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We show that the first and second Brauer--Thrall type theorems hold for the category of totally reflexive $R$-modules. More precisely, we prove that, for infinitely many integers $n$, there exists an indecomposable totally reflexive $R$-module of multiplicity $n$. Moreover, if the residue field of $R$ is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexiv"},"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":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1208.5730","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-08-28T18:04:04Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"4c52d477cea30fb7e39e3cbee7d45a7efdaf9aa84e20815aa7ca6ebdbc88c752","abstract_canon_sha256":"6bef412aa3ad8f835ee6dd760541b6ef8ce8d2a9bb113cc30fed4922a65f371b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:35.389885Z","signature_b64":"T75huhMt+knWFce0Uw157skHKETJYF4U/i6yVam9L2rYuYKI2KkRmpOMjL9DCLq9sfDimqX7scDfbtRLeTlpAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c852e6cf0f31a14462cefbc0745e0b5aed58ba159af4f664c8f5b1463ba97f50","last_reissued_at":"2026-05-18T00:53:35.389499Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:35.389499Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Brauer-Thrall for totally reflexive modules over local rings of higher dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.AC","authors_text":"Mohsen Gheibi, Olgur Celikbas, Ryo Takahashi","submitted_at":"2012-08-28T18:04:04Z","abstract_excerpt":"Let $R$ be a commutative Noetherian local ring. 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