FI-modules over Noetherian rings (1210.1854v2)

Abstract: FI-modules were introduced by the first three authors in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of S_n-representations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely generated FI-module is finitely generated. This lets us extend many of the results of [CEF] to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman's central stability for homology of congruence subgroups.

FI-modules over Noetherian rings

The paper, "FI-modules over Noetherian rings" by Thomas Church, Jordan S. Ellenberg, Benson Farb, and Rohit Nagpal, presents an advancement in understanding the structural properties of FI-modules when extended beyond fields of characteristic zero to arbitrary Noetherian rings. The authors set out to demonstrate that the Noetherian property holds for FI-modules over these rings, a concept central to extending representation stability and related applications to positive characteristic and integral settings.

The core contribution of the paper is proving that if V is a finitely-generated FI-module over a Noetherian ring R, then any sub-FI-module of V is also finitely-generated. This is analogous to the property previously established for FI-modules over fields of characteristic zero. The authors leverage this property to extend the applications of FI-modules theory, notably in areas such as the integral and mod p cohomology of configuration spaces, diagonal coinvariant algebras in positive characteristic, and a version of central stability concerning the homology of congruence subgroups.

Key Theoretical Results

The paper articulates several theorems underpinning the main results:

• Theorem A (Noetherian property): Establishes the Noetherian property by showing that any sub-FI-module of a finitely-generated FI-module over a Noetherian ring is finitely generated.
• Theorem B (Polynomial dimension): Proves the existence of a polynomial governing the dimensions of FI-modules over any field.
• Theorem C (Inductive description): Provides an inductive characterization for finitely-generated FI-modules by representing them as colimits over a poset of subsets.

Applications and Implications

The paper highlights crucial applications of these results in three specific areas:

1. Congruence Subgroups: Demonstrates that the FI-module describing homology with respect to congruence subgroups over a Noetherian ring exhibits representation stability, thereby refining previous results by Putman that depended on fields of sufficiently large characteristic.
2. Configuration Spaces: Extends results concerning the cohomology of configuration spaces of manifolds, proving the polynomial behavior of Betti numbers and providing inductive descriptions when the characteristic is arbitrary.
3. Diagonal Coinvariant Algebras: Shows that dimensions of multi-graded pieces of diagonal coinvariant algebras over fields also adhere to polynomial descriptions, expanding upon previous work restricted to fields of characteristic zero.

Theoretical and Practical Impact

The authors' generalization of FI-modules over arbitrary Noetherian rings opens avenues for tackling problems in algebra and topology involving families of symmetric group representations with integral coefficients and in non-zero characteristic settings. This work can potentially influence computational techniques and the understanding of algebraic structures in diverse mathematical fields.

By establishing foundational properties for FI-modules over more general rings, this paper enriches the mathematical toolkit available to researchers addressing questions involving inductive stability, homological growth, and representation theory. Future work might delve into deeper explorations of these modules in non-commutative rings or paper associated homological phenomena in broader contexts.

Overall, this research represents a significant step in the theoretical advancement of FI-modules, setting the stage for further exploration of their applications in representation theory, algebraic topology, and associated mathematical domains.