Homological degrees of representations of categories with shift functors · arXivDesk
1507.08023Jul 29, 2015Major changes include: A stronger upper bound for homological degrees of torsion modules; a new proof of the Koszulity of categories with shift functors; a new upper bound for homological degrees of FI-modules, etc
Homological degrees of representations of categories with shift functors
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
Nearby in the stack
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