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Write $N^t_E$ and 1 for the matrices $\\in \\Z^{(E_0\\times E_0\\setminus\\Sink(E))}$ which result from $N'^t_E$ and from the identity matrix after removing the columns corresponding to sinks. We consider the $K$-theory of the Leavitt algebra $L_R(E)=L_\\Z(E)\\otimes R$. We show that if $R$ is either a Noetherian regular ring or a stable $C^*$-algebra, then there is an exact sequence ($n\\in\\Z$) \\[ K_n(R)^{(E_0\\setminus"},"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":"0903.0056","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2009-02-28T09:25:42Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"147adb83956ceae36e2e24723da84d81ab32358eb363c3586b9e200fb122fa98","abstract_canon_sha256":"8e1b9ed3038dbb45cb3d8bd4c225b0cd9b467b5db938f9442d0f3dfd80ef4106"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:31.007311Z","signature_b64":"sYikZ2hz1p8PaZ3aNhRaWBlx7fnlWo6Y+eSgegx7AThCDMn2Q/RPSsCgBZlFm+xgpweT7tNvtp/r8BF6Zwr6Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09879117da9d3c9ed257316e4402029ce111c6f964271619281f576f094f56e6","last_reissued_at":"2026-05-18T04:16:31.006858Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:31.006858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$K$-theory of Leavitt path algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.KT","authors_text":"Guillermo Corti\\~nas, Miquel Brustenga, Pere Ara","submitted_at":"2009-02-28T09:25:42Z","abstract_excerpt":"Let $E$ be a row-finite quiver and let $E_0$ be the set of vertices of $E$; consider the adjacency matrix $N'_E=(n_{ij})\\in\\Z^{(E_0\\times E_0)}$, $n_{ij}=#\\{$ arrows from $i$ to $j\\}$. Write $N^t_E$ and 1 for the matrices $\\in \\Z^{(E_0\\times E_0\\setminus\\Sink(E))}$ which result from $N'^t_E$ and from the identity matrix after removing the columns corresponding to sinks. We consider the $K$-theory of the Leavitt algebra $L_R(E)=L_\\Z(E)\\otimes R$. 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