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Similarly, $L_\\infty$ and $L_\\infty\\otimes L_\\infty$ are distinguished by their Hochschild homologies and so they are not Morita equivalent either. By contrast, we show that $K$-theory cannot distinguish these algebras; we have $K_*(L_2)=K_*(L_2\\otimes L_2)=0$ and $K_*(L_\\infty)=K_*(L_\\infty\\otimes L_\\infty)=K_*(k)$."},"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":"1108.0352","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2011-08-01T16:16:46Z","cross_cats_sorted":["math.KT","math.OA"],"title_canon_sha256":"32f57cba68cdb5423d607473780ee83b3f680ddc0168f828133c0a06e6a3b48e","abstract_canon_sha256":"dd0c1fe2d4d386e35718bd00aadc632b3124a9571f8592649a0fb315a605be2e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:28.677797Z","signature_b64":"j1CjMf49/i9dPXkeVttbUaSO4AU/jmkmYRk/2gBVL5WlkAUBQKC9myimbtsFqKeUleoQ2Gi1BTGaitkI7JfOAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2537250ad09cd6ea899fd307a66c980038a2a024d6bb1dd9e0734da4ffd4266b","last_reissued_at":"2026-05-18T03:05:28.677350Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:28.677350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tensor products of Leavitt path algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT","math.OA"],"primary_cat":"math.RA","authors_text":"Guillermo Corti\\~nas, Pere Ara","submitted_at":"2011-08-01T16:16:46Z","abstract_excerpt":"We compute the Hochschild homology of Leavitt path algebras over a field $k$. As an application, we show that $L_2$ and $L_2\\otimes L_2$ have different Hochschild homologies, and so they are not Morita equivalent; in particular they are not isomorphic. Similarly, $L_\\infty$ and $L_\\infty\\otimes L_\\infty$ are distinguished by their Hochschild homologies and so they are not Morita equivalent either. By contrast, we show that $K$-theory cannot distinguish these algebras; we have $K_*(L_2)=K_*(L_2\\otimes L_2)=0$ and $K_*(L_\\infty)=K_*(L_\\infty\\otimes L_\\infty)=K_*(k)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0352","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1108.0352","created_at":"2026-05-18T03:05:28.677415+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.0352v3","created_at":"2026-05-18T03:05:28.677415+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.0352","created_at":"2026-05-18T03:05:28.677415+00:00"},{"alias_kind":"pith_short_12","alias_value":"EU3SKCWQTTLO","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"EU3SKCWQTTLOVCM7","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"EU3SKCWQ","created_at":"2026-05-18T12:26:28.662955+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EU3SKCWQTTLOVCM72MD2M3EYAA","json":"https://pith.science/pith/EU3SKCWQTTLOVCM72MD2M3EYAA.json","graph_json":"https://pith.science/api/pith-number/EU3SKCWQTTLOVCM72MD2M3EYAA/graph.json","events_json":"https://pith.science/api/pith-number/EU3SKCWQTTLOVCM72MD2M3EYAA/events.json","paper":"https://pith.science/paper/EU3SKCWQ"},"agent_actions":{"view_html":"https://pith.science/pith/EU3SKCWQTTLOVCM72MD2M3EYAA","download_json":"https://pith.science/pith/EU3SKCWQTTLOVCM72MD2M3EYAA.json","view_paper":"https://pith.science/paper/EU3SKCWQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.0352&json=true","fetch_graph":"https://pith.science/api/pith-number/EU3SKCWQTTLOVCM72MD2M3EYAA/graph.json","fetch_events":"https://pith.science/api/pith-number/EU3SKCWQTTLOVCM72MD2M3EYAA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EU3SKCWQTTLOVCM72MD2M3EYAA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EU3SKCWQTTLOVCM72MD2M3EYAA/action/storage_attestation","attest_author":"https://pith.science/pith/EU3SKCWQTTLOVCM72MD2M3EYAA/action/author_attestation","sign_citation":"https://pith.science/pith/EU3SKCWQTTLOVCM72MD2M3EYAA/action/citation_signature","submit_replication":"https://pith.science/pith/EU3SKCWQTTLOVCM72MD2M3EYAA/action/replication_record"}},"created_at":"2026-05-18T03:05:28.677415+00:00","updated_at":"2026-05-18T03:05:28.677415+00:00"}