Representation Stability and Finite Orthogonal Groups

In this paper, we prove homological stability results about orthogonal groups over finite commutative rings where 2 is a unit. Inspired by Putman and Sam (2017), we construct a category OrI ( R ) and prove a local Noetherianity theorem for the category of OrI ( R )-modules. This implies an asymptotic structure theorem for orthogonal groups. In addition, we show general homological stability theorems for orthogonal groups, with both untwisted and twisted coefficients.