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Wang) between $\\Lambda$ and $\\Gamma$ and which also induce an equivalence between the stable categories of finitely generated Gorenstein-projective modules $\\Lambda$-$\\underline{\\text{Gproj}}$ and $\\Gamma$-$\\underline{\\text{Gproj}}$. We prove that if $V$ is an indecomposable object in $\\Lambda$-$\\underline{\\text{Gproj}}$ with $\\underline{\\mathr"},"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":"2606.06648","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-04T18:55:24Z","cross_cats_sorted":[],"title_canon_sha256":"d2aa50f90253943f8de96bdab3431b280d884b8aeadba35bceac34db0284cbfb","abstract_canon_sha256":"24198a38d76db2dd2e53dc76437cea3a0f5c26fe8f869b74656e061e32b34ff3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:04:20.843441Z","signature_b64":"mRC2ov9/SLcGp4GtDL7lQZZAupn/74Xjwqwp3OnBoi+PKNbX9bBAkSJfv3TYezyQulbwfQJhi8GTJY2m9lXQCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac840d6ea5d775dc21139a9ae89d22eb0d23418e8ea512498f87ff77ef6f16fc","last_reissued_at":"2026-06-08T01:04:20.842496Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:04:20.842496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On universal deformation rings and stable equivalences of Gorenstein-projective modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Jose A. Velez-Marulanda, Shengyong Pan","submitted_at":"2026-06-04T18:55:24Z","abstract_excerpt":"Let $\\mathbf{k}$ be a field and let $\\Lambda$ and $\\Gamma$ finite dimensional $\\mathbf{k}$-algebras. Assume that ${_\\Gamma}X_\\Lambda$ and ${_\\Lambda}Y_\\Gamma$ are bimodules that define a singular equivalence of Morita type with level (in the sense of Z. Wang) between $\\Lambda$ and $\\Gamma$ and which also induce an equivalence between the stable categories of finitely generated Gorenstein-projective modules $\\Lambda$-$\\underline{\\text{Gproj}}$ and $\\Gamma$-$\\underline{\\text{Gproj}}$. 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