Filtered derived categories of curved deformations

We propose a solution to the ''curvature problem'' from arXiv: 1505. 03698 and arXiv: 0905. 3845 for infinitesimal deformations. Let k be a field, A a dg algebra over k and Aₙ = At/ (t^n+1) a cdg algebra over Rₙ = kt/ (t^n+1), n 0, with reduction Aₙ/tAₙ = A. We define the n-derived category Dⁿ (Aₙ) as the quotient of the homotopy category by the modules for which all quotients appearing in the associated graded object are acyclic. We prove this to be a compactly generated triangulated category with a semiorthogonal decomposition by n + 1 copies of D (A), in which Positselski's semiderived category embeds admissibly.