Vol. 9, No. 5, 2015

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Coherent analogues of matrix factorizations and relative singularity categories

Alexander I. Efimov and Leonid Positselski

Vol. 9 (2015), No. 5, 1159–1292

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 theorem is proven for coherent matrix factorizations and disproven for locally free matrix factorizations of finite rank. Contravariant (coherent) and covariant (quasicoherent) 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 curved differential graded modules (CDG-modules) over quasicoherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix.

matrix factorizations, relative singularities of Cartier divisors, triangulated categories of singularities, derived categories of the second kind, coderived categories, direct and inverse images, covariant Serre–Grothendieck duality, localization theory
Mathematical Subject Classification 2010
Primary: 14F05
Secondary: 13D09, 16G99
Received: 26 December 2014
Accepted: 27 March 2015
Published: 21 June 2015
Alexander I. Efimov
Department of Algebraic Geometry
Steklov Mathematical Institute of the Russian Academy of Sciences
Gubkina str., 8
and Laboratory of Algebraic Geometry
Higher School of Economics
7 Vavilova str.
Leonid Positselski
Mathematics Department
Technion – Israel Institute of Technology
32000 Haifa
and Sector of Algebra and Number Theory
Institute for Information Transmission Problems
and Laboratory of Algebraic Geometry
National Research University Higher School of Economics