{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/73373"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/73373","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Thom-Sebastiani and duality for matrix factorizations, and results on the higher structures of the Hochschild invariants","abstract":"The derived category of a hypersurface has an action by \"cohomology operations\" k[[beta]], deg[beta] = 2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the k[[beta]]-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[[beta]]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition). These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations. We also present a viewpoint on matrix factorizations in terms of (formal) groups actions on categories that is conducive to formulating functorial statements and in particular to the computation of higher algebraic structures on Hochschild invariants. 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 Knörrer Periodicity for eliminating metabolic quadratic bundles over a base.","abstract_html":"The derived category of a hypersurface has an action by "cohomology operations" k[[beta]], deg[beta] = 2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the k[[beta]]-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[[beta]]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition). These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations. We also present a viewpoint on matrix factorizations in terms of (formal) groups actions on categories that is conducive to formulating functorial statements and in particular to the computation of higher algebraic structures on Hochschild invariants. 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 Knörrer Periodicity for eliminating metabolic quadratic bundles over a base.","abstract_has_math":false,"creators":["Preygel, Anatoly"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Jacob A. Lurie."],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012","date_published":"2012","updated_at":"2026-07-22T22:21:13Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/73373","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Jacob A. Lurie."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/73373"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph. 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We also present a viewpoint on matrix factorizations in terms of (formal) groups actions on categories that is conducive to formulating functorial statements and in particular to the computation of higher algebraic structures on Hochschild invariants. 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 Knörrer Periodicity for eliminating metabolic quadratic bundles over a base."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Thom-Sebastiani and duality for matrix factorizations, and results on the higher structures of the Hochschild invariants"]}]}],"canonical_facts":{"dc:contributor.advisor":["Jacob A. Lurie."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. 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We prove a Thom-Sebastiani type Theorem, identifying the k[[beta]]-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[[beta]]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition). These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations. We also present a viewpoint on matrix factorizations in terms of (formal) groups actions on categories that is conducive to formulating functorial statements and in particular to the computation of higher algebraic structures on Hochschild invariants. 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 Knörrer Periodicity for eliminating metabolic quadratic bundles over a base."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/73373"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Thom-Sebastiani and duality for matrix factorizations, and results on the higher structures of the Hochschild invariants"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:13Z"}