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Coherent analogues of
matrix factorizations and relative singularity categories
Alexander I. Efimov and Leonid Positselski |
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Vol. 9 (2015), No. 5, 1159–1292
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Abstract |
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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 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. |
Keywords
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
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Mathematical Subject Classification 2010
Primary: 14F05
Secondary: 13D09, 16G99
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Milestones
Received: 26 December 2014
Accepted: 27 March 2015
Published: 21 June 2015
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Authors |
| Alexander I. Efimov | |
| Department of Algebraic
Geometry Steklov Mathematical Institute of the Russian Academy of Sciences Gubkina str., 8 Moscow 119991 Russia and Laboratory of Algebraic Geometry Higher School of Economics 7 Vavilova str. Moscow 117312 Russia |
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| Leonid Positselski | |
| Mathematics Department Technion – Israel Institute of Technology 32000 Haifa Israel and Sector of Algebra and Number Theory Institute for Information Transmission Problems Moscow 127994 Russia and Laboratory of Algebraic Geometry National Research University Higher School of Economics Moscow 117312 Russia |
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