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

TL;DR

This paper establishes key properties of motivic cohomology in mixed characteristic schemes, including vanishing, projective bundle formula, and comparison to Milnor K-theory, extending previous results to more general schemes.

Contribution

It proves that motivic cohomology in mixed characteristic satisfies expected properties, extending Elmanto–Morrow's results beyond schemes over fields.

Findings

Motivic cohomology satisfies Weibel's vanishing in mixed characteristic.

The projective bundle formula holds for these schemes.

Comparison to Milnor K-theory is established.

Abstract

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.