In this article, the authors define and investigate the triangulated category of relative singularities associated to a closed subscheme \(Z\) of a separated Noetherian scheme \(X\) with enough vector bundles, where \(\mathcal{O}_Z\) has finite flat dimension as an \(\mathcal{O}_X\)-module. It is given by the quotient of \(\text{D}^b(Z)\) by the thick subcategory generated by the image of the derived inverse image functor \(\mathbb{L}i^*: \text{D}^b(X) \to \text{D}^b(Z)\), and it is denoted by \(\text{D}^b_{\mathrm{Sing}}(Z/X)\). When \(X\) is regular, \(\text{D}^b_{\mathrm{Sing}}(Z/X)\) is precisely the singularity category of \(Z\) as defined by R.-O. Buchweitz in [“Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings”, unpublished manuscript (1987)].
In Section 1 of the article, the authors establish some general results regarding derived categories of the second kind (cf. [L. Positselski, Mem. Am. Math. Soc. 996, i-iii, 133 p. (2011; Zbl 1275.18002)]) associated to curved dg-modules over curved dg-rings. These results are applied later in the article to matrix factorizations, which may be considered as curved dg-modules over a certain curved dg algebra.
In Section 2, the authors prove what they refer to as their main result. Let \(X\) be as above, let \(\mathcal{L}\) be a line bundle on \(X\), and let \(w \in \mathcal{L}(X)\) be a section. Let \(X_0 \subseteq X\) denote the closed subscheme given by the zero locus of \(w\). Assume the morphism of sheaves \(w: \mathcal{O}_X \to \mathcal{L}\) is injective. Let \((X, \mathcal{L}, w)-\text{coh}\) denote the category of coherent matrix factorizations of \(w\); that is, the pair of \(\mathcal{O}_X\)-modules underlying the matrix factorization is allowed to be a pair of coherent modules, rather than locally free. The authors construct an equivalence \[ \text{D}^{\text{abs}} ((X, \mathcal{L}, w)-\text{coh})) \to \text{D}^b_{\mathrm{Sing}}(X_0/X), \] where \(\text{D}^{\text{abs}}(-)\) denotes a certain derived category of the second kind. When X is regular, this theorem recovers a well-known theorem of D. Orlov (Theorem 3.5 of [Math. Ann. 353, No. 1, 95–108 (2012; Zbl 1243.81178)]).
The authors also establish, in this section, what they refer to as covariant and contravariant Serre-Grothendieck duality theorems for matrix factorizations. In Section 3, the authors give some general results on ma- trix factorizations with a support condition, and also pushforwards and pullbacks of matrix factorizations. Hochschild (co)homology of dg categories of matrix factorizations is discussed in an appendix.