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As applications, we give new proofs for some results of D.Happel and L.Unger, and prove that every connected component in $\\mathscr{K}({A})$ has finite non-saturated points if $A$ is tame type, which gives a partially positive answer to the conjecture of"},"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":"1105.2994","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-05-16T02:44:57Z","cross_cats_sorted":[],"title_canon_sha256":"b2bd792d75af3b8e453ed548fdde846934d910c6ee0295aa114e2dff60c977ea","abstract_canon_sha256":"3f692fbf9fd6257fe871ac920d90ea5ebe180234e506ebcd6bb9dd36f8f8f0dc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:22:01.588977Z","signature_b64":"R5mpG6R1Yiz3GbIjFq8QETlsmJGb47u9hq6IkBbQeFdhc+ZNQWYZOM9P70paHVpMbOQhdIA2oQ79FHJaTLJsBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22cef8cb028191bd45fa94e233ef5f8cc9577bc21854b9653f20d168b3ab0c6e","last_reissued_at":"2026-05-18T04:22:01.588518Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:22:01.588518Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tilting modules over duplicated algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Guopeng Wang, Shunhua Zhang","submitted_at":"2011-05-16T02:44:57Z","abstract_excerpt":"Let $A$ be a finite dimensional hereditary algebra over a field $k$ and $A^{(1)}$ the duplicated algebra of $A$. 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