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Representation stability and finite linear groups. (English) Zbl 1408.18003
T. Church and B. Farb introduced the concept of representation stability in [Adv. Math. 245, 250–314 (2013; Zbl 1300.20051)], as a framework to describe representation-theoretic patterns observed in many areas of mathematics. In joint work with J. S. Ellenberg [T. Church et al., Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)], they investigated the $$FI$$-modules, the modules of the category of finite sets and injections, in which automorphism groups are symmetric groups, and established a Noetherian property and an asymptotic structure theorem for finitely generated $$FI$$-modules.
The current paper studies interesting analogues of $$FI$$-modules, where the role of the symmetric groups is played by the general linear groups and the symplectic groups over finite rings, and proved the local Noetherian property and asymptotic structure results for them. As a by-product, the authors give a a proof of the Lannes-Schwartz Artinian conjecture that the category $$V(R)$$ of finite-rank free $$R$$-modules whose morphisms are left-invertible linear maps is locally Noetherian when $$R$$ is a finite ring. As further applications, the authors establish homological stability theorems with twisted coefficients for the general linear and symplectic groups over finite rings, and representation-theoretic versions of homological stability for congruence subgroups of general linear groups, automorphism groups of free groups, symplectic groups over number rings, and the mapping class group. The final result shows the necessity of the finiteness condition on the ring $$R$$ in many results of the paper.

##### MSC:
 18A25 Functor categories, comma categories 11F75 Cohomology of arithmetic groups 16D90 Module categories in associative algebras 20C33 Representations of finite groups of Lie type
##### Keywords:
representation stability; FI-modules; Artinian conjecture
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