{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2003:2E3W4G6AKDRLLIWFXKV67KBFCD","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":"2defb5e27570fd9d47e0ce05f7f15850abe3d1ebf9bb25ce616c9402948559ed","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2003-10-10T12:38:58Z","title_canon_sha256":"97f170ba9bec197d1aafc06834bf3e83f6f7138ce841dbd950c461494c27ad9e"},"schema_version":"1.0","source":{"id":"math/0310146","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0310146","created_at":"2026-07-05T00:32:45Z"},{"alias_kind":"arxiv_version","alias_value":"math/0310146v1","created_at":"2026-07-05T00:32:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0310146","created_at":"2026-07-05T00:32:45Z"},{"alias_kind":"pith_short_12","alias_value":"2E3W4G6AKDRL","created_at":"2026-07-05T00:32:45Z"},{"alias_kind":"pith_short_16","alias_value":"2E3W4G6AKDRLLIWF","created_at":"2026-07-05T00:32:45Z"},{"alias_kind":"pith_short_8","alias_value":"2E3W4G6A","created_at":"2026-07-05T00:32:45Z"}],"graph_snapshots":[{"event_id":"sha256:828395100b22e4ffe0ec747787f5cd9338a458487bb8560c64ab9374b86bcf0e","target":"graph","created_at":"2026-07-05T00:32:45Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0310146/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This is a survey paper, based on lectures given at the Workshop on \"Structured ring spectra and their applications\" which took place January 21-25, 2002, at the University of Glasgow.\n  The term `Morita theory' is usually used for results concerning equivalences of various kinds of module categories. We focus on the covariant form of Morita theory, so the basic question is: When do two `rings' have `equivalent' module categories ?\n  We discuss this question in different contexts and illustrate it by examples:\n  (Classical) When are the module categories of two rings equivalent as categories ?\n","authors_text":"Stefan Schwede","cross_cats":[],"headline":"","license":"","primary_cat":"math.AT","submitted_at":"2003-10-10T12:38:58Z","title":"Morita theory in abelian, derived and stable model categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0310146","kind":"arxiv","version":1},"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":"sha256:01cfec53d35e25b4eca0ba221fa7f6cd4197178338d2513c56d2d7060ead6ec1","target":"record","created_at":"2026-07-05T00:32:45Z","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":"2defb5e27570fd9d47e0ce05f7f15850abe3d1ebf9bb25ce616c9402948559ed","cross_cats_sorted":[],"license":"","primary_cat":"math.AT","submitted_at":"2003-10-10T12:38:58Z","title_canon_sha256":"97f170ba9bec197d1aafc06834bf3e83f6f7138ce841dbd950c461494c27ad9e"},"schema_version":"1.0","source":{"id":"math/0310146","kind":"arxiv","version":1}},"canonical_sha256":"d1376e1bc050e2b5a2c5baabefa82510c33b8b3ed9f17fa7364f958c27f77b85","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d1376e1bc050e2b5a2c5baabefa82510c33b8b3ed9f17fa7364f958c27f77b85","first_computed_at":"2026-07-05T00:32:45.557405Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:32:45.557405Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q6WGwkpsZpnH2spsvwI/2uSVJN9BhmeblZhjmk/6QSEn1a1OtuTUYR7qVofyFhPo6Q5Rufqz7TFbm+TW93X3Bw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:32:45.557899Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0310146","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:01cfec53d35e25b4eca0ba221fa7f6cd4197178338d2513c56d2d7060ead6ec1","sha256:828395100b22e4ffe0ec747787f5cd9338a458487bb8560c64ab9374b86bcf0e"],"state_sha256":"d5859802b67439fad718f89c2634343f4f670d804f38fb665d1fd2430796d686"}