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For integers $k\\geq 0$, let $C{\\mathbb M}^k(X)$ denote the category of coherent ${\\CO}_X$-modules ${\\mathcal F}$, with locally free dimension $proj\\dim(\\CF)=k=grade({\\mathcal F})$. We prove that there is a zig-zag equivalence ${\\mathcal D}}^b\\left(C{\\mathbb M}^k(X)\\right) \\to {\\mathcal D}^k\\left({\\mathcal V}(X)\\right)$ of the derived categories. It follows that there is a sequence of zig-zag maps ${\\mathbb K}\\left(C{\\mathbb M}^{k+1}(X)\\right) \\"},"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":"1507.03978","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-07-14T19:40:48Z","cross_cats_sorted":["math.AG","math.KT"],"title_canon_sha256":"fa294b3df7a5e21be24f764c08b5451f9d7cfb154035ba2f5e36f7134233fdcf","abstract_canon_sha256":"fad54dc95240528a29e5d2504e48b2ecfc840beadc4618c23b1d42bffb0ef7f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:34.874368Z","signature_b64":"sowPIRi+WC9cB8i5VZCbwmOml9SH4zK7Vb5b85BjJv9QIK2IL0/Kv5bKP4fAKYsq309bxFqhHVlHHWtopkAFBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"83de4bfa3f9dfc88422120fb5f209d4879e33e8b68022cf62c836e2ec5723e10","last_reissued_at":"2026-05-18T01:33:34.873676Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:34.873676Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Witt, $GW$, $K$-theory of quasi-projective schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.KT"],"primary_cat":"math.AC","authors_text":"Satya Mandal","submitted_at":"2015-07-14T19:40:48Z","abstract_excerpt":"In this article we continue our investigation of the Derived Equivalences over noetherian quasi-projective schemes $X$, over affine schemes $\\spec{A}$. 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