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If, further, $\\mathsf{T}$ is $\\otimes$-triangulated, we say that $\\mathsf{H}$ is $\\otimes$-ideal if $F \\otimes G \\in \\mathsf{L}$ for all $G \\in \\mathsf{L}$ and all $F \\in \\mathsf{T}$. Analogously, a full triangulated subcategory $\\mathsf{C} \\subset \\mathsf{T}$ is colocalizing if it is stable for products. If, further, $\\mathsf{T}$ is closed, i.e. $\\otimes$-triangulated with internal homs (denoted $[-,-]$), we say that $\\mathsf{C}$ is $\\mathcal{H}$-"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2405.10383","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-05-16T18:23:51Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"c17d1c1ff0bb5e2c0e0828d5b237851c4b3af02f88e2e109930ec68f2eb8943f","abstract_canon_sha256":"2f49cd3f1d5192c66a39653e984a98cf639fdb17a5eceb8433cab8a5252a1dd5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:06:59.350793Z","signature_b64":"UEfTd0V7FuRYFE7L5Ktk/ORy7SQD/XKXZzipkJVwmcdEGnh4TqVBfOpLzSDsPGg4SMx6aNQuAL3cskjsIxBIAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ecea109c90ed51e41d356c2b261681772586c4026095f73e93facae274f678a","last_reissued_at":"2026-06-09T02:06:59.349800Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:06:59.349800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Localizing and colocalizing subcategories on schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AG","authors_text":"Ana Jerem\\'ias, Eduardo Loureiro, Leovigildo Alonso","submitted_at":"2024-05-16T18:23:51Z","abstract_excerpt":"A full triangulated subcategory $\\mathsf{L} \\subset \\mathsf{T}$ of triangulated category $\\mathsf{T}$ is localizing if it is stable for coproducts. If, further, $\\mathsf{T}$ is $\\otimes$-triangulated, we say that $\\mathsf{H}$ is $\\otimes$-ideal if $F \\otimes G \\in \\mathsf{L}$ for all $G \\in \\mathsf{L}$ and all $F \\in \\mathsf{T}$. Analogously, a full triangulated subcategory $\\mathsf{C} \\subset \\mathsf{T}$ is colocalizing if it is stable for products. 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