Coherent analogues of matrix factorizations and relative singularity categories

By Alexander I. Efimov, L. Positselski. 2011

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We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel of the direct image functor corresponding to the closed embedding of the zero locus and acting between the conventional (absolute) triangulated categories of singularities. Similar results are obtained for matrix factorizations of infinite rank; and two different "large" versions of the triangulated category of relative singularities, corresponding to the approaches of Orlov and Krause, are identified in the case of a Cartier divisor. A version of the Thomason-Trobaugh-Neeman localization theory is proven for coherent matrix factorizations and disproven for locally free matrix factorizations of finite rank. Contravariant (coherent) and covariant (quasi-coherent) versions of the Serre-Grothendieck duality theorems for matrix factorizations are established, and pull-backs and push-forwards of matrix factorizations are discussed at length. A number of general results about derived categories of the second kind for CDG-modules over quasi-coherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/43373677e791e661425522dcdb897547686d23e3

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Citations in the corpus

Matrix factorizations and Cohomological Field Theories (2011, methodology): “tribute some key ideas of their model to Witten. 3 ical string theories (see [27], [28], [29]). They are also related to an important invariant of the potential w, the singularity category (see [47], [59], [44]). The category of matrix factorizations for an isolated singularity ﬁts naturally into th”
Hochschild (co)homology of the second kind I (2010, methodology): “n checking the assertion of the lemma for Orlov’s cokernel functor Σ: H0(B-modcdg fgp ) −→Db Sing(X0), one can use the construction of the inverse functor Υ: Db Sing(X0) −→H0(B-mod cdg fgp ) given in [16], for which the desired compatibility is easy to establish. Alternatively, it suﬃces to use the ”
A category of kernels for equivariant factorizations and its implications for Hodge theory (2011, background): “ characterization of these factorization categories. This characterization is due, in various generality, to Eisenbud [Eis80], Buchweitz [Buc86], Orlov [Orl09, Orl12], Polishchuk-Vaintrob [PV10], and [Pos11]. Let us recall the deﬁnition of the singularity category. Deﬁnition 3.58. Let Y be a scheme ”
Global matrix factorizations (2011, background): “and so the lemma is proven. Remark 2.19. After receiving an earlier version of this article, Leonid Positselski has expanded on the ideas in this section to produce a new proof of Orlov’s equivalence [Pos2]. 3. Compactgeneratorsand Hochschild(co)homology Let X be as above a smooth variety over C, an”