## Algebra & Number Theory

### Stability in the homology of unipotent groups

#### Abstract

Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U n ( R ) ⊂ G L n ( R )$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U n ( R )$ vary with $n$ from the point of view of representation stability. Our main theorem asserts that if for each $n$ we have representations $M n$ of $U n ( R )$ over a ring $k$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $[ n ] ↦ H i ( U n ( R ) , M n )$ defines a finitely generated $O I$-module. As a consequence, if $k$ is a field then $dim H i ( U n ( R ) , k )$ is eventually equal to a polynomial in $n$. We also prove similar results for the Iwahori subgroups of $G L n ( 𝒪 )$ for number rings $𝒪$.

#### Article information

Source
Algebra Number Theory, Volume 14, Number 1 (2020), 119-154.

Dates
Received: 19 December 2018
Accepted: 18 August 2019
First available in Project Euclid: 7 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.ant/1586224821

Digital Object Identifier
doi:10.2140/ant.2020.14.119

Mathematical Reviews number (MathSciNet)
MR4076809

Zentralblatt MATH identifier
07180783

Subjects
Primary: 20J05: Homological methods in group theory
Secondary: 16P40: Noetherian rings and modules

#### Citation

Putman, Andrew; Sam, Steven V; Snowden, Andrew. Stability in the homology of unipotent groups. Algebra Number Theory 14 (2020), no. 1, 119--154. doi:10.2140/ant.2020.14.119. https://projecteuclid.org/euclid.ant/1586224821

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