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When applied to a homogeneous inclusion $A\\subseteq B$ of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of $A$ in terms of that of $B$ and of local $j$-multiplicities of hyperplane sections along ${\\rm Proj}\\,(B)$. Our formulas can be used to find the multiplicity of special fiber ring"},"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":"1101.2280","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-01-12T07:35:34Z","cross_cats_sorted":[],"title_canon_sha256":"a16cc3d991d8474837468da9dd0bf575fe384f91b9afb2ae821ec5754a56dbfc","abstract_canon_sha256":"861cc6e11626485bca4ba8d37de83cdf852e4fc0a5f9de445e972382affc9a47"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:43.211966Z","signature_b64":"aj1UQGdv+VaDZCPnpQ8Cc/FtSBvS0NXQcUNt1l9xtOjQ7QnSDQasUk6HA8mRA6Uxqy9JoHbDEE3AE4ROmxq3Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3fd5d82c4f64ce5f3f5d78e18c18bc969933225b856de209b33dab35fbe8b6db","last_reissued_at":"2026-05-18T04:31:43.211480Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:43.211480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Formulas for the multiplicity of graded algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Yu Xie","submitted_at":"2011-01-12T07:35:34Z","abstract_excerpt":"Let $R$ be a standard graded Noetherian algebra over an Artinian local ring. 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