Thom-Sebastiani & Duality for Matrix Factorizations

Abstract

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). 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). These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations. 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.

78 pages. Draft

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• Coherent Springer theory and the categorical Deligne-Langlands correspondence
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• Motivic Realizations of Singularity Categories and Vanishing Cycles
• Beilinson-Bernstein localization over the Harish-Chandra center
• Homological mirror symmetry for punctured spheres
• Calabi-Yau structures on categories of matrix factorizations
• Morita equivalence for convolution categories: Appendix to arXiv:0805.0157
• Cyclic symmetries of A_n-quiver representations
• The moduli stack and motivic Hall algebra for the bounded derived category
• Equivariant localization and completion in cyclic homology and derived loop spaces
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• Fukaya categories of higher-genus surfaces and pants decompositions
• Line fields on punctured surfaces and twisted derived categories
• The $\ell$-adic trace formula for dg-categories and Bloch's conductor conjecture
• A Note on Semi-Infinite Non-Commutative Hodge Theory and LG-Models
• Dirac Families for Loop Groups as Matrix Factorizations
• On D-modules of categories I