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We prove $S_{q^{1/2}}(M,\\mathcal{N})$ is a finitely generated $S_{1}(M,\\mathcal{N})$-module when $M$ is compact, which furthermore indicates the reduced stated skein module for the compact marked three manifold is finite dimensional. We also give an upper bound for the dimension of $S_{q^{1/2}}(M,\\mathcal{N})$ over $S_{1}(M,\\mathcal{N})$ when $M$ is compact.\n  For a pb surfa"},"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":"2309.10920","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2023-09-19T20:34:49Z","cross_cats_sorted":[],"title_canon_sha256":"3bc7b01f871c420913ad99f743071670ce3c438cce3cf56c7a01738a820f5c6f","abstract_canon_sha256":"bc6b6399af9d785dca13101b780cf14560075db57d6c1d619b537ea90182b9c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:02:53.639082Z","signature_b64":"F3JF/P6kJ1HrFoB4DBxcT48kIYzvZkwcpjSekfjlhdDweaeweual339e11AHSDSPP/sbM0nz1iJE+oghAk61Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cc6d7e98cda702503c272f094173e630ff5cc653fd5c38d5b40e0b325c321062","last_reissued_at":"2026-07-05T07:02:53.638516Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:02:53.638516Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Finiteness and dimension of stated skein modules over Frobenius","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Zhihao Wang","submitted_at":"2023-09-19T20:34:49Z","abstract_excerpt":"When the quantum parameter $q^{1/2}$ is a root of unity of odd order. The stated skein module $S_{q^{1/2}}(M,\\mathcal{N})$ has an $S_{1}(M,\\mathcal{N})$-module structure, where $(M,\\mathcal{N})$ is a marked three manifold. We prove $S_{q^{1/2}}(M,\\mathcal{N})$ is a finitely generated $S_{1}(M,\\mathcal{N})$-module when $M$ is compact, which furthermore indicates the reduced stated skein module for the compact marked three manifold is finite dimensional. 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