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

TL;DR

This paper establishes higher representation stability for the cohomology of configuration spaces using twisted commutative algebras, providing explicit bounds and conditions for freeness, thus advancing understanding of algebraic and topological stability phenomena.

Contribution

It introduces an iterative method for higher representation stability via TCAs, computes bounds for derived indecomposables, and proves freeness results under specific cohomological conditions.

Findings

All modules are finitely generated over TCAs.

Explicit bounds for derived indecomposables are provided.

Cohomology forms a free module over a TCA under certain conditions.

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.