HOMOLOGICAL DEGREES OF REPRESENTATIONS OF CATEGORIES WITH SHIFT FUNCTORS.

Let k be a commutative Noetherian ring and let 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, using the result of Church and Ellenberg [arXiv:1506.01022], we obtain another upper bound for homological degrees of finitely generated FI-modules. [ABSTRACT FROM AUTHOR]