Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras

Abstract

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.

Final version. To appear in Journal of Topology

References in corpus (10)

• FI-modules and stability for representations of symmetric groups
• Representation theory and homological stability
• Gröbner methods for representations of combinatorial categories
• Representation stability and finite linear groups
• Homological stability for configuration spaces of manifolds
• Discriminants in the Grothendieck Ring
• Higher enveloping algebras
• Stability in the homology of congruence subgroups
• Cohomology of generalized configuration spaces
• Homological stability and densities of generalized configuration spaces