Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras (2004.00252v4)

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.