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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of \(Z\) as defined by R.-O. Buchweitz in [“Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein rings”, unpublished manuscript (1987)].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.

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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