Motivic cohomology in mixed-characteristic

Motivic cohomology is a cohomology theory that can be defined internally within Grothendieck's category of motives. Voevodsky developed this theory for smooth varieties, demonstrating its profound connections to algebraic cycles and algebraic K-theory. However, its behaviour beyond the smooth case remains less well understood. Building upon recent advancements by Bachmann, Elmanto, Morrow, and Bouis, we establish its A^1-homotopty invariance for a broader class of "smooth" schemes. This is part of ongoing work in collaboration with Tess Bouis.