We explore some similarities between the theory of D-modules and that of quasi-coherent sheaves of categories. The original motivation is to better understand several results on the literature that relates vanishing cycles with some invariants of the category of matrix factorizations like its periodic cyclic homology or its etale cohomology.
We propose the following explanation: given a function f:X---> A^1, the category of matrix factorizations MF(X,f) can be thought as a higher categorical version of vanishing cycles for the sheaf of categories corresponding to Perf(X). Here, the 2-periodic structure on matrix factorizations corresponds to the monodromy of vanishing cycles.
A second goal is to find the general phenomena behind the work of A. Preygel that uses~derived algebraic geometry to construct MF(X,f) from a cohomological operation of degree 2 on Coh(X_0). We interpret this as the microlocalization of Perf(X) over T^*_0A^1 and extend it to arbitrary quasi-coherent sheaves of categories over smooth schemes.
To realize the above we need the theory of quasi-coherent sheaves of categories developed by D.Gaitsgory, J.Lurie, B.Toen and G.Vessozzi. The basic formalism is quite recent and contains pushforward and pullback. We slightly modify it to get an extraordinary pullback for complete intersection morphisms. We also introduce the matrix factorizations functor.
The main result of this thesis is a comparison between the usual and the extraordinary pullback. In rough terms, via Koszul duality, the extraordinary pullback localize over a conormal bundle and the usual pullback correspond to the part supported on the zero section.
We study the case of the closed immersion of a point in a smooth scheme. This defines for every quasi-coherent sheaf of categories its punctual singular support, analogous to the singular support of a D-module. In the end we give a deformation theory interpretation.
This is part of joint work with G. Stefanich. In future work we will define and study the global singular support of a quasi-coherent sheaf of categories. This is a closed conical subset of the cotangent bundle that measures the directions on which the sheaf is proper and smooth.

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Title

Matrix Factorizations for Quasi-Coherent Sheaves of Categories