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Keywords:
dg categories of relative singularities; matrix factorizations; non commutative algebraic geometry
dg categories of relative singularities; matrix factorizations; non commutative algebraic geometry
Summary:
In this paper we show that every object in the dg category of relative singularities ${\bf Sing}(B,\underline f)$ associated to a pair $(B,\underline f)$, where $B$ is a ring and $\underline f \in B^n$, is equivalent to an homotopy retract of a $K(B,\underline f)$-dg module concentrated in $n + 1$ degrees, where $K(B,\underline f)$ denotes the Koszul algebra associated to $(B,\underline f)$. When $n = 1$, we show that Orlov’s comparison theorem, which relates the dg category of relative singularities and that of matrix factorizations of an LG-model, holds true without any regularity assumption on the potential.
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