Recent Developments in Noncommutative Algebraic Geometry: "Filtered derived categories of curved deformations"

April 11, 2024
  • 14A22
The curvature problem in the deformation theory of dg algebras stems from the following observations: (1) According to the Hochschild complex, deformations of a dg algebra include \emph{curved} cdga’s and in general it is not possible to realise the full cohomology by means of dg Morita deformations (Keller-Lowen, 2009); (2) Since the differential of a cdga does not square to zero, there is no conventional derived category and second kind derived categories in the sense of Positselski may vanish for deformations (Keller-Lowen-Nicolás, 2009). In this talk, we propose an altogether different approach to (2) by introducing the \emph{filtered derived category} of an n-th order cdg deformation of a given dga A. This turns out to be a compactly generated triangulated category in which Positselski's semiderived category embeds admissibly. Further, it allows for a semiorthogonal decomposition into n+1 copies of D(A) and for smooth A it can be interpreted as a categorical resolution in the sense of Kuznetsov of the (in general) nonexistent classical derived category of the cdga. This is joint work with Alessandro Lehmann.