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Homological degrees of representations of categories with shift functors


Author: Liping Li
Journal: Trans. Amer. Math. Soc. 370 (2018), 2563-2587
MSC (2010): Primary 16E05, 16E10, 16E30
DOI: https://doi.org/10.1090/tran/7041
Published electronically: November 16, 2017
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Abstract: Let $ \mathbbm {k}$ be a commutative Noetherian ring and let $ {\underline {\mathscr {C}}}$ be a locally finite $ \mathbbm {k}$-linear category equipped with a self-embedding functor of degree 1. We show under a moderate condition that finitely generated torsion representations of $ {\underline {\mathscr {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 $ \mathbbm {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 $ {\mathscr {FI}}$-modules.


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Additional Information

Liping Li
Affiliation: Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education), College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, China
Email: lipingli@hunnu.edu.cn

DOI: https://doi.org/10.1090/tran/7041
Received by editor(s): August 29, 2015
Received by editor(s) in revised form: September 4, 2015, October 21, 2015, and July 21, 2016
Published electronically: November 16, 2017
Additional Notes: The author was supported by the National Natural Science Foundation of China 11771135, the Construct Program of the Key Discipline in Hunan Province, and the Start-Up Funds of Hunan Normal University 830122-0037.
Article copyright: © Copyright 2017 American Mathematical Society

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