论文标题
有序配置空间和扭曲的交换分解代数的较高表示稳定性
Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras
论文作者
论文摘要
使用与扭曲的交换代数（TCA）中的系数分解同源性，我们证明了方案/拓扑空间$ x $的（广义）配置空间的共同体的较高表示稳定性的两种口味。首先，我们提供了一种迭代程序，使用来自$ x $的共同体的动作来研究更高的表示稳定性，并证明所涉及的所有模块都是在相应的TCA上有限生成的。更定量地，我们根据Galatius-Bupers-randal-Williams的意义来计算派生的indecosables的明确界限。其次，当$ x $ VANISH的共同体学上的某些$ C_ \ Infty $ - 运行时，我们证明其配置空间的共同体学在仿载空间的配置空间内建立的TCA形成了一个免费的模块。这概括了教堂 - 埃伦贝格 - 法布（Church-Ellenberg-Farb）在$ \ mathrm {fi} $ - 模块中源于开放式流形的配置空间的共同体，并且在这种情况下解决了米勒·威尔森的各种构象。
Using factorization homology with coefficients in twisted commutative algebras (TCAs), we prove two flavors of higher representation stability for the cohomology of (generalized) configuration spaces of a scheme/topological space $X$. First, we provide an iterative procedure to study higher representation stability using actions coming from the cohomology of $X$ and prove that all the modules involved are finitely generated over the corresponding TCAs. More quantitatively, we compute explicit bounds for the derived indecomposables in the sense of Galatius-Kupers-Randal-Williams. Secondly, when certain $C_\infty$-operations on the cohomology of $X$ vanish, we prove that the cohomology of its configuration spaces forms a free module over a TCA built out of the configuration spaces of the affine space. This generalizes a result of Church-Ellenberg-Farb on the freeness of $\mathrm{FI}$-modules arising from the cohomology of configuration spaces of open manifolds and, moreover, resolves the various conjectures of Miller-Wilson under these conditions.
