{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:YCDVJ5AA4WAZCYG3YCAJ7N5MKZ","short_pith_number":"pith:YCDVJ5AA","canonical_record":{"source":{"id":"1210.4785","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-17T16:26:14Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"09093946a2a40a8e26bcd9b047bf58345d0433a786f7ebc8db6d03c2b0d548e6","abstract_canon_sha256":"d30f83108948a12122f29e9075986fd1885b198c8696a2a253d7f30e1bafe345"},"schema_version":"1.0"},"canonical_sha256":"c08754f400e5819160dbc0809fb7ac5643bb31b32292f115fb229bf833a9821f","source":{"kind":"arxiv","id":"1210.4785","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.4785","created_at":"2026-05-18T02:59:44Z"},{"alias_kind":"arxiv_version","alias_value":"1210.4785v2","created_at":"2026-05-18T02:59:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.4785","created_at":"2026-05-18T02:59:44Z"},{"alias_kind":"pith_short_12","alias_value":"YCDVJ5AA4WAZ","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YCDVJ5AA4WAZCYG3","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YCDVJ5AA","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:YCDVJ5AA4WAZCYG3YCAJ7N5MKZ","target":"record","payload":{"canonical_record":{"source":{"id":"1210.4785","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-17T16:26:14Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"09093946a2a40a8e26bcd9b047bf58345d0433a786f7ebc8db6d03c2b0d548e6","abstract_canon_sha256":"d30f83108948a12122f29e9075986fd1885b198c8696a2a253d7f30e1bafe345"},"schema_version":"1.0"},"canonical_sha256":"c08754f400e5819160dbc0809fb7ac5643bb31b32292f115fb229bf833a9821f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:59:44.983209Z","signature_b64":"+wDNJS0/sVrv636ydqMANsXhQwaMVqUuLcVnMvf51tAnBx5L2bR/mwW3t9ZsWpRTkQ/D2pBo6dKLOoG++c0cDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c08754f400e5819160dbc0809fb7ac5643bb31b32292f115fb229bf833a9821f","last_reissued_at":"2026-05-18T02:59:44.982349Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:59:44.982349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.4785","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:59:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B4QNfnmUW1vgovfZmYy8PJ3d0lvG4QoZRLVUWj138qJ6jeLpYWkvI0lkfc7pIwINOd30Xh852rdAx5Ya/O/mAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T10:19:03.357557Z"},"content_sha256":"aff9862aec1672bb906bcc627c1ca0d9f3cadd3f084800c9ae87c63ab7288d5e","schema_version":"1.0","event_id":"sha256:aff9862aec1672bb906bcc627c1ca0d9f3cadd3f084800c9ae87c63ab7288d5e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:YCDVJ5AA4WAZCYG3YCAJ7N5MKZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Projective dimension in filtrated K-theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.OA","authors_text":"Rasmus Bentmann","submitted_at":"2012-10-17T16:26:14Z","abstract_excerpt":"Under mild assumptions, we characterise modules with projective resolutions of length n in the target category of filtrated K-theory over a finite topological space in terms of two conditions involving certain Tor-groups. We show that the filtrated K-theory of any separable C*-algebra over any topological space with at most four points has projective dimension 2 or less. We observe that this implies a universal coefficient theorem for rational equivariant KK-theory over these spaces. As a contrasting example, we find a separable C*-algebra in the bootstrap class over a certain five-point space"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.4785","kind":"arxiv","version":2},"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":"2026-05-18T02:59:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kIF83wqmaq1h3/MsWs6xKv3vlaJmc4F452W2asSm4RUNuW61y8odAY+4mGtRPuq7E1XqApaOnGhJXLDLp9cQDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T10:19:03.357895Z"},"content_sha256":"3d5c32a7ff0142778af94c9e184c5acc4f9ccde4556c552f6c5c1162b4998f28","schema_version":"1.0","event_id":"sha256:3d5c32a7ff0142778af94c9e184c5acc4f9ccde4556c552f6c5c1162b4998f28"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YCDVJ5AA4WAZCYG3YCAJ7N5MKZ/bundle.json","state_url":"https://pith.science/pith/YCDVJ5AA4WAZCYG3YCAJ7N5MKZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YCDVJ5AA4WAZCYG3YCAJ7N5MKZ/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":"2026-06-12T10:19:03Z","links":{"resolver":"https://pith.science/pith/YCDVJ5AA4WAZCYG3YCAJ7N5MKZ","bundle":"https://pith.science/pith/YCDVJ5AA4WAZCYG3YCAJ7N5MKZ/bundle.json","state":"https://pith.science/pith/YCDVJ5AA4WAZCYG3YCAJ7N5MKZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YCDVJ5AA4WAZCYG3YCAJ7N5MKZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:YCDVJ5AA4WAZCYG3YCAJ7N5MKZ","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":"d30f83108948a12122f29e9075986fd1885b198c8696a2a253d7f30e1bafe345","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-17T16:26:14Z","title_canon_sha256":"09093946a2a40a8e26bcd9b047bf58345d0433a786f7ebc8db6d03c2b0d548e6"},"schema_version":"1.0","source":{"id":"1210.4785","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.4785","created_at":"2026-05-18T02:59:44Z"},{"alias_kind":"arxiv_version","alias_value":"1210.4785v2","created_at":"2026-05-18T02:59:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.4785","created_at":"2026-05-18T02:59:44Z"},{"alias_kind":"pith_short_12","alias_value":"YCDVJ5AA4WAZ","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YCDVJ5AA4WAZCYG3","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YCDVJ5AA","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:3d5c32a7ff0142778af94c9e184c5acc4f9ccde4556c552f6c5c1162b4998f28","target":"graph","created_at":"2026-05-18T02:59:44Z","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":"Under mild assumptions, we characterise modules with projective resolutions of length n in the target category of filtrated K-theory over a finite topological space in terms of two conditions involving certain Tor-groups. We show that the filtrated K-theory of any separable C*-algebra over any topological space with at most four points has projective dimension 2 or less. We observe that this implies a universal coefficient theorem for rational equivariant KK-theory over these spaces. As a contrasting example, we find a separable C*-algebra in the bootstrap class over a certain five-point space","authors_text":"Rasmus Bentmann","cross_cats":["math.KT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-17T16:26:14Z","title":"Projective dimension in filtrated K-theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.4785","kind":"arxiv","version":2},"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:aff9862aec1672bb906bcc627c1ca0d9f3cadd3f084800c9ae87c63ab7288d5e","target":"record","created_at":"2026-05-18T02:59:44Z","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":"d30f83108948a12122f29e9075986fd1885b198c8696a2a253d7f30e1bafe345","cross_cats_sorted":["math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OA","submitted_at":"2012-10-17T16:26:14Z","title_canon_sha256":"09093946a2a40a8e26bcd9b047bf58345d0433a786f7ebc8db6d03c2b0d548e6"},"schema_version":"1.0","source":{"id":"1210.4785","kind":"arxiv","version":2}},"canonical_sha256":"c08754f400e5819160dbc0809fb7ac5643bb31b32292f115fb229bf833a9821f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c08754f400e5819160dbc0809fb7ac5643bb31b32292f115fb229bf833a9821f","first_computed_at":"2026-05-18T02:59:44.982349Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:59:44.982349Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+wDNJS0/sVrv636ydqMANsXhQwaMVqUuLcVnMvf51tAnBx5L2bR/mwW3t9ZsWpRTkQ/D2pBo6dKLOoG++c0cDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:59:44.983209Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.4785","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aff9862aec1672bb906bcc627c1ca0d9f3cadd3f084800c9ae87c63ab7288d5e","sha256:3d5c32a7ff0142778af94c9e184c5acc4f9ccde4556c552f6c5c1162b4998f28"],"state_sha256":"03849f2afa5a787a4a1b62b3d10bddb27e38b1de6a354ef5405a811d8555f28e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Jq9l/DWkQAXXmYGwdLUg3HfkE6i1NF3bGVRc8w2Rl4ZXUZFoAkfd4Cy3xqTEKjstEmHGLuabFWglfbJbl7VcBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T10:19:03.359774Z","bundle_sha256":"c981cc98db240aad5a1952445a913689d1f8d0febea013bafd264803068a8d7a"}}