Weibel vanishing and the projective bundle formula for mixed characteristic motivic cohomology

We prove that the motivic cohomology of mixed characteristic schemes, introduced in our previous work, satisfies various expected properties of motivic cohomology, including a motivic refinement of Weibel’s vanishing in algebraic $K$-theory, the projective bundle formula, a comparison to Milnor $K$-theory, and a universal characterisation in terms of pro cdh descent. These results extend those of Elmanto–Morrow to schemes which are not necessarily defined over a field.