New paper on the arXiv, joint with Tom Church, Benson Farb, and UW grad student Rohit Nagpal. In our first paper on FI-modules (which I blogged about earlier this year) a crucial tool was the fact that the category of FI-modules over a field of characteristic 0 is Noetherian; that is, a submodule of a finitely generated FI-module is again finitely generated. But we didn’t know how to prove this over a more general ring, which limited the application of some of our results.
In the new paper, we show that the category of FI-modules is Noetherian over an arbitrary Noetherian ring. Sample consequence: if M is a manifold and Conf^n M is the configuration space of ordered n-tuples of distinct points on M, then we show that
dim_k H_i(Conf^n M, k)
is a polynomial function of n, for all n greater than some threshold. In our previous paper, we could prove this only when k had characteristic 0; now it works for mod p cohomology as well. We also discuss some of the results of Andy Putman’s paper on stable cohomology of congruence subgroups – it is a bad thing that I somehow haven’t blogged about this awesome paper! — showing how, at the expense of losing stable ranges, you can remove from his results some of the restrictions on the characteristic of the coefficient field.
Philosophically, this paper makes me happy because it brings me closer to what I wanted to do in the first place — talk about the representation theory of symmetric groups “for general n” without giving names to representations. In characteristic 0, this desire might seem a bit perverse, given the rich and beautiful story of the bijection between irreducible representations and partitions. But in characteristic p, the representation theory of S_n is much harder to describe. So it is pleasing to be able to talk, in a principled way, about what one might call “representation stability” in this context; I think that when V is a finitely generated FI-module over a finite field k it makes sense to say that the representations V_n of k[S_n] are “the same” for n large enough, even though I don’t have as nice a description of the isomorphism classes of such representations.