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Coherent analogues of matrix factorizations and relative singularity categories. (English) Zbl 1333.14018
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of $$Z$$ as defined by R.-O. Buchweitz in [“Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings”, unpublished manuscript (1987)].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.

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 13D09 Derived categories and commutative rings 16G99 Representation theory of associative rings and algebras
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