论文标题
cotangent复合物和汤姆光谱
The cotangent complex and Thom spectra
论文作者
论文摘要
通勤环地图的cotangent复合物是变形理论中的中心对象。自1990年代以来，它已被概括为$ e_ \ infty $ - 环光谱的同位环境。 在这项工作中，我们首先在$ \ infty $ - 类别的背景下建立，并使用Goodwillie的函数计算，以使文献中存在的$ e_ \ infty $ - 环光谱的cotangangent复合物的各种定义相等。然后，我们将注意力转向一个特定的例子。令$ r $为$ e_ \ infty $ -ring频谱，$ \ mathrm {pic}（r）$表示其picard $ e_ \ eftty $ -group。令$ mf $表示$ e_ \ infty $ - $ r $ -r $ -r $ -algebra的地图的$ e_ \ infty $ -groups $ f：g \ t to \ mathrm {pic}（pic}（r）$; $ MF $的例子由各种Coobordism Spectra提供。我们证明，$ r \ to MF $的Cotangent综合体等于$ MF $的Smash产品和与$ G $相关的结合频谱。
The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $E_\infty$-ring spectra in various ways. In this work we first establish, in the context of $\infty$-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of $E_\infty$-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let $R$ be an $E_\infty$-ring spectrum and $\mathrm{Pic}(R)$ denote its Picard $E_\infty$-group. Let $Mf$ denote the Thom $E_\infty$-$R$-algebra of a map of $E_\infty$-groups $f:G\to \mathrm{Pic}(R)$; examples of $Mf$ are given by various flavors of cobordism spectra. We prove that the cotangent complex of $R\to Mf$ is equivalent to the smash product of $Mf$ and the connective spectrum associated to $G$.
