Coherent analogues of matrix factorizations and relative singularity categories

Abstract

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.

LaTeX 2e with pb-diagram and xy-pic; 114 pages, 13 commutative diagrams. v.8: new sections 2.10, 3.1 and 3.7 inserted; v.9: appendix B added, remarks inserted in sections 2.10 and 2.7, section 1.8 expanded; v.10: new section 3.3 inserted, the whole paper has two authors now; v.11: small corrections, additions, and improvements -- this is intended as the final version

References in corpus (4)

• Nonhomogeneous quadratic duality and curvature
• Matrix factorizations for nonaffine LG-models
• Hochschild (co)homology of the second kind I
• Pushing forward matrix factorisations

Cited by in corpus (25)

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• Derived Kn"orrer periodicity and Orlov's theorem for gauged Landau-Ginzburg models
• Dedualizing complexes of bicomodules and MGM duality over coalgebras
• Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality
• Infinity-tilting theory
• Contramodules
• Equivariant Hodge theory and noncommutative geometry
• Flat morphisms of finite presentation are very flat
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Differential graded Koszul duality: an introductory survey
• Hori-mological projective duality
• Categorical wall-crossing formula for Donaldson-Thomas theory on the resolved conifold
• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• On flat generators and Matlis duality for quasicoherent sheaves
• The homological projective dual of Sym^2 P(V)
• Coderived and contraderived categories of locally presentable abelian DG-categories
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Local, colocal, and antilocal properties of modules and complexes over commutative rings
• Cut cotorsion pairs
• Homogeneity of cohomology classes associated with Koszul matrix factorizations
• Roos axiom holds for quasi-coherent sheaves
• Contraderived categories of CDG-modules
• Categorical Bockstein sequences