In this paper we study representation theory of the category FI^m introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI^m in a natural way. In particular, we prove the representation stability property of finitely generated FI^m -modules over fields of characteristic 0.
We prove by an inductive argument that any finitely generated FI^d -module over a commutative Noetherian ring has finite (Castelnuovo-Mumford) regularity. Our inductive argument is applicable also to the categories OI_d , FI^m , and OI_m.
In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI_G-modules in terms of torsion degrees of their local cohomology groups.
Let k be a commutative Noetherian ring and C−− be a locally finite k-linear category equipped with a self-embedding functor of degree 1. We show under a moderate condition that finitely generated torsion representations of C−− are super finitely presented (that is, they have projective resolutions each term of which is finitely generated). In the situation that these self-embedding functors are genetic functors, we give upper bounds for homological degrees of finitely generated torsion modules. These results apply to quite a few categories recently appearing in representation stability theory. In particular, when k is a field of characteristic 0, we obtain another upper bound for homological degrees of finitely generated FI-modules.
We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.