@article{Factorizations:2022,
      recid = {2022},
      author = {di Fiore, Carlos Ignacio},
      title = {Matrix Factorizations for Quasi-Coherent Sheaves of  Categories},
      publisher = {The University of Chicago},
      school = {Ph.D.},
      address = {2019-08},
      pages = {47},
      abstract = {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.},
      url = {http://knowledge.uchicago.edu/record/2022},
      doi = {https://doi.org/10.6082/uchicago.2022},
}