{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:YLFLLHDMQQKIHL2MAMV6FS3O66","short_pith_number":"pith:YLFLLHDM","canonical_record":{"source":{"id":"1004.0918","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2010-04-06T18:15:37Z","cross_cats_sorted":["math.AT","math.OA","math.RA"],"title_canon_sha256":"2824aa7b3ad16fa597a0bb9d6468cadac2ee32f311745b220ba4b6574702ddeb","abstract_canon_sha256":"bc7a08950924741a84d8fa275703b0c68ff321c34f6ba145f5ffd17eee1d0813"},"schema_version":"1.0"},"canonical_sha256":"c2cab59c6c841483af4c032be2cb6ef79a96e3d1371637e39d018c5096577a4a","source":{"kind":"arxiv","id":"1004.0918","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.0918","created_at":"2026-05-18T02:34:45Z"},{"alias_kind":"arxiv_version","alias_value":"1004.0918v5","created_at":"2026-05-18T02:34:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.0918","created_at":"2026-05-18T02:34:45Z"},{"alias_kind":"pith_short_12","alias_value":"YLFLLHDMQQKI","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YLFLLHDMQQKIHL2M","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YLFLLHDM","created_at":"2026-05-18T12:26:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:YLFLLHDMQQKIHL2MAMV6FS3O66","target":"record","payload":{"canonical_record":{"source":{"id":"1004.0918","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2010-04-06T18:15:37Z","cross_cats_sorted":["math.AT","math.OA","math.RA"],"title_canon_sha256":"2824aa7b3ad16fa597a0bb9d6468cadac2ee32f311745b220ba4b6574702ddeb","abstract_canon_sha256":"bc7a08950924741a84d8fa275703b0c68ff321c34f6ba145f5ffd17eee1d0813"},"schema_version":"1.0"},"canonical_sha256":"c2cab59c6c841483af4c032be2cb6ef79a96e3d1371637e39d018c5096577a4a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:34:45.284284Z","signature_b64":"VayHjXMXBnILKb0OLF2DbuJPbJJFfSEFpUc8q/p0Ki38xj/Bmfpu46gjD4boFH7YI0KuD02weKIgUaFGNmjSAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2cab59c6c841483af4c032be2cb6ef79a96e3d1371637e39d018c5096577a4a","last_reissued_at":"2026-05-18T02:34:45.283887Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:34:45.283887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.0918","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:34:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KmkRTrUISHUk94vn11MjO5mKaKLdSiiyZGRPgo67cJ3EccE0SEQCggUSrnOY1eoIS/43ON811dFFmVj9Fak0Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T03:39:55.832119Z"},"content_sha256":"52169bcaf6cdbee545e4a11a4f49865dc67053bb0f3fe3f407f547d78534f4ce","schema_version":"1.0","event_id":"sha256:52169bcaf6cdbee545e4a11a4f49865dc67053bb0f3fe3f407f547d78534f4ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:YLFLLHDMQQKIHL2MAMV6FS3O66","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algebraic Kasparov K-theory. I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.OA","math.RA"],"primary_cat":"math.KT","authors_text":"Grigory Garkusha","submitted_at":"2010-04-06T18:15:37Z","abstract_excerpt":"This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov $K$-theory spectra of $k$-algebras. These are shown to be homotopy invariant, excisive in each variable $K$-theories. We prove that the spectra represent universal unstable, Morita stable and stable bivariant homology theories respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0918","kind":"arxiv","version":5},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:34:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AZeA2pBRCoLhIBMH6ONSTxh8M0zmtHzA8rj4D2Ak48R09U/mdoQ6aPpCOX9Y/dPcdNZ/JV8pQlAVpc+uNBBeAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T03:39:55.832467Z"},"content_sha256":"13d19becc6dd7e405fdcbab8edd0ca13f393a3c36ba072c3b9877a33e49383e8","schema_version":"1.0","event_id":"sha256:13d19becc6dd7e405fdcbab8edd0ca13f393a3c36ba072c3b9877a33e49383e8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YLFLLHDMQQKIHL2MAMV6FS3O66/bundle.json","state_url":"https://pith.science/pith/YLFLLHDMQQKIHL2MAMV6FS3O66/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YLFLLHDMQQKIHL2MAMV6FS3O66/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-29T03:39:55Z","links":{"resolver":"https://pith.science/pith/YLFLLHDMQQKIHL2MAMV6FS3O66","bundle":"https://pith.science/pith/YLFLLHDMQQKIHL2MAMV6FS3O66/bundle.json","state":"https://pith.science/pith/YLFLLHDMQQKIHL2MAMV6FS3O66/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YLFLLHDMQQKIHL2MAMV6FS3O66/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:YLFLLHDMQQKIHL2MAMV6FS3O66","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"bc7a08950924741a84d8fa275703b0c68ff321c34f6ba145f5ffd17eee1d0813","cross_cats_sorted":["math.AT","math.OA","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2010-04-06T18:15:37Z","title_canon_sha256":"2824aa7b3ad16fa597a0bb9d6468cadac2ee32f311745b220ba4b6574702ddeb"},"schema_version":"1.0","source":{"id":"1004.0918","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.0918","created_at":"2026-05-18T02:34:45Z"},{"alias_kind":"arxiv_version","alias_value":"1004.0918v5","created_at":"2026-05-18T02:34:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.0918","created_at":"2026-05-18T02:34:45Z"},{"alias_kind":"pith_short_12","alias_value":"YLFLLHDMQQKI","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"YLFLLHDMQQKIHL2M","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"YLFLLHDM","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:13d19becc6dd7e405fdcbab8edd0ca13f393a3c36ba072c3b9877a33e49383e8","target":"graph","created_at":"2026-05-18T02:34:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper is to construct unstable, Morita stable and stable bivariant algebraic Kasparov $K$-theory spectra of $k$-algebras. These are shown to be homotopy invariant, excisive in each variable $K$-theories. We prove that the spectra represent universal unstable, Morita stable and stable bivariant homology theories respectively.","authors_text":"Grigory Garkusha","cross_cats":["math.AT","math.OA","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2010-04-06T18:15:37Z","title":"Algebraic Kasparov K-theory. I"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0918","kind":"arxiv","version":5},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:52169bcaf6cdbee545e4a11a4f49865dc67053bb0f3fe3f407f547d78534f4ce","target":"record","created_at":"2026-05-18T02:34:45Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"bc7a08950924741a84d8fa275703b0c68ff321c34f6ba145f5ffd17eee1d0813","cross_cats_sorted":["math.AT","math.OA","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2010-04-06T18:15:37Z","title_canon_sha256":"2824aa7b3ad16fa597a0bb9d6468cadac2ee32f311745b220ba4b6574702ddeb"},"schema_version":"1.0","source":{"id":"1004.0918","kind":"arxiv","version":5}},"canonical_sha256":"c2cab59c6c841483af4c032be2cb6ef79a96e3d1371637e39d018c5096577a4a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c2cab59c6c841483af4c032be2cb6ef79a96e3d1371637e39d018c5096577a4a","first_computed_at":"2026-05-18T02:34:45.283887Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:34:45.283887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VayHjXMXBnILKb0OLF2DbuJPbJJFfSEFpUc8q/p0Ki38xj/Bmfpu46gjD4boFH7YI0KuD02weKIgUaFGNmjSAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:34:45.284284Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.0918","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:52169bcaf6cdbee545e4a11a4f49865dc67053bb0f3fe3f407f547d78534f4ce","sha256:13d19becc6dd7e405fdcbab8edd0ca13f393a3c36ba072c3b9877a33e49383e8"],"state_sha256":"668ca3553c77074faa7a96af9b26403dc4d611de64df4f75a94918d376b50d74"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gR8hfObmouJFT1Bk1OqRuvmuS5ugtRo+GCD+dL43/pIeLbcF5uYnVj7xdZjdOujfgAS3mVvZgq91inSnYYSwDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T03:39:55.834306Z","bundle_sha256":"6c846af5b601474e29d7e6f9d7921e8f0f33e48e993f014678adfb16d613d339"}}