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The main purpose of this article is to investigate that the natural homomorphisms Tor$^R_c(k,H^c_I(M))\\to k\\otimes_R M$ and Ext$^{d}_R(k,H^c_I(M))\\to {\\rm Ext}^t_R(k, M)$ are non-zero where $d:=t-c$. In fact for a Cohen-Macaulay module $M$ we will show that the homomorphism Ext$^d_R(k,H^c_I(M))\\to {\\rm Ext}^t_"},"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":"1406.7461","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2014-06-29T04:58:51Z","cross_cats_sorted":[],"title_canon_sha256":"9a0be762d40f1ac682432e6df746ce57392c81041cf43cb01236d7d8131cb550","abstract_canon_sha256":"00b0ded544ba590f4b9454f747b6e4f22f44c58d3d193051a8b71e8b642f77b0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:45.243596Z","signature_b64":"xjXzmSOUj0kp3UmSoXcn00PpjVbY4ou1oSDMF+tmc0Qg0vekPBKomvHKtX9cE38S416pttzeS9lI+bDaQ/npCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ceb5bfb80079b088a40ed0e031bb88b7938766357f3dbb64da4f2d394738752","last_reissued_at":"2026-05-18T02:48:45.242973Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:45.242973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On natural homomorphisms of local cohomology modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Waqas Mahmood","submitted_at":"2014-06-29T04:58:51Z","abstract_excerpt":"Let $M$ be a non-zero finitely generated module over a finite dimensional commutative Noetherian local ring $(R,\\mathfrak{m})$ with dim$_R(M)=t$. 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