Thom-Sebastiani & Duality for Matrix Factorizations

By Anatoly Preygel. 2011

51 citations recorded by the source corpus.

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 Kn\"orrer Periodicity for eliminating metabolic quadratic bundles over a base.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/ef5b987d26d3d7dedc214e113fddc1debebab43b

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In Computational Theory and Mathematics

1. D. Pomerleano
2. M. Abouzaid
3. Kevin Lin
4. Dmitri Orlov
5. L. Katzarkov

Citations in the corpus

• Global matrix factorizations (2011, result): “aracteristic zero. We note that Anatoly Preygel has independently proved results similar to our results in this paper, using an exciting new and different approach involving derived algebraic geometry [Pre]. 2. Curved quasi-coherentsheavesand matrixfactorizations Let X be a smooth variety over Cand W”
• Homological mirror symmetry for punctured spheres (2011, background): “anifolds, and by recent work [2, 19], is known in some generality to be homologically smooth in the sense of Kontsevich [29] (homological smoothness also holds for categories of matrix factorizations [31, 35, 30]). In this paper, we give the first non-trivial verification that these categories are ind”