Let be a commutative Noetherian ring and let C be a locally finite -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.

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... ese results apply to quite a few categories recently appearing in representation stability theory. In particular, when 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. Contents 1. Introduction 2563 2. Preliminaries 2568 3. Super finitely presented property 2572 4. Upper bounds of homological degrees 2575 5. Applications in representation stability theory 2579 Acknowledgements 2585 References 2586