Mathematics – Algebraic Geometry
Scientific paper
2010-09-21
Mathematics
Algebraic Geometry
53 pages, LaTeX
Scientific paper
In this paper we prove a version of curved Koszul duality for Z/2Z-graded curved (dg) (co)algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the dg category of matrix factorizations MF(R,W). We show how Dyckerhoff's generating results fit into the framework of curved Koszul duality theory. One immediate application is the construction of a free dg algebra model for MF(R,W). As another application we clarify the relationship between the Borel-Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category MF(R,W). The same methods can also be used to study the dg category of equivariant or graded matrix factorizations. Both the Koszul duality property and its applications are generalized to include these cases as well. In particular we obtain an explicit set of (classical) generators for these categories. Our results in the graded case are closely related to Seidel's work on the derived category of coherent sheaves on Calabi-Yau hypersurfaces via the CY/LG correspondence.
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