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, f ) associated to a pair (B, f ), where B is a ring and f ∈ B n , is equivalent to a retract of a K(B, 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. Contents 1 Preliminaries 2 2 The structure of Sing(B,f) 9 3 Orlov's theorem 12Acknowledgments This paper's results are part of my PhD project under the supervision of B. Toën and G. Vezzosi. I would like to thank them both for uncountable many conversations on the subject. I would also like to acknowledge D. Beraldo, V. Melani and M. Robalo for many useful remarks and suggestions.