{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:CTBBU3EPMRWBFGEDY3F4HK3JPK","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":"2d88d26c7d9b043ffa94696e5c604a84d1c8aca5fa0bb32eede1b5726e2293c2","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2011-05-13T12:51:35Z","title_canon_sha256":"85a94e06f73fc136e0db53c0a32ac2a9ac658a4567877b109c6a62728f670315"},"schema_version":"1.0","source":{"id":"1105.2715","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.2715","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"arxiv_version","alias_value":"1105.2715v2","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.2715","created_at":"2026-05-18T02:58:00Z"},{"alias_kind":"pith_short_12","alias_value":"CTBBU3EPMRWB","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"CTBBU3EPMRWBFGED","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"CTBBU3EP","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:476905b5ac0ccce94814f49112766e542d981d846d9230795d3513e644ad9eda","target":"graph","created_at":"2026-05-18T02:58:00Z","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":"For a certain class of abelian categories, we show how to make sense of the \"Euler characteristic\" of an infinite projective resolution (or, more generally, certain chain complexes that are only bounded above), by passing to a suitable completion of the Grothendieck group. We also show that right-exact functors (or their left-derived functors) induce continuous homomorphisms of these completed Grothendieck groups, and we discuss examples and applications coming from categorification.","authors_text":"Catharina Stroppel, Pramod N. Achar","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2011-05-13T12:51:35Z","title":"Completions of Grothendieck groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2715","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:e5b9a1ea5a8d42afc7775a46e9cab812d8bae92519a28b5117fb7f32d2bbcac2","target":"record","created_at":"2026-05-18T02:58:00Z","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":"2d88d26c7d9b043ffa94696e5c604a84d1c8aca5fa0bb32eede1b5726e2293c2","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2011-05-13T12:51:35Z","title_canon_sha256":"85a94e06f73fc136e0db53c0a32ac2a9ac658a4567877b109c6a62728f670315"},"schema_version":"1.0","source":{"id":"1105.2715","kind":"arxiv","version":2}},"canonical_sha256":"14c21a6c8f646c129883c6cbc3ab697a838253d8ae122577405f5df2f8451d71","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14c21a6c8f646c129883c6cbc3ab697a838253d8ae122577405f5df2f8451d71","first_computed_at":"2026-05-18T02:58:00.903239Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:00.903239Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BAXCQNN630vy5CJ4OIxHChqm9YboG6/EFgaYHswFaRu7xkeBVBVH9Pse36K9oxRrdiqXCdWMV3O85rBJe3W7Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:00.903717Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.2715","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e5b9a1ea5a8d42afc7775a46e9cab812d8bae92519a28b5117fb7f32d2bbcac2","sha256:476905b5ac0ccce94814f49112766e542d981d846d9230795d3513e644ad9eda"],"state_sha256":"66cc5f91fa1409fa345652bd3419827fd998e6756327031c5f132cb8cbc1b95c"}