  Consider the polynomial ring in countably infinitely many variables over a
field of characteristic zero, together with its natural action of the infinite
general linear group G. We study the algebraic and homological properties of
finitely generated modules over this ring that are equipped with a compatible
G-action. We define and prove finiteness properties for analogues of Hilbert
series, systems of parameters, depth, local cohomology, Koszul duality, and
regularity. We also show that this category is built out of a simpler, more
combinatorial, quiver category which we describe explicitly.
  Our work is motivated by recent papers in the literature which study
finiteness properties of infinite polynomial rings equipped with group actions.
(For example, the paper by Church, Ellenberg and Farb on the category of
FI-modules, which is equivalent to our category.) Along the way, we see several
connections with the character polynomials from the representation theory of
the symmetric groups. Several examples are given to illustrate that the
invariants we introduce are explicit and computable.
