On the structure of dg categories of relative singularities

Abstract

In this paper we show that every object in the dg category of relative singularities Sing$(B,\underline{f})$ associated to a pair $(B,\underline{f})$, where $B$ is a ring and $\underline{f}\in B^n$, is equivalent to a retract of a $K(B,\underline{f})$-dg module concentrated in $n+1$ degrees. When $n=1$, we show that Orlov's comparison theorem, which relates the dg category of relative singularities and that of matrix factorizations of an LG-model, holds true without any regularity assumption on the potential.