相对Calabi-Yau完备
Relative Calabi-Yau completions
摘要：
本文将变形的$n$ -Calabi-Yau完备化的Keller构造\cite{Kel11}推广到相对的情形。这给出了一个构造，扩展任何给定的dg函子$F : A \rightarrow B$之间的光滑dg范畴的dg函子$\widetilde{F} : \widetilde{A} \rightarrow \widetilde{B}$，连同一个家庭的变形$(\widetilde{A},\widetilde{B},\widetilde{F})$参数化的相对负循环同调类$\widetildeη \in HN_{n-2}(B, A)$。我们证明了这些扩张在\cite{BD19}意义下允许相对$n$ -Calabi-Yau结构.特别地，这个证明覆盖了\cite{Kel11}的原始绝对情况。
英文摘要：
We generalize Keller's construction \cite{Kel11} of deformed $n$-Calabi-Yau completions to the relative context. This gives a construction that extends any given dg functor $F : A \rightarrow B$ between smooth dg categories to a dg functor $\widetilde{F} : \widetilde{A} \rightarrow \widetilde{B}$, together with a family of deformations of $(\widetilde{A},\widetilde{B},\widetilde{F})$ parametrized by relative negative cyclic homology classes $\widetildeη \in HN_{n-2}(B, A)$. We prove that these extensions admit relative $n$-Calabi-Yau structures in the sense of \cite{BD19}. In particular, this proof covers the original absolute case of \cite{Kel11}.
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