Let be a (not necessarily commutative) ring whose additive group is finitely generated and let be the group of upper-triangular unipotent matrices over . We study how the homology groups of vary with from the point of view of representation stability. Our main theorem asserts that if for each we have representations of over a ring that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule defines a finitely generated -module. As a consequence, if is a field then is eventually equal to a polynomial in . We also prove similar results for the Iwahori subgroups of for number rings .
"Stability in the homology of unipotent groups." Algebra Number Theory 14 (1) 119 - 154, 2020. https://doi.org/10.2140/ant.2020.14.119