{"paper":{"title":"Thom-Sebastiani & Duality for Matrix Factorizations","arxiv_id":"1101.5834","primary_cat":"math.AG","submitted_at":"2011-01-30T23:34:40+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.99,"id":"s0","own":false,"role":"background","text":"The derived category of a hypersurface has an action by \"cohomology operations\" k[t], deg t=-2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.12,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.53,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.01,"background":0.99,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.7,"id":"s1","own":true,"role":"result","text":"We prove a Thom-Sebastiani type Theorem, identifying the k[t]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[t]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.79,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.7,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.3,"interpretation":0.0}},{"i":2,"p":0.88,"id":"s2","own":true,"role":"interpretation","text":"These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.83,"main_claim_p":0.18,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.12,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.88}},{"i":3,"p":0.74,"id":"s3","own":true,"role":"result","text":"Some applications include: we refine and establish the expected computation of 2-periodic Hochschild invariants of matrix factorizations; we show that the category of matrix factorizations is smooth, and is proper when the critical locus is proper; we show how Calabi-Yau structures on matrix factorizations arise from volume forms on the total space; we establish a version of Kn\\\"orrer Periodicity for eliminating metabolic quadratic bundles over a base.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.1,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.74,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.25,"interpretation":0.01}}]},"compiled":{"main_claim":{"p":0.98,"text":"We prove a Thom-Sebastiani type Theorem, identifying the k[t]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[t]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition).","location":"abstract s1","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"OpenRouter","model":"classifier-1.13","endpoint":"https://openrouter.ai/api/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":3882,"cost_usd_reported":0.000163},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13-20260917","abstract_states":{"claims_first":0.05,"states_limitation":0.12,"human_participants":0.01,"controlled_comparison":0.01,"randomized_assignment":0.01,"states_quantitative_result":0.06,"compares_with_prior_results":0.25,"mentions_code_or_data_availability":0.01},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"theoretical result","probabilities":{"review":0.0,"method_or_tool":0.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":1.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":1.0,"text":"We prove a Thom-Sebastiani type Theorem, identifying the k[t]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[t]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition).","location":"abstract s1"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":4,"sentences_in_abstract":4},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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