Algebra over FI

FI is the category of finite sets with injections. An FI something F is a functor from FI to the category of somethings. In particular, for S in FI, the symmetric group Sym(S) acts on F(S) by something automorphisms of F(S). FI-somethings arise in several areas of mathematics; for instance, FI-modules arise as cohomology groups of configuration spaces of n labelled points on a given manifold (with n varying), and FI-algebras arise as coordinate rings of certain algebraic-statistical models, where the number of observed random variables varies. This talk aims to be a gentle introduction into algebra over FI; no prior exposure to FI-maths is required.

Sometimes, FI-somethings inherit good properties of the category of somethings (and indeed the same applies when FI is replaced by other suitable base categories). I will present several, by now, classical examples of this phenomenon, primarily due to Church, Ellenberg, Farb, Daniel Cohen, Aschenbrenner, Hillar, Sullivant, Sam, and Snowden. For instance, a finitely generated FI-module M over a Noetherian ring is Noetherian, and if the ring is a field, then the dimension of M(S) is eventually a polynomial in |S|.

After this overview, I will zoom in on new joint work with Rob Eggermont and Azhar Farooq that says that if X(S) < K^{c x S} is avariety of c x S-matrices and for every injection S -> T the natural map K^{c x T} -> K^{c x S} maps X(T) into X(S), then the number of Sym(S) orbits on the set of irreducible components of X(S) is a *quasi*polynomial in |S| for |S| sufficiently large.