On the structure of dg categories of relative singularities

TL;DR

This paper characterizes objects in the dg category of relative singularities as retracts of specific dg modules and extends Orlov's comparison theorem to cases without regularity assumptions.

Contribution

It demonstrates that all objects in the dg category of relative singularities are retracts of certain dg modules and generalizes Orlov's theorem to non-regular potentials.

Findings

Objects in Sing(B,f) are retracts of K(B,f)-dg modules in n+1 degrees

Orlov's comparison theorem holds without regularity assumptions for n=1

Extension of the theorem broadens applicability to more general LG-models

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.