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2020 Stability in the homology of unipotent groups
Andrew Putman, Steven V Sam, Andrew Snowden
Algebra Number Theory 14(1): 119-154 (2020). DOI: 10.2140/ant.2020.14.119

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 .

Citation

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Andrew Putman. Steven V Sam. Andrew Snowden. "Stability in the homology of unipotent groups." Algebra Number Theory 14 (1) 119 - 154, 2020. https://doi.org/10.2140/ant.2020.14.119

Information

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

zbMATH: 07180783
MathSciNet: MR4076809
Digital Object Identifier: 10.2140/ant.2020.14.119

Subjects:
Primary: 20J05
Secondary: 16P40

Rights: Copyright © 2020 Mathematical Sciences Publishers

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