Unstable Gersten complexes

I will present joint work with Rakesh Pawar aimed at understanding the truncated Cousin complexes for unramified sheaves of groups introduced in the very first chapter of Morel's fundamental book on motivic homotopy theory. Drawing on techniques due to Bousfield and Kan, we put this idea into a general formalism of unstable exact couples, leading to a notion of unstable Gersten complexes. Applications range from unstable motivic homotopy theory to adelic descriptions of torsors and Artin–Mazur étale homotopy types.