2n}(K) for Sp_{2n}(R), analogous but more involved than Lee-Szczarba's for SL_n, and applies it to compute top cohomology of principal level-p congruence subgroups over Euclidean rings.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The construction of the projective resolution is valid and produces a genuine resolution of the Steinberg module as an Sp_{2n}(R)-module for general number rings R (with the cohomology application restricted to Euclidean R).","pith_extraction_headline":"The symplectic Steinberg module admits an explicit projective resolution over Sp_{2n}(R)."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.06499/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T18:01:19.747748Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T12:37:05.404757Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"379dc27d93bcb17177d8a16369403ab09e12aa1d3dadcca898e93c47131f6f1d"},"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"}