Abstract:

A guiding principle of the field of equivariant commutative algebra is that non-noetherian rings behave like noetherian rings up to the action of a large group of symmetries. One of the central examples of this principle is the noetherianity of the infinite variable polynomial ring R=C[x_1,x_2,...] with the action of the infinite symmetric group \frakS permuting variables. In this thesis, we further develop the equivariant commutative algebra of R by studying symmetric R-modules, i.e., R-modules equipped with a compatible \frakS-action. In particular, we study the local behavior of such modules over the \frakS-prime ideals h_n=(x_i^n:i\geq 1), and prove a number of structural results on the category of symmetric R/h_n-modules, including giving a semi-orthogonal decomposition and generators for the bounded derived category of modules.

An important tool for our study is representation theory of various combinatorial categories, such as a generalization of FI to a weighted setting. By proving an equivalence between weighted FI-modules and modules over infinite variable polynomial rings that are equivariant with respect to parabolic subgroups of the infinite general linear group, we give finiteness properties of weighted FI-modules, such as the rationality of an analogue of the Hilbert series and finite generation of local cohomology. Along the way, we relate the polynomial representation theory of these parabolic subgroups to generalizations of Schur functors from categories consisting of flags of vector spaces.