{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:TUS2TE5OG7AZOJBTVQEKXE5LTP","short_pith_number":"pith:TUS2TE5O","schema_version":"1.0","canonical_sha256":"9d25a993ae37c1972433ac08ab93ab9bff0f8e5701162f7230730832904627c3","source":{"kind":"arxiv","id":"2605.13029","version":1},"attestation_state":"computed","paper":{"title":"On the additivity of projective presentations of maximal rank","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The modules which have a projective presentation of maximal rank are exactly the τ-regular modules.","cross_cats":[],"primary_cat":"math.RT","authors_text":"Grzegorz Bobi\\'nski, Jan Schr\\\"oer","submitted_at":"2026-05-13T05:34:06Z","abstract_excerpt":"We study projective presentations of finite-dimensional modules over finite-dimensional algebras. We discuss if projective presentations of maximal rank behave additively. More precisely, we ask if the direct sum of copies of a projective presentation of maximal rank is again of maximal rank. The modules which have a projective presentation of maximal rank are exactly the $\\tau$-regular modules. This class of modules can be seen as a generalization of modules of projective dimension at most one, and of $\\tau$-rigid modules. The $\\tau$-regular modules form open subsets of module varieties. Thei"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.13029","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-05-13T05:34:06Z","cross_cats_sorted":[],"title_canon_sha256":"a0ddac85afb54ceb7bd19bed26c523facf0ae1430f5ff1286796822f8d5255b9","abstract_canon_sha256":"9a8a84a5b1a375d7287984d44344865f23162b1243d9be6d6482ab8e5333e2a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:59.807119Z","signature_b64":"02bjD96H43R6N2lO0c6Ot15C743Ey4KKm390fT9+SqUy8EZMvrDwdlaiqKuihqfSpC8ZBVYZt0UjPgdOdlO5Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d25a993ae37c1972433ac08ab93ab9bff0f8e5701162f7230730832904627c3","last_reissued_at":"2026-05-18T03:08:59.806304Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:59.806304Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the additivity of projective presentations of maximal rank","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The modules which have a projective presentation of maximal rank are exactly the τ-regular modules.","cross_cats":[],"primary_cat":"math.RT","authors_text":"Grzegorz Bobi\\'nski, Jan Schr\\\"oer","submitted_at":"2026-05-13T05:34:06Z","abstract_excerpt":"We study projective presentations of finite-dimensional modules over finite-dimensional algebras. We discuss if projective presentations of maximal rank behave additively. More precisely, we ask if the direct sum of copies of a projective presentation of maximal rank is again of maximal rank. The modules which have a projective presentation of maximal rank are exactly the $\\tau$-regular modules. This class of modules can be seen as a generalization of modules of projective dimension at most one, and of $\\tau$-rigid modules. The $\\tau$-regular modules form open subsets of module varieties. Thei"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The modules which have a projective presentation of maximal rank are exactly the τ-regular modules.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The algebra and modules are finite-dimensional, allowing definition of module varieties and the τ functor from prior representation theory.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"τ-regular modules are those with projective presentations of maximal rank; they form open subsets of module varieties whose closures are generically τ-regular components, with additivity of maximal rank tied to reduction to projective dimension at most one.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The modules which have a projective presentation of maximal rank are exactly the τ-regular modules.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"727467d688e0de50f34859df2c0b615502f5266e548ce19cd4aedf116dac637d"},"source":{"id":"2605.13029","kind":"arxiv","version":1},"verdict":{"id":"540c318b-3fd1-434b-b169-74b60e2ed693","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T01:59:08.730074Z","strongest_claim":"The modules which have a projective presentation of maximal rank are exactly the τ-regular modules.","one_line_summary":"τ-regular modules are those with projective presentations of maximal rank; they form open subsets of module varieties whose closures are generically τ-regular components, with additivity of maximal rank tied to reduction to projective dimension at most one.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The algebra and modules are finite-dimensional, allowing definition of module varieties and the τ functor from prior representation theory.","pith_extraction_headline":"The modules which have a projective presentation of maximal rank are exactly the τ-regular modules."},"references":{"count":24,"sample":[{"doi":"","year":2014,"title":"T. Adachi, O. Iyama, I. Reiten, -tilting theory. Compos. Math. 150 (2014), no. 3, 415--452","work_id":"b653fe0a-9d47-4c4f-b695-f931358ad9bc","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"C. Amiot, O. Iyama, I. Reiten, G. Todorov, Preprojective algebras and c -sortable words. Proc. Lond. Math. Soc. (3) 104 (2012), no. 3, 513--539","work_id":"f975634f-07de-4d94-b74f-dff977b1c329","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Asai, The wall-chamber structures of the real Grothendieck groups","work_id":"db0b55e3-7ed5-46e1-89b1-202b9058d15d","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"I. Assem, D. Simson, A. Skowro\\'nski, Elements of the representation theory of associative algebras. Vol1. Techniques of representation theory. London Math. Soc. Stud. Texts, 65 Cambridge University P","work_id":"821e301f-aa65-4174-beff-7c094bbc7b45","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1977,"title":"M. Auslander, I. Reiten, Representation theory of Artin algebras. V. Methods for computing almost split sequences and irreducible morphisms. Comm. Algebra 5 (1977), no. 5, 519--554","work_id":"3b1c1b1b-2dd3-4115-9f3c-316b9835c8c6","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":24,"snapshot_sha256":"4ad5b4a2ab897520e6cddcccd098fdab30d0f0b6fb9292c46fb88369e62a8772","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.13029","created_at":"2026-05-18T03:08:59.806444+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.13029v1","created_at":"2026-05-18T03:08:59.806444+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13029","created_at":"2026-05-18T03:08:59.806444+00:00"},{"alias_kind":"pith_short_12","alias_value":"TUS2TE5OG7AZ","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"TUS2TE5OG7AZOJBT","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"TUS2TE5O","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TUS2TE5OG7AZOJBTVQEKXE5LTP","json":"https://pith.science/pith/TUS2TE5OG7AZOJBTVQEKXE5LTP.json","graph_json":"https://pith.science/api/pith-number/TUS2TE5OG7AZOJBTVQEKXE5LTP/graph.json","events_json":"https://pith.science/api/pith-number/TUS2TE5OG7AZOJBTVQEKXE5LTP/events.json","paper":"https://pith.science/paper/TUS2TE5O"},"agent_actions":{"view_html":"https://pith.science/pith/TUS2TE5OG7AZOJBTVQEKXE5LTP","download_json":"https://pith.science/pith/TUS2TE5OG7AZOJBTVQEKXE5LTP.json","view_paper":"https://pith.science/paper/TUS2TE5O","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.13029&json=true","fetch_graph":"https://pith.science/api/pith-number/TUS2TE5OG7AZOJBTVQEKXE5LTP/graph.json","fetch_events":"https://pith.science/api/pith-number/TUS2TE5OG7AZOJBTVQEKXE5LTP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TUS2TE5OG7AZOJBTVQEKXE5LTP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TUS2TE5OG7AZOJBTVQEKXE5LTP/action/storage_attestation","attest_author":"https://pith.science/pith/TUS2TE5OG7AZOJBTVQEKXE5LTP/action/author_attestation","sign_citation":"https://pith.science/pith/TUS2TE5OG7AZOJBTVQEKXE5LTP/action/citation_signature","submit_replication":"https://pith.science/pith/TUS2TE5OG7AZOJBTVQEKXE5LTP/action/replication_record"}},"created_at":"2026-05-18T03:08:59.806444+00:00","updated_at":"2026-05-18T03:08:59.806444+00:00"}