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 = A[t]/(tⁿ⁺¹) a cdg algebra over Rₙ = k[t]/(tⁿ⁺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.