{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:XVNULHUVYBZI32PXARTHB32MX7","short_pith_number":"pith:XVNULHUV","canonical_record":{"source":{"id":"1304.2520","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-04-09T10:33:45Z","cross_cats_sorted":[],"title_canon_sha256":"7a75b45dd5f05f8a5704fee2fefe049c2917d046e2e09a2add5e9bf417426c05","abstract_canon_sha256":"4b3462e61b66007371ad12c4387920cd3dc08468c9f11750bc861f587719635d"},"schema_version":"1.0"},"canonical_sha256":"bd5b459e95c0728de9f7046670ef4cbfe8b722b21b53b19f64feb807fc5493e5","source":{"kind":"arxiv","id":"1304.2520","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.2520","created_at":"2026-05-18T00:45:35Z"},{"alias_kind":"arxiv_version","alias_value":"1304.2520v2","created_at":"2026-05-18T00:45:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.2520","created_at":"2026-05-18T00:45:35Z"},{"alias_kind":"pith_short_12","alias_value":"XVNULHUVYBZI","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XVNULHUVYBZI32PX","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XVNULHUV","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:XVNULHUVYBZI32PXARTHB32MX7","target":"record","payload":{"canonical_record":{"source":{"id":"1304.2520","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-04-09T10:33:45Z","cross_cats_sorted":[],"title_canon_sha256":"7a75b45dd5f05f8a5704fee2fefe049c2917d046e2e09a2add5e9bf417426c05","abstract_canon_sha256":"4b3462e61b66007371ad12c4387920cd3dc08468c9f11750bc861f587719635d"},"schema_version":"1.0"},"canonical_sha256":"bd5b459e95c0728de9f7046670ef4cbfe8b722b21b53b19f64feb807fc5493e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:35.449951Z","signature_b64":"VT3me2Wb82wDmLD2JNEn67se7VN+6Im9DvdLYJz5rjP0tFZhraD53KrtL4btUVDFETd9GDLwQu5xwK28e/GxBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd5b459e95c0728de9f7046670ef4cbfe8b722b21b53b19f64feb807fc5493e5","last_reissued_at":"2026-05-18T00:45:35.449364Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:35.449364Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.2520","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":"2026-05-18T00:45:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e1xxh4ENtQ6Z6PBF17yC+aYgTbLCJGCvxVgZhbShOQB6RdSXWaaT1385Zy4BM0U7Xohl2jyACtVAiF+bqqXjDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T01:56:14.784876Z"},"content_sha256":"87b0ec3be525440b6061105f934de97cde0e8d94ddad067da6c4610188bc659d","schema_version":"1.0","event_id":"sha256:87b0ec3be525440b6061105f934de97cde0e8d94ddad067da6c4610188bc659d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:XVNULHUVYBZI32PXARTHB32MX7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A functorial formalism for quasi-coherent sheaves on a geometric stack","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Ana Jeremias, Leovigildo Alonso, Maria J. Vale, Marta Perez","submitted_at":"2013-04-09T10:33:45Z","abstract_excerpt":"A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category.\n  We associate, in a 2-functorial way, to a 1-morphism of geometric stacks $f$, an adjunction $(f^*, f_*)$ for the corresponding categories of quasi-coherent sheaves that agrees"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.2520","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":"2026-05-18T00:45:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AxIAzJx8Nk5SMFThgLtuLHzFmC6qNqraO26fOKlHNg1SA52a8NC/cYripKBsuGMBf94NSoAT8ednO0HtZu7ZAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T01:56:14.785565Z"},"content_sha256":"3e937368dc072dad952d7718260b3fae42f96b32d2fdd7f2f88d2d0945ca9fc9","schema_version":"1.0","event_id":"sha256:3e937368dc072dad952d7718260b3fae42f96b32d2fdd7f2f88d2d0945ca9fc9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XVNULHUVYBZI32PXARTHB32MX7/bundle.json","state_url":"https://pith.science/pith/XVNULHUVYBZI32PXARTHB32MX7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XVNULHUVYBZI32PXARTHB32MX7/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":"2026-06-05T01:56:14Z","links":{"resolver":"https://pith.science/pith/XVNULHUVYBZI32PXARTHB32MX7","bundle":"https://pith.science/pith/XVNULHUVYBZI32PXARTHB32MX7/bundle.json","state":"https://pith.science/pith/XVNULHUVYBZI32PXARTHB32MX7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XVNULHUVYBZI32PXARTHB32MX7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XVNULHUVYBZI32PXARTHB32MX7","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":"4b3462e61b66007371ad12c4387920cd3dc08468c9f11750bc861f587719635d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-04-09T10:33:45Z","title_canon_sha256":"7a75b45dd5f05f8a5704fee2fefe049c2917d046e2e09a2add5e9bf417426c05"},"schema_version":"1.0","source":{"id":"1304.2520","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.2520","created_at":"2026-05-18T00:45:35Z"},{"alias_kind":"arxiv_version","alias_value":"1304.2520v2","created_at":"2026-05-18T00:45:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.2520","created_at":"2026-05-18T00:45:35Z"},{"alias_kind":"pith_short_12","alias_value":"XVNULHUVYBZI","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XVNULHUVYBZI32PX","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XVNULHUV","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:3e937368dc072dad952d7718260b3fae42f96b32d2fdd7f2f88d2d0945ca9fc9","target":"graph","created_at":"2026-05-18T00:45:35Z","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":"A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category.\n  We associate, in a 2-functorial way, to a 1-morphism of geometric stacks $f$, an adjunction $(f^*, f_*)$ for the corresponding categories of quasi-coherent sheaves that agrees","authors_text":"Ana Jeremias, Leovigildo Alonso, Maria J. Vale, Marta Perez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-04-09T10:33:45Z","title":"A functorial formalism for quasi-coherent sheaves on a geometric stack"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.2520","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":"sha256:87b0ec3be525440b6061105f934de97cde0e8d94ddad067da6c4610188bc659d","target":"record","created_at":"2026-05-18T00:45:35Z","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":"4b3462e61b66007371ad12c4387920cd3dc08468c9f11750bc861f587719635d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-04-09T10:33:45Z","title_canon_sha256":"7a75b45dd5f05f8a5704fee2fefe049c2917d046e2e09a2add5e9bf417426c05"},"schema_version":"1.0","source":{"id":"1304.2520","kind":"arxiv","version":2}},"canonical_sha256":"bd5b459e95c0728de9f7046670ef4cbfe8b722b21b53b19f64feb807fc5493e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd5b459e95c0728de9f7046670ef4cbfe8b722b21b53b19f64feb807fc5493e5","first_computed_at":"2026-05-18T00:45:35.449364Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:35.449364Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VT3me2Wb82wDmLD2JNEn67se7VN+6Im9DvdLYJz5rjP0tFZhraD53KrtL4btUVDFETd9GDLwQu5xwK28e/GxBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:35.449951Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.2520","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:87b0ec3be525440b6061105f934de97cde0e8d94ddad067da6c4610188bc659d","sha256:3e937368dc072dad952d7718260b3fae42f96b32d2fdd7f2f88d2d0945ca9fc9"],"state_sha256":"fdea588f39d1337bc954c569181615282c7cc6f63985fc2141b46a5bd2e809ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Vqz3LBRcIBjtTM5UnEN/rQB7gRVQGJTA/xDdgyHz/q1V1ffP4z7yV9AnKCGlecKLET8G272SfGl8oWoguipTDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T01:56:14.788561Z","bundle_sha256":"9c5d32000cb497e1cc3c737b0a95b64acb0a78ae3159561d72dc7b36a79b1374"}}