Mixed characteristic motivic cohomology satisfies Weibel vanishing, the projective bundle formula, comparison to Milnor K-theory, and pro cdh descent.
On the relative Gersten conjecture for Milnor K-theory in the smooth case
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abstract
We show that the Gersten complex for the (improved) Milnor K-sheaf on a smooth scheme over an excellent discrete valuation ring is exact except at the first place and that exactness at the first place may be checked at the discrete valuation ring associated to the the generic point of the special fiber. This complements results of Gillet and Levine for K-theory, Geisser for motivic cohomology and Schmidt and Strunk and the author for \'etale cohomology.
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Weibel vanishing and the projective bundle formula for mixed characteristic motivic cohomology
Mixed characteristic motivic cohomology satisfies Weibel vanishing, the projective bundle formula, comparison to Milnor K-theory, and pro cdh descent.