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In this paper we study the graded structure of the $i$-th local cohomology module of $M$ defined by a pair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. More precisely, we discuss finiteness property and vanishing of the graded components $H^{i}_{R_{+},J}(M)_{n}$.\n  Also, we study the Artinian property and tameness of certain submodules and quotient modules of $H^{i}_{R_{+},J}(M)$."},"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":"1502.04970","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-02-17T17:36:40Z","cross_cats_sorted":[],"title_canon_sha256":"5bb4789088657498f407e0af169b257635e50767d7c6dad09a490511b51f5c70","abstract_canon_sha256":"cf860df37fbda131418a6a8cdf47fcb55adcfd5576a9d32323c37c8186189ba9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:53.410464Z","signature_b64":"FqfV8sb2LQVZjLq5EEauKn6RjkS+yNGsrCoEs7XK/m7cXweum9h9oHxSfCU3f27IkaSB5pUtCXxulVAHbUc3AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"632488f39b282995ce455e27596b6ebb47858b150edcf9ea21d03fa725db89ed","last_reissued_at":"2026-05-18T02:26:53.410065Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:53.410065Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On graded local cohomology modules defined by a pair of ideals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Kh. 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