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 $A_n=A[t]\mathrm{/}(t^{n+1})$ a cdg algebra over $R_n=k[t]\mathrm{/}(t^{n+1})$, $n\ge 0$, with reduction $A_n\mathrm{/}t{A}_n=A$. We define the $n$-derived category $D^n(A_n)$ 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.