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Furthermore, by using the classification result and the theory of Balmer's tensor triangular geometry, we show that the spectrum of the tensor triangulated category $({\\rm DMF}(X,L,W), \\otimes^{\\frac{1}{2}})$ is homeomorphic to the relative singular locus ${\\rm Sing}(X_0/X)$, introduced in this paper, of the z"},"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":"1701.02573","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-01-10T13:11:10Z","cross_cats_sorted":["math.CT","math.RT"],"title_canon_sha256":"cc183a87d6115459a73f76305797a5c930e20b71592267dd78780e6d09a7cc8e","abstract_canon_sha256":"3d73fb6e83afc6ddc60dd225d8979006e859d24baa55d5919ba66f4075165ccc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:39.680904Z","signature_b64":"7Mpasip+Z4rFHOLiodihkpyfAgnC8woMnIM+h8V24SI7VnUmeCESQzv+AjQ2ruCAeo8tSqpSyzHfA6CfDwXMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85b7c15eccb6579fd2a2579aab53dafb3199574c1d37c2af2be79b1bbad7520d","last_reissued_at":"2026-05-18T00:25:39.680189Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:39.680189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Relative singular locus and Balmer spectrum of matrix factorizations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.RT"],"primary_cat":"math.AG","authors_text":"Yuki Hirano","submitted_at":"2017-01-10T13:11:10Z","abstract_excerpt":"For a separated Noetherian scheme $X$ with an ample family of line bundles and a non-zero-divisor $W\\in\\Gamma(X,L)$ of a line bundle $L$ on $X$, we classify certain thick subcategories of the derived matrix factorization category ${\\rm DMF}(X,L,W)$ of the Landau-Ginzburg model $(X,L,W)$. 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