Abstract
For a finite ring R, not necessarily commutative, we prove that the category of \({{\,\mathrm{\texttt {VIC}}\,}}(R)\)-modules over a left Noetherian ring \(\mathbf {k}\) is locally Noetherian, generalizing a theorem of the authors that dealt with commutative R. As an application, we prove a very general twisted homology stability for \({{\,\mathrm{GL}\,}}_n(R)\) with R a finite noncommutative ring.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The purpose of considering \(R^n\) as a right R-module is that \({{\,\mathrm{GL}\,}}_n(R)\) acts on \(R^n\) on the left by right R-module automorphisms.
The fact that the target of the Peirce embedding is \({{\,\mathrm{Mat}\,}}_{\mu }(R)\) is just a matter of convenience so we do not have to precisely define a ring of “matrices” whose entries all lie in different places. Later on we will identify the various \(e_i R e_j\) with division rings \(\mathbb {D}_k\) and even additive groups \(\mathbb {L}_{hk}\), and we suggest to the reader that they not focus too much on how these lie inside R.
This is not a typo – we are using the fact that the left and right actions of R on itself commute, i.e., that R is an (R, R)-bimodule.
Note that \(\phi (e_j) = \phi (e_j^2) = \phi (e_j) e_j = e_i r e_j\) for some \(r \in R\).
We cannot require \(g' = \widehat{g}'\) since we need \(g'' \circ g' = \text {id}\), which requires changing the dependent rows.
Recall that for words \(s_1 \cdots s_p\) and \(t_1 \cdots t_q\) in \(\Sigma ^{*}\), we have \(s_1 \cdots s_p \preceq t_1 \cdots t_q\) if there exists a strictly increasing function \(\lambda :\{1,\ldots ,p\} \rightarrow \{1,\ldots ,q\}\) with the following two properties:
-
\(s_i = t_{\lambda (i)}\) for \(1 \le i \le p\), and
-
for all \(1 \le j \le q\), there exists some \(1 \le i \le p\) such that \(\lambda (i) \le j\) and \(t_{\lambda (i)} = t_j\).
-
References
Church, T., Ellenberg, J. S., Farb, B.: FI-modules and stability for representations of symmetric groups. Duke Math. J. 164(9), 1833–1910 (2015). arxiv:1204.4533v4
Church, T., Ellenberg, J. S., Farb, B., Nagpal, R.: FI-modules over Noetherian rings. Geom. Topol. 18(5), 2951–2984 (2014). arxiv:1210.1854v2
Church, T., Farb, B.: Representation theory and homological stability. Adv. Math. 245, 250–314 (2013). arxiv:1008.1368v3
Draisma, J., Kuttler, J.: Bounded-rank tensors are defined in bounded degree. Duke Math. J. 163(1), 35–63 (2014). arxiv:1103.5336v2
Dwyer, W.G.: Twisted homological stability for general linear groups. Ann. Math. 111(2), 239–251 (1980)
Farb, B.: Representation stability. In: Proceedings of the 2014 International Congress of Mathematicians. Volume II, 1173–1196. arxiv:1404.4065v1
Franjou, V., Touzé, A. (eds.): Lectures on Functor Homology, Progress in Mathematics, 311. Birkhäuser/Springer, Cham (2015)
Gan, W. L., Li, L.: Noetherian property of infinite EI categories, New York J. Math. 21, 369–382 (2015). arxiv:1407.8235v3
Harman, N.: Effective and infinite-rank superrigidity in the context of representation stability. Preprint 2019, arxiv:1902.05603v1
Higman, G.: Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. 3(2), 326–336 (1952)
Kuhn, N.J.: Generic representations of the finite general linear groups and the Steenrod algebra II. K-Theory 8(4), 395–428 (1994)
Lam, T.Y.: A First Course in Noncommutative Rings, Second edition, Graduate Texts in Mathematics, 131. Springer, New York (2001)
Lam, T. Y.: Serre’s problem on projective modules, Springer Monographs in Mathematics, Springer, Berlin (2006)
Putman, A.: A new approach to twisted homological stability, with applications to congruence subgroups, preprint 2021, arxiv:2109.14015
Putman, A., Sam, S. V.: Representation stability and finite linear groups. Duke Math. J. 166(13), 2521–2598 (2017). arxiv:1408.3694v3
Putman, A., Sam, S. V., Snowden, A.: Stability in the homology of unipotent groups. Algebra Number Theory 14(1), 119–154 (2020). arxiv:1711.11080v4
Randal-Williams, O., Wahl, N.: Homological stability for automorphism groups. Adv. Math. 318, 534–626 (2017). arxiv:1409.3541
Richter, G.: Noetherian semigroup rings with several objects. In: Group and Semigroup Rings (Johannesburg, 1985), 231–246, North-Holland Math. Stud., 126, Notas Mat., 111, North-Holland, Amsterdam
V Sam, S., Snowden, A.: Gröbner methods for representations of combinatorial categories. J. Am. Math. Soc. 30, 159–203 (2017). arxiv:1409.1670v3
Scorichenko, A.: Stable K-theory and functor homology over a ring, thesis (2000)
van der Kallen, W.: Homology stability for linear groups. Invent. Math. 60(3), 269–295 (1980)
Wilson, J. C. H.: \(\text{FI}_\text{ W }\)-modules and stability criteria for representations of classical Weyl groups. J. Algebra 420, 269–332 (2014). arxiv:1309.3817v2
Acknowledgements
We would like to thank Benson Farb and Andrew Snowden for helpful comments, and Peter Patzt for pointing out a small mistake in an earlier version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported in part by NSF Grant DMS-1811322.
Supported in part by NSF Grant DMS-1849173.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Putman, A., Sam, S.V. \({{\,\mathrm{\texttt {VIC}}\,}}\)-modules over noncommutative rings. Sel. Math. New Ser. 28, 88 (2022). https://doi.org/10.1007/s00029-022-00799-7
Accepted:
Published:
DOI: https://doi.org/10.1007/s00029-022-00799-7