论文标题
一个点的派生变形理论
The derived deformation theory of a point
论文作者
论文摘要
我们为在差分级代数上对模块$ x $变形的非交通派生变形问题提供了一个前代表的对象。粗略地，我们表明，相应的变形函子是由$ x $的衍生内态代数上的双杆构造在衍生的内态代数上的同型。当$ x $在基本场上是一维时，我们专门研究案例，并介绍了框架变形的概念，这些变形略微固定了问题，并允许我们获得Ed Segal论文结果的派生类似物。我们的主要技术工具是Pridham和Lurie对衍生变形理论的解释之后的Koszul二元性。在此过程中，我们证明了一大批DGA对其Koszul Double Dual是准同形的，我们将其解释为派生的完成函数。这改善了lu-palmieri-wu-zhang的定理。我们还将结果适应了多点变形理论的设定，此外，我们还对通用产物形成了分析。作为应用程序，我们为Braun-Chuang Lazarev的衍生商提供了变形理论解释。
We provide a prorepresenting object for the noncommutative derived deformation problem of deforming a module $X$ over a differential graded algebra. Roughly, we show that the corresponding deformation functor is homotopy prorepresented by the dual bar construction on the derived endomorphism algebra of $X$. We specialise to the case when $X$ is one-dimensional over the base field, and introduce the notion of framed deformations, which rigidify the problem slightly and allow us to obtain derived analogues of the results of Ed Segal's thesis. Our main technical tool is Koszul duality, following Pridham and Lurie's interpretation of derived deformation theory. Along the way we prove that a large class of dgas are quasi-isomorphic to their Koszul double dual, which we interpret as a derived completion functor; this improves a theorem of Lu-Palmieri-Wu-Zhang. We also adapt our results to the setting of multi-pointed deformation theory, and furthermore give an analysis of universal prodeformations. As an application, we give a deformation-theoretic interpretation to Braun-Chuang-Lazarev's derived quotient.
