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We prove that if $A$ and $B$ are quadratic, then $C$ is quadratic if and only if the associated graded twisting map has a property we call the unique extension property. We show that $A$ and $B$ being Koszul does not imply $C$ is Koszul (or even quadratic), and we establish sufficient conditions under which $C"},"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":"1708.02514","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2017-08-08T15:09:40Z","cross_cats_sorted":[],"title_canon_sha256":"166ec0e84b5f9554cd59f8639648efa1e0f7b03d2acaeef791b2fc7d85c83ddd","abstract_canon_sha256":"98f3688c6e65e04e266123dce5210a551c4a54ee338c234f1344df7be19946ae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:16.369333Z","signature_b64":"1L7C2xqVLGnSkKQYshN3iGFFN9Lo9FrXpi4sZ1QxqFru9utdh3Q2oZXpf8WE5hF+i2sLI4VE7TrDn/D5JaldCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"abef8fda691fd13f6854d517abb63b8d652a96eb1cb573021799b222a008a289","last_reissued_at":"2026-05-18T00:14:16.368743Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:16.368743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Koszul Property for Graded Twisted Tensor Products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Andrew Conner, Peter Goetz","submitted_at":"2017-08-08T15:09:40Z","abstract_excerpt":"Let $k$ be a field. 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